Stability results, almost global generalized Beltrami fields and applications to vortex structures in the Euler equations
SSTABILITY RESULTS, ALMOST GLOBAL GENERALIZED BELTRAMI FIELDSAND APPLICATIONS TO VORTEX STRUCTURES IN THE EULER EQUATIONS
ALBERTO ENCISO, DAVID POYATO, AND JUAN SOLER
Abstract.
Strong Beltrami fields, that is, vector fields in three dimensions whose curl is the productof the field itself by a constant factor, have long played a key role in fluid mechanics and magnetohy-drodynamics. In particular, they are the kind of stationary solutions of the Euler equations where onehas been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarilycomplicated topology. On the contrary, there are very few results about the existence of generalizedBeltrami fields, that is, divergence-free fields whose curl is the field times a non-constant function. Infact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) have beenrecently shown to be rare, in the sense that for “most” proportionality factors there are no nontrivialBeltrami fields of high enough regularity (e.g., of class C ,α ), not even locally.Our objective in this work is to show that, nevertheless, there are “many” Beltrami fields withnon-constant factor, even realizing arbitrarily complicated vortex structures. This fact is relevant inthe study of turbulent configurations. The core results are an “almost global” stability theorem forstrong Beltrami fields, which ensures that a global strong Beltrami field with suitable decay at infinitycan be perturbed to get “many” Beltrami fields with non-constant factor of arbitrarily high regularityand defined in the exterior of an arbitrarily small ball, and a “local” stability theorem for generalizedBeltrami fields, which is an analogous perturbative result which is valid for any kind of Beltrami field(not just with a constant factor) but only applies to small enough domains.The proof relies on an iterative scheme of Grad–Rubin type. For this purpose, we study the Neumannproblem for the inhomogeneous Beltrami equation in exterior domains via a boundary integral equationmethod and we obtain H¨older estimates, a sharp decay at infinity and some compactness properties forthese sequences of approximate solutions. Some of the parts of the proof are of independent interest. Contents
1. Introduction 22. Neumann problem for the inhomogeneous Beltrami equation and radiation conditions 52.1. Inhomogeneous Helmholtz equation in the exterior domain 62.2. Inhomogeneous Beltrami equation in the exterior domain 132.3. Well-posedness of the boundary integral equation 222.4. Optimal fall-off in exterior domains 243. An iterative scheme for strong Beltrami fields 253.1. Further notation and preliminaries 253.2. Iterative scheme 303.3. Linear transport problem 323.4. Limit of the approximate solutions 344. Knotted and linked stream lines and tubes in generalized Beltrami fields 394.1. Knots and links in strong Beltrami fields 394.2. Knots and links in almost global generalized Beltrami fields 405. Local stability of generalized Beltrami fields 425.1. A local stability theorem 426. Potential theory techniques for inhomogeneous integral kernels 476.1. Inhomogeneous volume and single layer potentials 476.2. Regularity of the boundary integral operator 58Appendix A. Gradient, curl and divergence on surfaces 66Appendix B. Obstructions to the existence of generalized Beltrami fields 68References 69
This work has been partially supported by the MINECO-Feder (Spain) research grant, the project MTM2014-53406-R (D.P. and J.S.), the Junta de Andaluc´ıa (Spain) Project FQM 954 (D.P. and J.S.), the MECD (Spain) research grantFPU2014/06304 (D.P.) and by the ERC Starting Grant 633152 and the ICMAT–Severo Ochoa grant SEV-2011-0087 (A.E.). a r X i v : . [ m a t h . A P ] S e p ALBERTO ENCISO, DAVID POYATO, AND JUAN SOLER Introduction
Beltrami fields , that is, three dimensional vector fields whose curl is proportional to the field, are aparticularly important class of smooth stationary solutions of the three-dimensional incompressible Eulerequations: ∂ t u + ( u · ∇ ) u = −∇ p , div u = 0 . In a way, what makes them so special is the celebrated structure theorem of Arnold [3], which assertsthat, under suitable technical hypotheses, the velocity field of a smooth stationary solution to the Eulerequations is either a Beltrami field or “laminar”, in the sense that it admits a regular first integral whosesmooth level sets provide “layers” to which the fluid flow is tangent. In fluid mechanics, a Beltrami fieldis interpreted as a fluid whose velocity is parallel to its vorticity.Understanding the knot and link type of stream lines and tubes in stationary fluids has also attractedthe attention of many researchers, both from the theoretical and the experimental points of view [18,19, 28, 44], because knotted stationary vortex structures turned out to play a key role in the so calledLagrangian theory of turbulence. From a numerical point of view, the description of the flows in theliterature that allow for arbitrary vortex structures is mainly based on an active vector formulation ofEuler’s equations (see [11] and the references therein). The existence of knotted and linked vortex linesand tubes in stationary solutions to the Euler equations has been recently established in [18, 19] using strong Beltrami fields , that is, Beltrami fields with a constant proportionaly factor:curl u = λu , λ ∈ R \{ } . (1)Notice that the Beltrami fields in [18, 19] can be assumed to fall off as 1 / | x | at infinity, and that this decayrate is optimal (see the global obstructions in the form of a Liouville type theorem in [7, 36]). Concreteexamples of Beltrami fields with constant proportionality factor are the ABC flows, whose analysis hasyielded considerable insight into the aforementioned phenomenon of Lagrangian turbulence [17].The main objective of this paper is to study the existence, regularity and stability results of generalizedBeltrami fields (i.e., Beltrami fields with nonconstant proportionality factor). This vector fields play afundamental role in the understanding of turbulence. The idea that turbulent flows can be understoodas a superposition of Beltrami flows has already been proposed in [12, 39]. They are also relevant inmagnetohydrodynamics in the context of vanishing Lorentz force ( force-free fields ) and they can be usedto model magnetic relaxation, which is relevant in some astrophysical applications [27, 29, 34, 35]. Indeed,to the best of our knowledge there are just a handful of explicit examples, all of which have Euclideansymmetries, and the analysis of Beltrami fields with nonconstant factor has proved to be extremely hard.The heart of the matter is that, as it was recently proved in [20], the equation for a generalized Beltramifield, curl u = f u , div u = 0 , (2)does not admit any nontrivial solution, even locally, for a “generic” nonconstant function f . In a veryprecise sense, it shows that Beltrami fields with a nonconstant factor are rare and such obstruction is of apurely local nature. These results have been carefully stated in Appendix B for the reader’s convenience.One of the aims of this paper is to show that, although generalized Beltrami fields are indeed rare,one can still prove some kind of partial stability result. Specifically, we will show that for each nontrivialBeltrami field, there are “many” close enough nonconstant proportionality factor that enjoy close non-trivial generalized Beltrami fields. The stabilility result is “partial” in the sense that a “full” stabilityresult cannot be expected since the space of factors that enjoy nontrivial generalized Beltrami fields doesnot contain any ball in the C k,α norm by the above-mentioned obstructions.The analysis of stability canbe crucial to shed some light on the interactions between the different scales in the study of relevantconfigurations in a fully turbulent state.More concretely, we will prove two stability results for generalized Beltrami fields. The first one(Theorem 3.7) is an “almost global” perturbation result for strong Beltrami fields defined on R . Roughlyspeaking, it asserts that given any nontrivial solution of (1) on R with optimal fall-off at infinity (i.e.,1 / | x | ) and any arbitrarily small ball G , there are infinitely many nonconstant factors f , as close tothe constant λ as one wishes in C k,α ( R ), such that the corresponding equation (2) admits nontrivialsolutions on the complement R \ G . This can be combined with the results in [18, 19] to construct almostglobal Beltrami fields with a nonconstant factor that feature vortex lines and vortex tubes of arbitrarilycomplicated topology (Theorem 4.2). The second stability result (Theorem 5.3) states an analogue forperturbations of nontrivial Beltrami fields with constant or nonconstant factor defined in a small enoughopen set where the field does not to vanish. The point of these stability results is that the perturbation ofthe initial proportionality factor is defined by recursively propagating a two-variable function along the ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 3 integral curves of a velocity vector field, so that is the flexibility in choosing the proportionality factorthat is granted by the method of proof. Notice that the idea of constructing the proportionality factor bydragging along the integral curves of a field is somehow inherent to the problem, as the incompressibilitycondition div u = 0 implies that, if it is nonconstant, the factor must be a first integral of the generalizedBeltrami field, i.e., u · ∇ f = 0 . Let us outline the key aspects of the proofs. For concreteness, since all the ideas involved in theproof of the local partial stability result are essentially present in that of the almost global theorem,we shall only discuss the latter result in this Introduction. As we have already mentioned, the pointof the partial stability result is to develop a perturbation technique allowing us to deform the initialfactor f , which for the purpose of this discussion can be taken to be a nonzero constant λ . This requiresanalyzing a related boundary value problem, namely, the Neumann boundary value problem for theinhomogeneous Beltrami equation with constant proportionality factor λ in exterior domains. To ourbest knowledge, this problem has not been directly studied in the literature. Our analysis is based ona boundary integral equation method for complex-valued solutions which requires some potential theoryestimates for generalized volume and single layer potentials and an analysis of the decay properties andradiation conditions of the solutions. They will be determined through the natural connections betweenthe complex-valued solutions of the Beltrami, Helmholtz and Maxwell systems.In [27], the authors show that one can perturb a harmonic field (i.e., a Beltrami field with λ = 0) definedin an exterior domain to construct a generalized Beltrami field with a nonconstant factor. However, theperturbed fields and factors are of low regularity (of class C ,α and C ,α , respectively). In view of therelevance and important applications of Beltrami fields with nonzero λ , we have striven to extend theresult for harmonic fields to general Beltrami fields, and also to show the existence of perturbations ofarbitrarily high regularity (the field will be in C k +1 ,α and the factor in C k,α for any fixed integer k ). Itshould be stressed that the passing from λ = 0 to nonzero λ is not a trivial matter, since the behaviorof the equations at infinity is completely different (oversimplifying a little, for λ = 0 the behavior of thefields at infinity is that of a harmonic function, so one gets uniqueness simply from a decay condition,while for nonzero λ , Beltrami fields solve Helmhotz’s equation, so radiation conditions must be specifiedto obtain uniqueness.) We will present a detailed treatment of these topics (Section 2 and Appendix 6),since we consider that they are of independent interest.The gist of the proof of the almost global partial stability result for strong Beltrami fields is to studythe convergence in C k,α of an iterative scheme that takes the form (cid:26) ∇ ϕ n · u n = 0 , x ∈ Ω ,ϕ n = ϕ , x ∈ Σ , (cid:26) curl u n +1 − λu n +1 = ϕ n u n , x ∈ Ω ,u n +1 · η = u · η, x ∈ S. Here, Ω stands for an exterior domain with smooth boundary S , η is its outward unit normal vector fieldand Σ is some open subset of the boundary. This is a modified Grad–Rubin method (see [1, 5] for theoriginal Grad–Rubin method in the setting of force-free fields perturbations of harmonic fields), which wewill start up with a strong Beltrami field u of constant proportionality factor λ (which can be assumedto exhibit knotted and linked vortex structures) and prescribes the value ϕ of the perturbation of theproportionality factor λ over Σ. Notice that { ϕ n } n ∈ N and { u n } n ∈ N are taken in a consistent way so thatwhenever they have limits ϕ and u in some sense, then ϕ is a global first integral of u and such vectorfield verifies the Beltrami equation (2) with f = λ + ϕ .Our approach will be based again on the analysis of stationary transport equations along stream tubesand a sequence of inhomogeneous problems of div-curl type that we will call inhomogeneous Beltramiequations and which are intimately linked to the Helmhotz equation. In fact, we will start with thecomplex-valued fundamental solution of the Helmholtz equation in R Γ λ ( x ) = e iλ | x | π | x | , x ∈ R \{ } , and will arrive at a representation formula of Helmholtz–Hodge type for its complex-valued solutions.Then, it is necessary to specify the optimal decay and radiation conditions that allow dealing withgeneralized volume and single layer potentials, namely, (cid:90) ∂B R (0) | u ( x ) | d x S = o ( R ) , R → + ∞ , (3) (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) i xR × u ( x ) − u ( x ) (cid:12)(cid:12)(cid:12) d x S = o ( R ) , R → + ∞ . (4) ALBERTO ENCISO, DAVID POYATO, AND JUAN SOLER
Here, (3) is nothing but a weak decay condition of the velocity field u in L and (4) will be called the L Silver–M¨uller–Beltrami radiation condition ( L SMB) and will be deduced from both the classicalSommerfeld and Silver–M¨uller radiation conditions, whose connections with the Helmholtz equation andthe Maxwell system are classical.Summing up, we will be interested in analyzing the existence and uniqueness of complex-valued smoothsolution with high order H¨older-type regularity of the general
Neumann boundary value problem for theinhomogeneous Beltrami equation (NIB) curl u − λu = w, x ∈ Ω ,u · η = g, x ∈ Ω , + L decay property (3) , + L SMB radiation condition (4) . (5)Notice that although we were originally interested in real-valued Beltrami fields, we will be concernedwith complex-valued solutions to (5) and we will then take real parts to obtain the real-valued ones. Thereason to do it is twofold. Firstly, this will allow us to employ a representation formula for complex-valued radiating fields. Secondly, this presents no problems related to the application to knotted structures asone can realize the fields in [18, 19] as the real parts of complex-valued radiating Beltrami fields. Problem(5) was previously studied in [29], who proved C regularity results in bounded domains. We introducesome potential theory estimates of high order for generalized potentials associated with inhomogeneouskernels in exterior domains and adapt the boundary integral method to the unbounded setting. We willalso improve regularity from C to C k +1 ,α .Consequently, we will rely on the complex-valued counterpart of the modified Grad–Rubin method: (cid:26) ∇ ϕ n · u n = 0 , x ∈ Ω ,ϕ n = ϕ , x ∈ Σ , curl v n +1 − λv n +1 = ϕ n u n , x ∈ Ω ,v n +1 · η = u · η, x ∈ S, + L Decay property (3) , + L SBM radiation condition (4) , (6)where u n = (cid:60) v n are the real parts of the complex-valued solutions v n . The compactness of { u n } n ∈ N in C k +1 ,α (Ω , R ) follows from some Schauder estimates of Equation (5) in H¨older spaces. Similarly, { ϕ n } n ∈ N will be shown to be compact in C k,α (Ω) too. Concerning the application to solutions u withknotted vortex structures of the kind constructed in [18, 19], we will see that the solution u inherits theknotted vortex structures from u (up to a small deformation) by virtue of structural stability. This is astraightforward consequence of the fact that u can be chosen close to u as long as the prescribed value ϕ is small enough.The paper is organized as follows. Section 3 is devoted to study the iterative scheme (6). First, weanalize the linear transport equations in the right hand side and the convergence of the iterative schemewill then follow from the analysis of NIB (5). Such problem will be studied in Section 2 by extendingthe results in [29, 38, 43]. By comparison with the vector-valued divergence-free Helmhotz equation , the reduced Maxwell system and the Beltrami equation, we will deduce the appopriate radiation and decayconditions. The SBM radiation condition (4) will then be connected with the classical Silver–M¨ullerand Sommerfeld radiation conditions and we will then present a representation formula of Helmholtz–Hodge type which involves these radiation conditions and that will be extremely useful to obtain ourexistence, uniqueness and regularity results. In Section 4 we combine the above results to constructsmall perturbations of the constant proportionality factor λ leading to nontrivial generalized Beltramifields that exhibit the same kind of knots and links and so to construct stationary solutions to the Eulerequations. In order to support the above regularity results, Section 6 will focus on obtaining H¨olderestimates of high order for volume and single layer potentials associated with the kernel Γ λ ( x ). Theunderlying ideas can be adapted to many other general inhomogeneous kernels with a controlled decayat infinity. The local partial stability result for generalized Beltrami fields will be discussed in Section 5.Finally, Appendix A summarizes some geometric results that will be used throughout the paper andAppendix B recalls, for the benefit of the reader, the results on the generic non-existence of generalizedBeltrami fields proved in [20]. ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 5
Notation.
Let us conclude this Introduction by summing up some notation that will be used throughoutthe paper without further notice. The notation regarding the domains can be stated as follows: • G is a C k +5 bounded domain homeomorphic to an Euclidean ball and containing theorigin, i.e., 0 ∈ G . • Ω := R \ G is its exterior domain and S := ∂ Ω = ∂G is the boundary surface. • η denotes the outward unit normal vector field of S . (7)Although most of our results hold under weaker assumption on the boundary regularity (specifically C k +1 ,α boundaries), there are certain results concerning a singular boundary integral equations whichneed S to be at least C k +5 because higher order derivatives of the normal vector field η are involved (seefor instance Theorem 6.11).Concerning functional spaces, we will essentially use the same notation as in [23]. Let us agree tosay that C k (Ω) is the space of functions of class C k on Ω with finite C k norm (meaning that all theirderivatives up to order k are bounded). We will replace Ω by Ω when the function and all its derivativesup to order k can be continuously extended to the closure of Ω. The space C k,α (Ω) is the inhomogeneousH¨older space with exponent α ∈ (0 ,
1) and k -th order regularity. We will use similar notation C k ( S ), C k,α ( S ) for functions defined on S . Vector-valued analogues of these spaces are denoted in the usualfashion, e.g. C k,α (Ω , R ).2. Neumann problem for the inhomogeneous Beltrami equation and radiation conditions
In this section we analyze the existence and uniqueness of solutions in C k +1 ,α of the NIB problems(5) arising in the modified Grad–Rubin iterative method (6). The key tool is a representation formula ofHelmholtz–Hodge type for its solutions, which we will combine with the well-posedness of the underlyingboundary integral equation for the tangential components in the space of C k +1 ,α tangent vector fieldsto the boundary. For this we will need to improve some regularity results for high order derivativesof generalized volume and single layer potentials arising in the classical potential theory, which willrequire some potential-theoretic estimates for inhomogeneous singular integral kernels that are relegatedto Section 6 for simplicity of exposition. Regarding the representation formula, we will introduce anddiscuss in detail the weakest decay and radiation conditions under which this formula holds (namely,(3) and (4)), as this topic is of independent interest. Notice that many other radiation conditions havebeen used in the literature for related models: the natural one for the scalar complex-valued Helmholtzequation is the Sommerfeld radiation condition and those of the reduced Maxwell system are called the
Silver–M¨uller radiation conditions (SM) (see e.g. [9, 10, 37, 45]).Let us first recall some previous results in the literature on the exterior NIB boundary value problem(5). Although the same problem is studied in [29] for bounded domains and C vector fields by means ofa related approach [29] (which also establishes a Helmholtz–Hodge like representation formula for suchfields and employs boundary integral equations), the technique that we present in this section has notbeen studied in the case of exterior domains and C k,α -regularity. We recall that in [29] it was essential toassume that λ is “regular” with respect to the interior problem. This is the case when λ is not a Dirichleteigenvalue of the Laplacian in the interior domain, or if it is a simple eigenvalue whose eigenfunction hasnon-zero mean, so this condition holds generically (as it can be seen e.g. by considering arbitrarily smallrescalings of the domain).Related results for exterior domains are proved in [38]. Indeed, the technique used in bounded domainby [43] and [29] (for λ = 0 and λ (cid:54) = 0, respectively) goes through to the case of λ = 0 and exterior domainsvia sharp estimates of harmonic volume and single layer potentials in C ,α . Roughly speaking, the maintechnical difference that we will encounter here is that in our case λ is a nonzero constant, which leads toinhomogeneous kernels where the above estimates in unbounded domains are much harder to obtain. Infact, while these estimates are standard for λ = 0, only estimates for the first order derivatives have beenderived in the case of nonzero λ (see [9]). In fact, [29] only considers C estimates even for the (easier)interior problem.There is some literature regarding Laplace’s equation in less regular settings (e.g. L p data and Lipschitzdomains). For C domains, [14, 15] solved it via the analysis of harmonic measures and [22] introduceda method of layer potentials. The latter looks like the method that we propose and is supported byFredholm’s theory: some boundary singular integral operator is shown to be compact and one to onein the C setting, leading to biyectivity and an useful lower estimate that entails the well posedness.For purely Lipschitz domains, compactness does no longer hold [21] whilst biyectivity is preserved [16].Regarding non-symmetric elliptic operators L = − div A ( x ) ∇ in the half-space ( x, t ) ∈ R n × R + , the well ALBERTO ENCISO, DAVID POYATO, AND JUAN SOLER posedness of the Dirichlet problem with L p data [26] follows from the method of “ ε -approximability” andthe absolute continuity of the L -harmonic measure with respect to the surface measure.This section is organized as follows. In the first part, we analyze the representation formula, theradiation conditions and some existence and uniqueness results for the scalar complex-valued Helmholtzequation. We will introduce there some classical notation and powerful tools like the far field pattern of a radiating solution not only in the homogeneous setting but also in the inhomogeneous one. Inthe second part we move to the Beltrami problem and try to carry out the same program as withHelmholtz equation. We will introduce the natural SM radiation conditions of the reduced Maxwellsystem and will link them with the natural radiation conditions both for the inhomogeneous Beltramiequation and an intimately related model: the divergence-free Helmholtz equation. Then, we prove theaforementioned representation formula and our existence and uniqueness results, which follow from thegeneralized potential theory estimates in Section 6 along with the analysis of the well-posedness for theboundary integral equation for the tangential components. This will be studied in the last paragaph ofthis section.2.1. Inhomogeneous Helmholtz equation in the exterior domain.
The Helmholtz equation withwave number λ ∈ R in the exterior domain Ω stands for the elliptic PDE∆ a + λ a = 0 , x ∈ Ω , where the unknown is a possibly complex-valued scalar function a ∈ C (Ω , C ). This equation arises inacoustic and electromagnetic mathematics [10, 37] and in the study of high energy eigenvalue asymptotics.The Helmholtz equation also appears in the study of Beltrami fields arising either from the incompressibleEuler equation or from the force-free field system of magnetohydrodynamicscurl u = λu, x ∈ Ω . Taking curl for λ (cid:54) = 0 one arrives at the following vector-valued equation ∇ (div u ) − ∆ u = λ u, x ∈ Ω . Since Beltrami fields are divergence-free, then one recovers the vector-valued Helmholtz equation in thedomain Ω. This relation with the Beltrami equation suggests to study the representation formulas,radiation conditions and uniqueness lemmas for the Helmholtz equation in the literature.First of all, let us define the next hierarchy of radiation conditions for a complex-valued scalar function a ∈ C (Ω , C ). Definition 2.1. (1) L Sommerfeld radiation condition (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) ∇ a ( y ) · yR − iλa ( y ) (cid:12)(cid:12)(cid:12) d y S = o ( R ) , R → + ∞ . (8)(2) L Sommerfeld radiation condition (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) ∇ a ( y ) · yR − iλa ( y ) (cid:12)(cid:12)(cid:12) d y S = o (1) , R → + ∞ . (9)(3) ( L ∞ ) Sommerfeld radiation condition sup y ∈ ∂B R (0) (cid:12)(cid:12)(cid:12) ∇ a ( y ) · yR − iλa ( y ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , R → + ∞ . (10)The following chain of implications is obvious: L ∞ Sommerfeld = ⇒ L Sommerfeld = ⇒ L Sommerfeld . Originally, only the strongest one (10) was considered. However, several authors [10, 37] came to theconclusion that a weaker radiation condition (9) may be assumed to obtain representation formulas andcertain uniqueness results. Although we follow the same approach, we weaken the radiation conditionto an even weaker one (8) by assuming some kind of decay at infinity that will be much weaker than | x | − though. As it will be shown later, both decay and radiation conditions can be recovered from the L Sommerfeld radiation condition for solution to the Helmholtz equation. Before showing that thisradiation conditions leads to a representation formula of Stokes type, let us analyze them in the case ofthe fundamental solution to the 3-D Helmholtz equation,Γ λ ( x ) = e iλ | x | π | x | = (cid:18) cos( λ | x | )4 π | x | + i sin( λ | x | )4 π | x | (cid:19) . (11) ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 7
Since ∇ Γ λ ( x ) = (cid:18) iλ − | x | (cid:19) Γ λ ( x ) x | x | , (12)a straightforward inductive argument shows that all the partial derivatives of Γ λ ( x ) up to second orderverify an even stronger version of the Sommerfeld radiation condition (10). Hence we easily infer: Proposition 2.2.
The fundamental solution of the Helmholtz equation, together with its partial deriva-tives up to order satisfy the identities ∇ Γ λ ( x ) · x | x | − iλ Γ λ ( x ) = − Γ λ ( x ) | x | , ∇ (cid:18) ∂ Γ λ ∂x i (cid:19) ( x ) · x | x | − iλ ∂ Γ λ ∂x i ( x ) = (cid:18) | x | − iλ (cid:19) Γ λ ( x ) x i | x | , ∇ (cid:18) ∂ Γ λ ∂x i ∂x j (cid:19) ( x ) · x | x | − iλ ∂ Γ λ ∂x i ∂x j ( x ) = −∇ (cid:18) ∂ Γ λ ∂x i (cid:19) ( x ) · ∂∂x j (cid:18) x | x | (cid:19) + ∂∂x j (cid:18)(cid:18) | x | − iλ (cid:19) Γ λ ( x ) x i | x | (cid:19) , for every i, j ∈ { , , } . Consequently, sup x ∈ ∂B R (0) (cid:12)(cid:12)(cid:12) ∇ ( D γ Γ λ )( x ) · xR − iλD γ Γ λ ( x ) (cid:12)(cid:12)(cid:12) = O (cid:18) R (cid:19) , for R → + ∞ , for every multi-index with | γ | ≤ . In particular, Γ λ ( x ) together with its partial derivatives up to order two verify the Sommerfeld radiationcondition (10). It is then an easy task to obtain new complex-valued solutions to the homogeneousHelmholtz equation enjoying such radiation condition through the definition of the generalized volumeand single layer potentials associated with the kernel Γ λ ( x ). Proposition 2.3.
Let a be the generalized single layer potential with density ζ ∈ C ( S ) associated withthe Helmholtz equation, i.e., a ( x ) := ( S λ ζ )( x ) = (cid:90) S Γ λ ( x − y ) ζ ( y ) d y S, for every x ∈ Ω . Then, a and all its partial derivatives up to second order are solutions to the Helmholtzequation which verify the Sommerfeld radiation condition (10).Proof. Taking derivatives under the integral sign, one checks that a solves the complex-valued Helmholtzequation in Ω. In order to check Sommerfeld radiation condition (10), let us use the preceding propertiesin Proposition 2.2. Fixing z ∈ R and taking derivatives under the integral sign, we have ∇ a ( x ) · x − z | x − z | − iλa ( x ) = (cid:90) S (cid:18) ∇ x Γ λ ( x − y ) x − z | x − z | − iλ Γ λ ( x − y ) (cid:19) ζ ( y ) d y S = (cid:90) S (cid:18) ∇ x Γ λ ( x − y ) x − y | x − y | − iλ Γ λ ( x − y ) (cid:19) ζ ( y ) d y S + (cid:90) S ∇ x Γ λ ( x − y ) (cid:18) x − z | x − z | − x − y | x − y | (cid:19) ζ ( y ) d y S. Multiplying the first term by | x − z | and assuming that | x − z | is big enough (Proposition 2.2), we find | x − z | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S (cid:18) ∇ x Γ λ ( x − y ) x − y | x − y | − iλ Γ λ ( x − y ) (cid:19) ζ ( y ) d y S (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x − z | ( | x − z | − d ) (cid:107) ζ (cid:107) L ( S ) , where d stands for max {| y − z | : y ∈ S } . Therefore, this term vanishes for | x − z | → + ∞ . Regarding thesecond term, it is easily checked that ∇ Γ λ ( x ) = O ( | x | − ), when | x | → + ∞ . To conclude the proof of thisresult, let us obtain some extra decay from the difference in the middle, which can be upper boundedthrough the next straightforward reasonings involving the mean value theorem (cid:12)(cid:12)(cid:12)(cid:12) x i − z i | x − z | − x i − y i | x − y | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ddθ x i − ( θy i + (1 − θ ) z i ) | x − ( θy + (1 − θ ) z ) | dθ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) | x − ( θy + (1 − θ ) z ) | ( z i − y i ) − ( x i − ( θy i + (1 − θ ) z i )) x − ( θy +(1 − θ ) z ) | x − ( θy +(1 − θ ) z ) | · ( z − y ) | x − ( θy + (1 − θ ) z ) | dθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | y − z | (cid:90) | x − ( θy + (1 − θ ) z ) | dθ ≤ d | x − z | − d . (13) ALBERTO ENCISO, DAVID POYATO, AND JUAN SOLER
Therefore, | x − z | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ∇ x Γ λ ( x − y ) (cid:18) x − z | x − z | − x − y | x − y | (cid:19) ζ ( y ) d y S (cid:12)(cid:12)(cid:12)(cid:12) ≤ C d | x − z | ( | x − z | − d ) (cid:107) ζ (cid:107) L ( S ) , whose limit also vanishes as | x − z | → + ∞ . Consequently, a verifies the Sommerfeld radiation conditioncentered at any z ∈ R . In particular, the above assertion also holds for z = 0. A similar reasoning withthe partial derivatives of a up to second order also holds according to Proposition 2.2. (cid:3) The same result remains true for generalized volume potential with compactly supported densities. Inthis case, radiating solutions for the inhomogeneous complex-valued Helmholtz equation can be obtained.The proof is identical, with the only distinction that we must change the constant d in the lower boundsof the denominators from d = max {| y − z | : y ∈ S } to d = max {| y − z | : y ∈ supp ζ } . Proposition 2.4.
Let a be the generalized volume potential with density ζ ∈ C c (Ω) associated with theHelmholtz equation, i.e., a ( x ) := ( N λ ζ )( x ) = (cid:90) Ω Γ λ ( x − y ) ζ ( y ) d y S, for every x ∈ Ω . Then, a solves the inhomogeneous Helmholtz equation − (∆ a + λ a ) = − ζ, in the exterior domain Ω . Moreover, a and all its partial derivatives up to second order verify theSommerfeld radiation condition (10). To establish the representation formula for the inhomogeneous Helmholtz equation, we study theradiation conditions for the volume and single layer potentials, as well as its decay properties at infinity.We will need the
Hardly–Littlewood–Sobolev estimates of fractional integrals [42, Theorem 1.2.1], whichwe state not in terms of L p integrability conditions but in terms of pointwise decay at infinity. For theconvenience of the reader, we include a simple derivation of this form of the estimates: Theorem 2.5.
Consider any dimension N and exponent < α < N . Define the associated Rieszpotential by R α ( x ) := 1 | x | α , x ∈ R N . For any measurable function f : R N −→ R , we have that (1) the decay property | ( R α ∗ f )( x ) | ≤ C (cid:107)| x | ρ f (cid:107) L ∞ ( R N ) | x | α − ( N − ρ ) , holds for every x ∈ R N as long as f = O ( | x | − ρ ) for | x | → + ∞ and ρ is any nonnegative exponentsuch that N − α < ρ < N. Here, C stands for a positive constant that depends on N , α and ρ but do not depend on f . (2) the optimal decay | x | − α is obtained in the compactly supported case, i.e., | ( R α ∗ f )( x ) (cid:107) ≤ C (cid:107) f (cid:107) L ∞ ( R N ) | x | α , for every x ∈ R N , as long as f ∈ L ∞ ( R N ) has compact support inside some ball B R (0) . Now,not only does C depend on N and α but also on the size R > of the support.Proof. Let us begin with the first item. Fix any constant 0 < R < R = 1 /
2) and split the integralwe are interested in into the next two parts (cid:90) R | x − y | α f ( y ) dy = I + I , where I = (cid:90) B R | x | (0) | x − y | α f ( y ) dy, I = (cid:90) B cR | x | | x − y | α f ( y ) dy. In order to estimate I , notice that y ∈ B R | x | (0) = ⇒ | x − y | ≥ (1 − R ) | x | . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 9
Therefore, I is bounded by (cid:90) B R | x | (0) | x − y | α | f ( y ) | dy ≤ K (1 − R ) α | x | α (cid:90) B R | x | (0) | y | ρ dy = Kω N (1 − R ) α | x | α (cid:90) R | x | r N − r ρ dr = Kω N N − ρ R N − ρ (1 − R ) α | x | α − ( N − ρ ) . Here K := (cid:107)| x | ρ f (cid:107) L ∞ ( R ) and ω N stands for the ( N − R N . Itis worth remarking that we are dealing with finite integrals as a consequence of the hypothesis ρ < N .Similarly, the second integral, I , can also be split as follows (cid:90) B R | x | (0) c | x − y | α | f ( y ) | dy = (cid:90) B R | x | ( x ) \ B R | x | (0) | x − y | α | f ( y ) | dy + (cid:90) ( B R | x | (0) ∪ B R | x | ( x )) c | x − y | α | f ( y ) | dy. An analogous argument can be used to obtain the next upper bound of the first term (cid:90) B R | x | ( x ) \ B R | x | (0) | x − y | α | f ( y ) | dy ≤ (cid:90) B R | x | ( x ) | x − y | α | f ( y ) | dy ≤ K (cid:90) B R | x | ( x ) | x − y | α | y | ρ dy = K (cid:90) B R | x | (0) | x − y | ρ | y | α dy = Kω N N − α R N − α (1 − R ) ρ | x | α − ( N − ρ ) . This time, finite integrals are involved due to the hypothesis N − α < ρ . Regarding the second term, letus decompose the integral into two parts once more. The appropriate subdomains to be considered are A = { y ∈ ( B R | x | (0) ∪ B R | x | ( x )) c : | x − y | ≤ | y |} ,B = { y ∈ ( B R | x | (0) ∪ B R | x | ( x )) c : | x − y | > | y |} . Let us complete the proof of the first inequality with the following estimates for the integrals over A and B , which follow from the same reasoning involving the hypothesis N − α < ρ : (cid:90) A | x − y | α | f ( y ) | dy ≤ K (cid:90) A | x − y | α | y | ρ dy ≤ K (cid:90) A | x − y | α + ρ dy ≤ K (cid:90) B R | x | ( x ) c | x − y | α + ρ dy = Kω N (cid:90) + ∞ R | x | r N − r α + ρ dr = Kω N α − ( N − ρ ) 1 R α − ( N − ρ ) | x | α − ( N − ρ ) , (cid:90) B | x − y | α | f ( y ) | dy ≤ K (cid:90) B | x − y | α | y | ρ dy ≤ K (cid:90) B | y | α + ρ dy ≤ K (cid:90) B R | x | (0) c | y | α + ρ dy = Kω N (cid:90) + ∞ R | x | r N − r α + ρ dr = Kω N α − ( N − ρ ) 1 R α − ( N − ρ ) | x | α − ( N − ρ ) . Let us now pass to the second item. Let us start with | x | > R , so that | ( R α ∗ f )( x ) | ≤ (cid:90) B R (0) | x − y | α | f ( y ) | dy. Notice that whenever y ∈ B R (0), then one has | x − y | ≥ | x | − | y | ≥ | x | − R = (cid:18) − R | x | (cid:19) | x | ≥ | x | . Therefore | ( R α ∗ f )( x ) | ≤ α | x | α (cid:107) f (cid:107) L ( R ) ≤ α | B R (0) | (cid:107) f (cid:107) L ∞ ( R N ) | x | α . The case | x | ≤ R is easier since y ∈ B R (0) = ⇒ | x − y | ≤ | x | + | y | < R , and consenquently, Young inequality for the convolution of L p functions leads to | ( R α ∗ f )( x ) | ≤ (cid:90) B R ( x ) | x − y | α | f ( y ) | dy = (cid:90) B R (0) | f ( x − y ) | | y | α dy = | f | ∗ (cid:16) χ B R (0) R α (cid:17) ( x ) ≤ (cid:107) R α (cid:107) L ( B R (0)) (cid:107) f (cid:107) L ∞ ( R N ) ≤ (2 R ) α (cid:107) R α (cid:107) L ( B R (0)) (cid:107) f (cid:107) L ∞ ( R N ) | x | α , where 1 ≤ R | x | has been used in the last inequality. (cid:3) The above results permit obtaining a Stokes-type formula to represent the solutions to the inhomoge-neous Helmholtz equation. Now, we deal with the weakest radiation condition, namely, the L Sommerfeldradiation condition and some property of weak decay at infinity in L . Since the proof is completely anal-ogous to the more important result for complex-valued solutions of the inhomogeneous Beltrami equationthat we present in the next subsection (Theorem 2.12), we will skip the proof. A detailed proof withthe more restrictive L Sommerfeld radiation condition (9) can be found in [10, Theorem 2.4] and [37,Theorem 3.1.1].
Theorem 2.6.
Let a ∈ C (Ω , C ) ∩ C (Ω , C ) be any function which verifies the L Sommerferld radiationcondition (8) and the following decay property at infinity (cid:90) ∂B R (0) | a ( y ) | d y S = o ( R ) , when R → + ∞ . (14) Assume that ∆ a + λ a = O ( | x | − ρ ) when | x | → + ∞ , for some exponent < ρ < . Then, a ( x ) = − (cid:90) Ω Γ λ ( x − y )(∆ a ( y ) + λ a ( y )) dy (15)+ (cid:90) S ∂ Γ λ ( x − y ) ∂η ( y ) a ( y ) d y S − (cid:90) S Γ λ ( x − y ) ∂a∂η ( y ) d y S, for every x ∈ Ω and, as a consequence, a = O ( | x | − ( ρ − ) , when | x | → + ∞ . Indeed, when ∆ a + λ a has compact support, one obtains the optimal decay at infinity, namely, a = O ( | x | − ) , when | x | → + ∞ . The properties follow from Theorem 2.5 and they may also be found in [10, 37]. Notice that the decayrates | x | − ( ρ − (for the inhomogeneous equation) and | x | − (for the homogeneous one) are straighforwardconsequences of the representation formula.Let us now show the link between our L version an the L version ([10, 37]). To this end, we shallnext see that (9) ⇒ (8) + (14) in the homogeneous case. Indeed, let a ∈ C (Ω , C ) ∩ C (Ω , C ) be anysolution to the complex-valued homogeneous Helmholtz equation in the exterior domain fulfilling the L Sommerfeld radiation condition (9). Computing the square in such radiation condition, we arrive at (cid:12)(cid:12)(cid:12) ∇ a ( x ) · xR − iλa ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∇ a ( x ) · xR (cid:12)(cid:12)(cid:12) + λ | a ( x ) | − (cid:60) (cid:16) iλa ( x ) ∇ a ( x ) · xR (cid:17) = (cid:12)(cid:12)(cid:12) ∇ a ( x ) · xR (cid:12)(cid:12)(cid:12) + λ | a ( x ) | + 2 λ (cid:61) (cid:16) a ( x ) ∇ a ( x ) · xR (cid:17) , (16)for any x ∈ ∂B R (0), where (cid:60) and (cid:61) mean the real and imaginary parts of the corresponding complexnumbers. Consider any positive radius R such that G ⊆ B R (0) and define the subdomains Ω R := B R (0) \ G , for each R > R . Therefore, the homogeneous Helmholtz equation and Green’s formula leadto − λ (cid:90) Ω R | a ( x ) | dx = (cid:90) Ω R a ( x )∆ a ( x ) dx = (cid:90) ∂ Ω R a ( x ) ∇ a ( x ) · ν ( x ) d x S − (cid:90) Ω R ∇ a ( x ) · ∇ a ( x ) dx. Let us split the boundary integral into the boundary’s connected components (cid:90) ∂B R (0) a ( x ) ∇ a ( x ) · xR d y S − (cid:90) S a ( x ) ∇ a ( x ) · η ( x ) d x S = (cid:90) Ω R |∇ a ( x ) | dx − λ (cid:90) Ω R | a ( x ) | dx. and take imaginary parts in the preceding equation to arrive at (cid:61) (cid:32)(cid:90) ∂B R (0) a ( x ) ∇ a ( x ) · xR d x S (cid:33) = (cid:61) (cid:18)(cid:90) S a ( x ) ∇ a ( x ) · η ( x ) d x S (cid:19) . (17) ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 11
Combining equations (16) and (17) along with the L Sommerfeld radiation condition (9), one obtainslim R → + ∞ (cid:90) ∂B R (0) (cid:18)(cid:12)(cid:12)(cid:12) ∇ a ( x ) · xR (cid:12)(cid:12)(cid:12) + λ | a ( x ) | (cid:19) d x S = − λ (cid:61) (cid:18)(cid:90) S a ( x ) ∇ a ( x ) · η ( x ) d x S (cid:19) . (18)Consequently, (cid:90) ∂B R (0) | a ( x ) | d x S = O (1) when R → + ∞ , (19)and Cauchy-Schwarz inequality ensures that (cid:90) ∂B R (0) | a ( x ) | d x S = O ( R ) , when R → + ∞ . In particular, the weak decay property (14) holds.Notice that the L version in [10, 37] of the representation formula for the homogeneous equation is adirect consequence of our L version in Theorem 2.6 and the preceding discussion: Corollary 2.7.
Let a ∈ C (Ω , C ) ∩ C (Ω , C ) be any solution to the complex-valued homogeneous Helmholtzequation in the exterior domain which verifies the L Sommerferld radiation condition (9). Then, a ( x ) = (cid:90) S ∂ Γ λ ( x − y ) ∂η ( y ) a ( y ) d y S − (cid:90) S Γ λ ( x − y ) ∂a∂η ( y ) d y S, for every x ∈ Ω . As a consequence, a = O ( | x | − ) , when | x | → + ∞ . An immediate consequence of the representation formulas in Theorem 2.6 and Corollary 2.7 is thata far field pattern at infinity exists for each solution to the Helmholtz equation (see [10] for details).The far field pattern of a solution to the Helmholtz equation is a very powerful tool since it providesa description of the asymptotic behavior at infinity. It gives, for instance, easy uniqueness criteria forradiating solutions. A related inverse problem has also been widely studied, as it is interesting to knowwhether a fixed function over the unit sphere is the far field pattern of some radiating solution to theHelmholtz equation.Although most of the literature is only devoted to far field patterns of complex-valued radiatingsolutions to the homogeneous Helmholtz equation, our problem clearly concerns the inhomogeneoussetting. Thus, we revisit the theory of far field patterns and its relation to the general inhomogeneousHelmholtz equation in the particular case of compactly supported inhomogeneities (it would not be hardto extend it to more general inhomogeneous terms suitable decay at infinity). For this, consider anysolution a ∈ C (Ω , C ) ∩ C (Ω , C ) to the inhomogeneous Helmholtz equation − (∆ a + λ a ) = f, x ∈ Ω , where f is compactly supported in Ω and a verifies both the decay condition (14) and the L Sommerfeldradiation condition (8). Then, Theorem 2.6 leads to a ( x ) = (cid:90) Ω Γ λ ( x − y ) f ( y ) dy + (cid:90) S ∂ Γ λ ( x − y ) ∂η ( y ) a ( y ) d y S − (cid:90) S Γ λ ( x − y ) ∂a∂η ( y ) d y S. Consider the compact subset K := supp f and notice the asymptotic behaviorΓ λ ( x − y ) = Γ λ ( x ) (cid:26) e − iλ x | x | · y + O (cid:18) | x | (cid:19)(cid:27) , when | x | → + ∞ ,∂ Γ λ ( x − y ) ∂η ( y ) = Γ λ ( x ) (cid:40) ∂e − iλ x | x | · y ∂η ( y ) + O (cid:18) | x | (cid:19)(cid:41) , when | x | → + ∞ , where O (cid:0) | x | − (cid:1) is uniform in y ∈ K ∪ S in the first formula and uniform in y ∈ S in the second one.From here we deduce the asymptotic behavior a ( x ) = Γ λ ( x ) (cid:26) a ∞ (cid:18) x | x | (cid:19) + O (cid:18) | x | (cid:19)(cid:27) when | x | → + ∞ , (20)where a ∞ is called the far field pattern of a , and reads as a ∞ ( σ ) = (cid:90) Ω e − iλσ · y f ( y ) dy + (cid:90) S ∂e − iλσ · y ∂η ( y ) a ( y ) d y S − (cid:90) S e − iλσ · y ∂a∂η ( y ) d y S, for each point σ ∈ ∂B (0). It is apparent that a ∞ is uniquely determined from formula (20). Hence, we can define the followingwell-defined linear and one to one map D ∞ −→ C ∞ ( ∂B (0)) a (cid:55)−→ a ∞ , (21)where the domain of the far field pattern mapping is D ∞ := { a ∈ C (Ω , C ) ∩ C (Ω , C ) : ∆ a + λ a has compact support and (8) and (14) hold } . A similar reasoning leads to an explicit formula for the far field pattern of the derivatives of a , namely,( ∇ a ) ∞ ( σ ) = iλa ∞ ( σ ) σ, ∀ σ ∈ ∂B (0) . (22)The splitting in (20) ensures thatlim R → + ∞ (cid:90) ∂B R (0) | a ( x ) | dx = 14 π (cid:90) ∂B (0) | a ∞ ( σ ) | d σ S. (23)The celebrated Rellich lemma [10, Lemma 2.11] states that the only complex-valued solution a ∈ C (Ω , C )to the exterior homogeneous Helmholtz equation such that the limit in the left hand side of the precedingformula becomes zero is the zero function identically. Therefore, whenever a solution to the homogeneousHelmholtz equation has a well-defined far field pattern and it vanishes (i.e., a ∞ ≡ a vanisheseverywhere.The following uniqueness result is of great interest to deal with Dirichlet and Neumann boundary valueproblems in the exterior domain. It is an immediate consequence of the Rellich lemma and the discussionleading to Corollary 2.7, and it can be found in [10, Theorem 2.12]: Lemma 2.8.
Consider any solution a ∈ C (Ω , C ) ∩ C (Ω , C ) to the complex-valued homogeneous Helmholtzequation in the exterior domain Ω fulfilling the L Sommerfeld radiation condition (9). Then, a verifiesthe inequality λ (cid:61) (cid:18)(cid:90) S a ( x ) ∂a∂η ( x ) d x S (cid:19) ≤ . If the equality holds, then a vanishes everywhere in Ω . Before moving to the Beltrami problem, notice that the preceding results for scalar solutions toHelmholtz equation also work for vector-valued solutions. In this case, the decay property and ra-diation conditions can be considered componentwise. For instance, given any vector-valued solution u ∈ C (Ω , C ) ∩ C (Ω , C ) to the Helmholtz equation, − (∆ u + λ u ) = F, x ∈ Ω , where F is compactly supported, then, the decay property and the L Sommerfeld radiation conditionsread as (cid:90) ∂B R (0) | u ( x ) | dx = o ( R ) , when R → + ∞ , (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) Jac u ( x ) xR − iλu ( x ) (cid:12)(cid:12)(cid:12) dx = o ( R ) , when R → + ∞ . One can wonder whether there are more natural radiation conditions for vector-valued solutions toHelmholtz equation. The general answer is given in [9, Theorem 4.13] and [45, Section 5, Theorem2], although we will be mostly interested in the divergence-free case:
Remark 2.9.
If the vector-valued solution to the preceding Helmholtz equation verifies the above condi-tions, then u and its first order partial derivatives enjoy the strong Sommerfeld radiation condition andhave well defined far field patterns. Let us write the Helmholtz equation in the following equivalent way curl(curl u ) − ∇ (div u ) − λ u = F in Ω , and take curl to obtain curl(curl(curl u )) − λ curl u = curl F in Ω . Now, far field patterns in both equations give F ∞ = (curl(curl u ) − ∇ (div u ) − λ u ) ∞ = iλσ × (curl u ) ∞ − iλ (div u ) ∞ σ − λ u ∞ , F ) ∞ = (cid:0) curl(curl(curl u )) − λ curl u (cid:1) ∞ = − iλ σ × ( σ × ( σ × u ∞ )) − λ (curl u ) ∞ = iλ σ × u ∞ − λ (curl u ) ∞ . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 13
Div–Free Helmholtz Reduced MaxwellBeltrami B := u, E := − curl uiλ u := B SMH SMSMB u B := uE := − curl uiλ Figure 1.
Sketch of the connections between the three related models: divergence-freeHelmholtz equation , reduced Maxwell system and Beltrami equation . The picture in theleft shows the bonds between such models whilst the picture in the right exhibits theassociated relations between its natural radiation conditions.
Notice that compactly supported functions, such as F and curl F , have vanishing far fields patterns thanksto (23). Through the definition of far field patterns we arrive at the following two decompositions x | x | × curl u ( x ) − div u ( x ) x | x | + iλu ( x )= Γ λ ( x ) (cid:26)(cid:18) x | x | × (curl u ) ∞ (cid:18) x | x | (cid:19) − (div u ) ∞ (cid:18) x | x | (cid:19) + iλu ∞ (cid:18) x | x | (cid:19)(cid:19) + O (cid:18) | x | (cid:19)(cid:27) ,λ x | x | × u ( x ) + i curl u ( x )= Γ λ ( x ) (cid:26)(cid:18) λ x | x | × u ∞ (cid:18) x | x | (cid:19) + i (curl u ) ∞ (cid:19) + O (cid:18) | x | (cid:19)(cid:27) , when | x | → + ∞ . Consequently, the terms associated to the far field patterns vanish and we obtain theradiation conditions sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) xR × curl u ( x ) − div u ( x ) xR + iλu ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ , (24)sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) λ xR × u ( x ) + i curl u ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ . (25) When u is a divergence-free solution to the Helmholtz equation (as in our case), the radiation conditionare simpler and read sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) xR × curl u ( x ) + iλu ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ , (26)sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) λ xR × u ( x ) + i curl u ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ . (27)2.2. Inhomogeneous Beltrami equation in the exterior domain.
Now, we move to the complex-valued inhomogeneous Beltrami equation. In order to understand where the natural radiation condition(4) comes form, let us extend the arguments in Remark 2.9 in the homogeneous case. To this end, wewill connect three different systems that will provide an appropriate terminology. The heuristic idea issummarized in Figure 1. Through the relations between the vector fields u and B in the left hand side ofsuch pictures, we find (see [10, Theorem 6.4] and [45]) that the divergence-free Helmholtz equation andthe reduced Maxwell system [10, Definition 6.5] are completely equivalent, i.e., (cid:26) ∆ u + λ u = 0 , x ∈ Ω , div u = 0 , x ∈ Ω . ⇐⇒ (cid:26) curl E − iλB = 0 , x ∈ Ω , curl B + iλE = 0 , x ∈ Ω . In order that the solutions to this system could be represented through the classical
Stratton–Chu formulas [10, Theorem 6.6], the
Silver–M¨uller radiation conditions (SM) have to be considered:sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) B ( x ) × xR − E ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ , sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) E ( x ) × xR + B ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ . Due to our choice of B and E , the SM radiation conditions leads to (26)–(27) again. Thus, the naturalradiation conditions for the divergence-free vector-valued Helmholtz equation are actually a consequenceof the SM radiation conditions for the reduced Maxwell system. Therefore, we will call them the Silver–M¨uller–Helmholtz radiation conditions (SMH).Let us now consider the case of the Beltrami equationcurl u − λu = 0 , x ∈ Ω . When λ (cid:54) = 0, then u is a solution to the divergence-free Helmholtz equation, and consequently it alsosolves the reduced Maxwell system. Therefore, one may want to transfer the SMH or the original SMradiation condition to the Beltrami framework. An easy substitution in (26) and (27) leads to a singleradiation condition for Beltrami fields, which we will call it the Silver–M¨uller–Beltrami radiation condition (SMB): sup x ∈ B R (0) (cid:12)(cid:12)(cid:12) i xR × u ( x ) − u ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , when R → + ∞ . It might seem that the only connection between the Beltrami equation and the divergence-free vector-valued Helmholtz equation is the first implication sketched in Figure 1, but the connection is actuallymuch stronger. The reason is the following. Given any solution u to the Beltrami equation, it is obviouslya solution to the divergence-free Helmholtz equation. The point is that, conversely, given any solution (cid:98) u to the divergence-free Helmholtz equation, u := curl (cid:98) u + λ (cid:98) u λ . (28)is a solution to the Beltrami equation, and all the solutions can be constructed this way.In view of this converse relation, it is natural to wonder about the radiation conditions that one shouldassume on (cid:98) u in order for u to verify the SMB radiation condition. For this, notice that i xR × u ( x ) − u ( x ) = i λ (cid:16) xR × curl (cid:98) u ( x ) + iλ (cid:98) u ( x ) (cid:17) + i λ (cid:16) λ xR × (cid:98) u ( x ) + i curl (cid:98) u ( x ) (cid:17) , for every x ∈ ∂B R (0). Therefore, the SMB radiation condition on u is recovered form the SMH radiationconditions on (cid:98) u , so all the possible links between the three models and its corresponding radiationconditions in Figure 1 follow. Remark 2.10.
The complex-valued Beltrami fields u satisfying the SMB radiation condition take theform (28) for some solution (cid:98) u of the divergence-free Helmholtz equation satisfying the SMH radiationconditions. As in the Sommerfeld radiation condition, let us consider the following hierarchy of SMB radiationconditions.
Definition 2.11. (1) L Silver–M¨uller–Beltrami (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) i xR × u ( x ) − u ( x ) (cid:12)(cid:12)(cid:12) d x S = o ( R ) , R → + ∞ . (29)(2) L Silver–M¨uller–Beltrami (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) i xR × u ( x ) − u ( x ) (cid:12)(cid:12)(cid:12) d x S = o (1) , R → + ∞ . (30)(3) ( L ∞ ) Silver–M¨uller–Beltrami sup x ∈ ∂B R (0) (cid:12)(cid:12)(cid:12) i xR × u ( x ) − u ( x ) (cid:12)(cid:12)(cid:12) = o (cid:18) R (cid:19) , R → + ∞ . (31)The next theorem shows the desired decomposition theorem of Helmholtz–Hodge type under the above L decay and radiation hypotheses, that were already mentioned in the Introduction (see (3) and (4)): Theorem 2.12.
Let u ∈ C (Ω , C ) be any vector field which verifies the L SMB condition (29) and thefollowing decay property at infinity (cid:90) ∂B R (0) | u ( x ) | d x S = o ( R ) , when R → + ∞ (32) ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 15 b x G ε R Ω( x, ε, R ) b Figure 2.
Domain Ω( x, ε, R ). Assume that div u, curl u − λu = O ( | x | − ρ ) when | x | → + ∞ for some exponent < ρ < . Then, u canbe decomposed as u ( x ) = −∇ φ ( x ) + curl A ( x ) + λA ( x ) , for every x ∈ Ω , where φ and A are the scalar and vector fields φ ( x ) = (cid:90) Ω Γ λ ( x − y ) div u ( y ) dy + (cid:90) S Γ λ ( x − y ) η ( y ) · u ( y ) d y S,A ( x ) = (cid:90) Ω Γ λ ( x − y )(curl u ( y ) − λu ( y )) dy + (cid:90) S Γ λ ( x − y ) η ( y ) × u ( y ) d y S. As a consequence, u = O ( | x | − ( ρ − ) , when | x | → + ∞ . Indeed, when both div u and curl u − λu are compactly supported, one obtains the optimal decay at infinity,namely, u = O ( | x | − ) , when | x | → + ∞ , and u satisfies the Sommerfeld radiation condition (10) componentwise.Proof. Consider any x ∈ Ω and fix any couple of radii ε , R > B ε ( x ) ⊆ Ω and B ε ( x ) ∪ G ⊆ B R (0) . Define the subdomain Ω( x, ε, R ) := Ω ∩ ( B R (0) \ B ε ( x )) for R > R and ε > ε , as in Figure 2.The main difference between the formula of Stokes type for the scalar Helmholtz equation in thepreceding paragaph and the formula of Helmholtz–Hodge type for the Beltrami equation here is that theformer holds true by virtue of the scalar Green’s first formula while the later needs some sort of vectorformula of Green type to be derived.Let us fix any vector e ∈ C . Since Γ λ solves the scalar homogeneous Helmholtz equation outside theorigin, then Γ λ e is a solution to the vector-valued homogeneous Helmholtz equation too. Therefore, thefollowing identity holds0 = − (cid:90) Ω( x,ε,R ) (∆(Γ λ ( x − y ) e ) + λ (Γ λ ( x − y ) e )) · u ( y ) dy. As in the classical Helmholtz–Hodge theorem, it is essential to bear the next formula in mindcurl(curl) = ∇ (div) − ∆ , which allows writing the above identity in the following way0 = − (cid:90) Ω( x,ε,R ) ∇ y (div x (Γ λ ( x − y ) e )) · u ( y ) dy + (cid:90) Ω( x,ε,R ) curl y (curl x (Γ λ ( x − y ) e )) · u ( y ) dy + λ (cid:90) Ω( x,ε,R ) (Γ λ ( x − y ) e ) · u ( y ) dy. (33) Hence Equation (33) can be written as0 = − (cid:90) Ω( x,ε,R ) div y (div x (Γ λ ( x − y ) e ) u ( y )) dy + (cid:90) Ω( x,ε,R ) div x (Γ λ ( x − y ) e ) div u ( y ) dy + (cid:90) Ω( x,ε,R ) div y (curl x (Γ λ ( x − y ) e ) × u ( y )) dy + (cid:90) Ω( x,ε,R ) curl x (Γ λ ( x − y ) e ) · curl u ( y ) dy + λ (cid:90) Ω( x,ε,R ) (Γ λ ( x − y ) e ) · u ( y ) dy. (34)Now, we can apply the divergence theorem to the first and third terms in (34) and standard vectorcalculus identities to find0 = − (cid:90) ∂ Ω( x,ε,R ) ( ∇ x Γ λ ( x − y ) ν ( y ) · u ( y )) · e d y S + (cid:90) Ω( x,ε,R ) ( ∇ x Γ λ ( x − y ) div u ( y )) · e dy + (cid:90) ∂ Ω( x,ε,R ) ( ∇ x Γ λ ( x − y ) × ( ν ( y ) × u ( y ))) · e d y S − (cid:90) Ω( x,ε,R ) ( ∇ x Γ λ ( x − y ) × curl u ( y )) · e dy + λ (cid:90) Ω( x,ε,R ) (Γ λ ( x − y ) u ( y )) · e dy. (35)Let us now remove the dot product by e (notice that (35) holds for any constant vector e ∈ C ) andsubtract and add the appropriate terms to obtain the following formula0 = − (cid:90) ∂ Ω( x,ε,R ) ∇ x Γ λ ( x − y ) ν ( y ) · u ( y ) d y S + (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) div u ( y ) dy + (cid:90) ∂ Ω( x,ε,R ) ∇ x Γ λ ( x − y ) × ( ν ( y ) × u ( y )) d y S − (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) × (curl u ( y ) − λu ( y )) dy + λ (cid:32) − (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) × u ( y ) dy + λ (cid:90) Ω( x,ε,R ) Γ λ ( x − y ) u ( y ) dy (cid:33) . We can write the last term in terms of curl u − λu and ν × u using integration by parts:0 = − (cid:90) ∂ Ω( x,ε,R ) ∇ x Γ λ ( x − y ) ν ( y ) · u ( y ) d y S + (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) div u ( y ) dy + (cid:90) ∂ Ω( x,ε,R ) ∇ x Γ λ ( x − y ) × ( ν ( y ) × u ( y )) d y S − (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) × (curl u ( y ) − λu ( y )) dy + λ (cid:32) − (cid:90) Ω( x,ε,R ) Γ λ ( x − y )(curl u ( y ) − λu ( y )) dy + (cid:90) ∂ Ω( x,ε,R ) Γ λ ( x − y ) ν ( y ) × u ( y ) d y S (cid:33) . (36)Let us finally take limits when ε → R → + ∞ in the preceding identities. We start with the volumeintegrals, that obviously converges to the integral over the whole exterior domain due to the dominatedconvergence theorem, the Hardy–Littewood–Sobolev theorem of fractional integration (Theorem 2.5) andthe hypotheses on div u and curl u − λu : (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) div u ( y ) dy −→ (cid:90) Ω ∇ x Γ λ ( x − y ) div u ( y ) dy, (cid:90) Ω( x,ε,R ) ∇ x Γ λ ( x − y ) × (curl u ( y ) − λu ( y )) dy −→ (cid:90) Ω ∇ x Γ λ ( x − y ) × (curl u ( y ) − λu ( y )) dy, (cid:90) Ω( x,ε,R ) Γ λ ( x − y )(curl u ( y ) − λu ( y )) dy −→ (cid:90) Ω Γ λ ( x − y )(curl u ( y ) − λu ( y )) dy, when ε → R → + ∞ .Regarding the boundary integrals, it is worth splitting them into the three connected components ofthe boundary surface of Ω( x, ε, R ): ∂ Ω( x, ε, R ) = S ∪ ∂B ε ( x ) ∪ ∂B R (0) . Since the integrals over S are not relevant in the limit ε → R → + ∞ , we focus on the two remainingterms. On the one hand, the boundary terms over the sphere ∂B ε ( x ) can be written as I ε := (cid:90) ∂B ε ( x ) ∇ x Γ λ ( x − y ) y − xε · u ( y ) d y S − (cid:90) ∂B ε ( x ) ∇ x Γ λ ( x − y ) × (cid:18) y − xε × u ( y ) (cid:19) d y S ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 17 − λ (cid:90) ∂B ε ( x ) Γ λ ( x − y ) y − xε × u ( y ) d y S. Notice that the derivative formula (12) for Γ λ ( x ) now reads ∇ x Γ λ ( x − y ) = (cid:18) iλ − | x − y | (cid:19) e iλ | x − y | π | x − y | x − y | x − y | . This identity and
Lagrange’s formula v = ( e · v ) e − e × ( e × v ) , for any unit vector e and any general vector v show that I ε := − (cid:18) iλ − ε (cid:19) e iλε πε (cid:90) ∂B ε ( x ) u ( y ) d y S − λ e iλε πε (cid:90) ∂B ε ( x ) y − xε × u ( y ) d y S = iλ e iλε πε (cid:90) ∂B ε ( x ) (cid:18) i y − xε × u ( y ) − u ( y ) (cid:19) d y S + e iλε πε (cid:90) ∂B ε ( x ) u ( y ) d y S. Consequently, the first term converges to zero as ε → u ( x ) due tothe properties of the mean value over spheres of continuous functions.In addition, the boundary terms over ∂B R (0) may also be written in a similar way I R := − (cid:90) ∂B R (0) ∇ x Γ λ ( x − y ) yR · u ( y ) d y S + (cid:90) ∂B R (0) ∇ x Γ λ ( x − y ) × (cid:16) yR × u ( y ) (cid:17) d y S + λ (cid:90) ∂B R (0) Γ λ ( x − y ) yR × u ( y ) d y S = (cid:90) ∂B R (0) (cid:18) iλ − | x − y | (cid:19) e iλ | x − y | π | x − y | y − x | y − x | yR · u ( y ) d y S − (cid:90) ∂B R (0) (cid:18) iλ − | x − y | (cid:19) e iλ | x − y | π | x − y | y − x | y − x | × (cid:16) yR × u ( y ) (cid:17) d y S + λ (cid:90) ∂B R (0) e iλ | x − y | π | x − y | yR × u ( y ) d y S. This time, the reasoning is slightly different because Lagrange’s formula for the triple vector productcannot be directly applied since B R (0) is not centered at x . See Remark 2.13 below for the behavior ofthis boundary integrals if we had defined Ω( x, ε, R ) = Ω ∩ B R ( x ) ∩ ( R \ B ε ( x )) instead of Ω( x, ε, R ) =Ω ∩ B R (0) ∩ ( R \ B ε ( x )). Let us add and subtract the appropriate terms in order to obtain a moresuggestive equation where Lagrange’s formula for the triple vector product can be applied I R := − iλ (cid:90) ∂B R (0) e iλ | x − y | π | x − y | (cid:16) i yR × u ( y ) − u ( y ) (cid:17) d y S − (cid:90) ∂B R (0) e iλ | x − y | π | x − y | u ( y ) d y S + (cid:90) ∂B R (0) (cid:18) iλ − | x − y | (cid:19) e iλ | x − y | π | x − y | (cid:18) y − x | y − x | − yR (cid:19) yR · u ( y ) d y S − (cid:90) ∂B R (0) (cid:18) iλ − | x − y | (cid:19) e iλ | x − y | π | x − y | (cid:18) y − x | y − x | − yR (cid:19) × (cid:16) yR × u ( y ) (cid:17) d y S. Then, the same argument as in (13) leads to the following bound of the norm of I R for R > | x || I R | ≤ | λ | π ( R − | x | ) (cid:90) ∂B R (0) (cid:12)(cid:12)(cid:12) i yR × u ( y ) − u ( y ) (cid:12)(cid:12)(cid:12) d y S (37)+ 14 π ( R − | x | ) (cid:90) ∂B R (0) | u ( y ) | d y S + 2 C | x | π ( R − | x | ) (cid:90) ∂B R (0) | u ( y ) | d y S. Thereby, I R → R → + ∞ , thanks to the L SMB radiation condition (29) and the L decayproperty (32).Now that we then have the representation formula in the statement of the theorem, the asymptoticbehavior at infinity follows from Theorem 2.5 and the componentwise Sommerfeld radiation condition inthe compactly supported case is a direct consequence of Propositions 2.3 and 2.4. (cid:3) Remark 2.13.
Consider Ω( x, ε, R ) = Ω ∩ B R ( x ) ∩ ( R \ B ε ( x )) instead of Ω( x, ε, R ) = Ω ∩ B R (0) ∩ ( R \ B ε ( x )) in Eq. (36). We can argue in the same way both for the boundary terms over ∂B ε ( x ) andfor those over ∂B R ( x ) . Then, the former has already been studied in the above proof and the later reads I R := (cid:18) iλ − R (cid:19) e iλR πR (cid:90) ∂B R ( x ) u ( y ) d y S + λ e iλR πR (cid:90) ∂B R ( x ) y − xR × u ( y ) d y S (38)= − iλ e iλR πR (cid:90) ∂B R ( x ) (cid:18) i y − xε × u ( y ) − u ( y ) (cid:19) d y S − e iλR πR (cid:90) ∂B R ( x ) u ( y ) d y S. Therefore, the same representation theorem might have been obtained from the following radiation anddecay conditions (cid:90) ∂B R ( x ) (cid:18) i y − xε × u ( y ) − u ( y ) (cid:19) d y S = o ( R ) , when R → + ∞ , (cid:90) ∂B R ( x ) u ( y ) d y S = o ( R ) , when R → + ∞ , for every x ∈ Ω . The hypotheses are stronger than (29) and (32) in the sense that they have to be assumedon every x ∈ Ω . However, they are weaker in the sense that norms can be removed here. Therefore, onemight take advantage of certain geometric cancellations of our vector fields to ensure these conditions.An obvious but interesting feature of the above boundary terms is that in both cases, when Ω( x, ε, R ) =Ω ∩ B R (0) ∩ ( R \ B ε ( x )) (37) and Ω( x, ε, R ) = Ω ∩ B R ( x ) ∩ ( R \ B ε ( x )) (38), the harmonic case λ = 0 doesnot need to prescribe any radiation condition at infinity, as it is the case in the classical Helmholtz–Hodgetheorem and in [38, 43] . Remark 2.14.
Analogously to the case of the Helmholtz equation, the L SMB radiation condition implyboth the L SMB radiation condition and the weak decay property in L : (30) ⇒ (29) + (32) . Indeed,let u ∈ C (Ω , C ) be any solution to the complex-valued homogeneous Beltrami equation in the exteriordomain which satisfies the L SMB radiation condition (30). Let us compute the square in the radiationcondition as follows (cid:12)(cid:12)(cid:12) i xR × u ( x ) − u ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) xR × u ( x ) (cid:12)(cid:12)(cid:12) + | u ( x ) | − (cid:60) (cid:16) iu ( x ) · (cid:16) xR × u ( x ) (cid:17)(cid:17) = (cid:12)(cid:12)(cid:12) xR × u ( x ) (cid:12)(cid:12)(cid:12) + | u ( x ) | + 2 (cid:61) (cid:16) u ( x ) · (cid:16) xR × u ( x ) (cid:17)(cid:17) , (39) for any x ∈ ∂B R (0) . For any positive radius such that G ⊆ B R (0) , we define the subdomains Ω R := B R (0) \ G , for each R > R . Elementary computations involving the Beltrami equation leads to − λ (cid:90) Ω R | u ( x ) | dx = (cid:90) Ω R u ( x ) · curl u ( x ) dx = (cid:90) Ω R div( u × u ) dx + (cid:90) Ω R u ( x ) · curl u ( x ) dy. Let us compute the mean value of the two preceding equalities and obtain, thanks to the divergence theorem, − λ (cid:90) Ω R | u ( x ) | dx = (cid:60) (cid:18)(cid:90) Ω R u ( x ) · curl u ( x ) dx (cid:19) + 12 (cid:90) ∂B R (0) ( u ( x ) × u ( x )) · xR d x S − (cid:90) S ( u ( x ) × u ( x )) · η ( y ) d y S. Taking imaginary parts in the above equation leads to (cid:61) (cid:32)(cid:90) ∂B R (0) ( u ( x ) × u ( x )) · xR d x S (cid:33) = (cid:61) (cid:18)(cid:90) S ( u ( x ) × u ( x )) · η ( x ) d x S (cid:19) , (40) for each R > R . Finally, (39), (40) along with the L SMB radiation condition (30) lead to lim R → + ∞ (cid:90) ∂B R (0) (cid:18)(cid:12)(cid:12)(cid:12) xR × u ( x ) (cid:12)(cid:12)(cid:12) + | u ( x ) | (cid:19) dx = 2 (cid:61) (cid:18)(cid:90) S u ( x ) · ( η ( x ) × u ( x )) d x S (cid:19) . (41) As a consequence, (cid:90) ∂B R (0) | u ( x ) | dx = O (1) , when R → + ∞ , and Cauchy-Schwarz inequality leads to the L decay property (cid:90) ∂B R (0) | u ( x ) | dx = O ( R ) , when R → + ∞ . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 19
In particular, one gets (32).
This remark is useful because in order to check the hypotheses in Theorem 2.12 it is sometimes simplerto check that the L SMB radiation condition holds. Furthermore, it can be combined with the Rellichlemma [10, Lemma 2.11] to obtain a uniqueness result, which is similar to that for the reduced Maxwellsystem in [10, Theorem 6.10]:
Lemma 2.15.
Consider any solution u ∈ C (Ω , C ) to the complex-valued homogeneous Beltrami equa-tion in the exterior domain satisfying the L SMB radiation condition (30). Then, u verifies the inequality (cid:61) (cid:18)(cid:90) S u ( x ) · ( η ( x ) × u ( x )) d x S (cid:19) ≥ . If the equality holds, then u vanishes everywhere in Ω . To conclude, let us state the existence result for the complex-valued homogeneous Beltrami equationthat will be needed in the modified Grad-Rubin iterative scheme in Section 3. Since this iterative methodonly involves compactly supported inhomogeneities, we will focus on this case although it is easy to extendit to general inhomogeneous terms with an appropriate fall off at infinity. Hereafter we will denote by X k,α ( S ) ≡ X k,α ( S, R ) the real vector space of all tangent vector fields on S of regularity C k,α , i.e., X k,α ( S ) := { ξ ∈ C k,α ( S, R ) : ξ · η = 0 on S } . Its complex counterpart will be denoted by X k,α ( S, C ). Theorem 2.16.
Let (cid:54) = λ ∈ R be any constant that is not a Dirichlet eigenvalue of the Laplace operatorin the interior domain, w ∈ C k,αc (Ω , C ) and g ∈ C k +1 ,α ( S, C ) such that div w ∈ C k,α (Ω , C ) and thefollowing compatibility condition (cid:90) S ( λg + w · η ) dS = 0 (42) is satisfied. Consider any solution ξ ∈ X k +1 ,α ( S, C ) to the boundary integral equation (cid:18) I − T λ (cid:19) ξ = µ, x ∈ S, (43) where T λ ξ and µ are defined by ( T λ ξ )( x ) = (cid:90) S η ( x ) × ( ∇ x Γ λ ( x − y ) × ξ ( y )) d y S + λ (cid:90) S Γ λ ( x − y ) η ( x ) × ξ ( y ) d y S, (44) µ ( x ) = 1 λ (cid:90) Ω η ( x ) × ∇ x Γ λ ( x − y ) div w ( y ) dy − (cid:90) S η ( x ) × ∇ x Γ λ ( x − y ) g ( y ) d y S + (cid:90) Ω η ( x ) × ( ∇ x Γ λ ( x − y ) × w ( y )) dy + λ (cid:90) S Γ λ ( x − y ) η ( x ) × w ( y ) d y S. (45) Define the complex-valued vector field u ( x ) := −∇ φ ( x ) + curl A ( x ) + λA ( x ) , x ∈ Ω , (46) where φ and A stand for the scalar and vector fields φ ( x ) = − λ (cid:90) Ω Γ λ ( x − y ) div w ( y ) dy + (cid:90) S Γ λ ( x − y ) g ( y ) d y S, (47) A ( x ) = (cid:90) Ω Γ λ ( x − y ) w ( y ) dy + (cid:90) S Γ λ ( x − y ) ξ ( y ) d y . (48) Then, u is a complex-valued solution to the exterior NIB problem curl u − λu = w, x ∈ Ω ,u · η = g, x ∈ Ω , + L SMB radiation condition (29) , + L decay property (32) . (49) Furthermore, the decay and radiation conditions are stronger since u actually behaves as O (cid:0) | x | − (cid:1) atinfinity and verifies the Sommerfeld radiation condition (10) componentwise. Proof.
Since the divergence of any solution u can be recovered from the equation through the identitydiv u = − λ div w , then one arrives at the next expression for the candidate to be a solution to (49) u ( x ) = −∇ φ ( x ) + curl A ( x ) + λA ( x ) , where φ and A are defined as follows φ ( x ) = − λ (cid:90) Ω Γ λ ( x − y ) div w ( y ) dy + (cid:90) S Γ λ ( x − y ) g ( y ) d y S,A ( x ) = (cid:90) Ω Γ λ ( x − y ) w ( y ) dy + (cid:90) S Γ λ ( x − y ) η ( y ) × u + ( y ) d y S. Consider ξ := η × u + , where u ± denotes the limits of u at S from Ω and G respectively. In order toobtain a more manageable formula for ξ , one can use the well known jump relations for the derivatives ofa single layer potential associated with the fundamental solution to the Helmholtz equation, Γ λ ( x ) (seee.g. [9]). This formulas lead to the following identity u ± ( x ) = 1 λ (cid:90) Ω ∇ x Γ λ ( x − y ) div w ( y ) dy − PV (cid:90) S ∇ x Γ λ ( x − y ) g ( y ) d y S (50)+ (cid:90) Ω ∇ x Γ λ ( x − y ) × w ( y ) dy + PV (cid:90) S ∇ x Γ λ ( x − y ) × ξ ( y ) d y S + λ (cid:90) Ω Γ λ ( x − y ) w ( y ) dy + λ (cid:90) S Γ λ ( x − y ) ξ ( y ) d y S ± η ( x ) g ( x ) ∓ η ( x ) × ξ ( x ) , where PV stands for the Cauchy principal value integral. It is clear that the terms in the last lineare actually ± u ± ( x ). Consequently, one can take cross products by η ( x ) and arrive at the boundaryintegral equation in (43) for the tangential component ξ . There, we have intentionally avoided the PVsigns because the η ( x ) factor in such integrals provides certain geometrical cancellations (see Section 6)leading to absolutely convergent integrals.Now, let us show that the field u thus defined is a solution to (49) as long as ξ solves the boundaryintegral equation (43). We will prove later that ξ is unique and, consequently, (49) is uniquely solvable.First, let us obtain some PDEs for the potentials φ and A both in the interior and the exterior domain.Since volume and single layer potentials are indeed complex-valued solutions to such PDEs, we have∆ φ + λ φ = (cid:26) λ div w, x ∈ Ω0 , x ∈ G ∆ A + λ A = (cid:26) − w, x ∈ Ω0 , x ∈ G (51)Therefore, curl u − λu = ∇ (div A ) − ∆ A + λ curl A + λ ∇ φ − λ curl A − λ A = − (∆ A + λ A ) + ∇ (div A + λφ ) (cid:124) (cid:123)(cid:122) (cid:125) a . A direct substitution of (51) into the previous formula leads to the following PDE for u at any side ofthe boundary surface S : curl u − λu = (cid:26) w + ∇ a, x ∈ Ω , ∇ a, x ∈ G. (52)In order to show that u solves (49), it remains to check that ∇ a is zero in the exterior domain and u satisfies the boundary condition u + · η = g (the decay and radiation conditions will be studied later). Tothis end, it might be useful to find first a PDE for a . The same reasoning as above shows that a solvesthe homogeneous Helmholtz equation, specifically∆ a + λ a = div(∆ A ) + λ ∆ φ + λ div A + λ φ = div(∆ A + λ A ) + λ (∆ φ + λ φ ) = 0 . (53)Let us show first the jump relations for the scalar potential a . Straightforward computations on theexplicit formulas for φ and A involving the divergence theorem lead to a ( x ) = div A ( x ) + λφ ( x )= (cid:90) Ω ∇ x Γ λ ( x − y ) · w ( y ) dy + (cid:90) S ∇ x Γ λ ( x − y ) · ξ ( y ) d y S − (cid:90) Ω Γ λ ( x − y ) div w ( y ) dy + λ (cid:90) S Γ λ ( x − y ) g ( y ) d y S ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 21 = − (cid:90) Ω div y (Γ λ ( x − y ) w ( y )) dy + (cid:90) S ∇ x Γ λ ( x − y ) · ξ ( y ) d y S + λ (cid:90) S Γ λ ( x − y ) g ( y ) d y S = (cid:90) S Γ λ ( x − y )( λg ( y ) + w ( y ) · η ( y )) d y S + (cid:90) S ∇ x Γ λ ( x − y ) · ξ ( y ) d y S. Finally, notice that ∇ x Γ λ ( x − y ) · ξ ( y ) = − ( ∇ S ) y [Γ λ ( x − y )] · ξ ( y ) for every y ∈ S because of ξ being atangent vector field along S . Hence, the integration by parts formula over S (see Appendix A) yields thenext simpler expression for a : a ( x ) = (cid:90) S Γ λ ( x − y ) ( λg ( y ) + w ( y ) · η ( y ) + div S ξ ( y )) d y S, i.e., a is just a new single layer potential. As such, the first and second jumps relations read a + − a − ≡ , (cid:18) ∂a∂η (cid:19) + − (cid:18) ∂a∂η (cid:19) − ≡ − ( λg + w · η + div S ξ ) , (54)on the surface S . In particular, a is continuous across S but its normal derivative exhibits a jumpdiscontinuity with height λg + w · η + div S ξ . The same kind of reasoning yields the jump relation for uu + − u − = g η − η × ξ, x ∈ S. (55)Consequently, the boundary integral equation (43) along with the jump relation (55) ensure that η × u + = ξ, η × u − = 0 , (56)on S . Regarding a , let us show that it is indeed constant on S and to this end, define the next vectorfield in the interior domain G : v := λu + ∇ a, x ∈ G. Notice that v is a strong Beltrami field with factor λ by virtue of (52). Then, one can repeat the samekind of uniqueness criterion as in Lemma 2.15 in the simpler bounded setting, specifically λ (cid:90) G | v | dx = (cid:90) G v · curl v dx = (cid:90) G div( v × v ) dx = (cid:90) S ( η × v ) · v dS. Now, notice that we can substitute both v and v in the above formula with its tangential parts thanksto the presence of a cross product by the unit normal vector field η and − η × ( η × v ) = − λη × ( η × u − ) + ∇ S a = ∇ S a, by virtue of (56). Thereby, the integration by parts formula in Appendix A leads again to λ (cid:90) G | v | dx = (cid:90) S ( η × ∇ S a ) · ∇ a dS = − (cid:90) S a curl S ( ∇ S a ) dS = 0 , where the well know formula curl S ∇ S = 0 (see Proposition A.1) has been used in the last step. Conse-quently, v vanishes everywhere in G and, in particular, ∇ S a ≡
0, i.e., a ± ≡ a = const on S .We will next prove that a vanishes everywhere in the exterior domain Ω using the uniqueness resultin Lemma 2.8. Notice that since a can be written as a sum of volume and single layer potentialswith compactly supported densities together with its first order partial derivatives, then a satisfies theSommerfeld radiation condition (10) due to Propositions 2.3 and 2.4. Consequently, this lemma can beapplied. We therefore want to show that (cid:61) (cid:32)(cid:90) S a + (cid:18) ∂a∂η (cid:19) + dS (cid:33) = 0 . (57)To derive (57), let us first pass from the exterior trace values to the interior ones thanks to the jumprelations (54) (cid:90) S a + (cid:18) ∂a∂η (cid:19) + dS = I + II, where both terms read I := − a (cid:90) S ( λg + w · η + div S ξ ) dS,II := (cid:90) S a − (cid:18) ∂a∂η (cid:19) − dS. On the one hand, I becomes zero because of the divergence theorem over surfaces and the compatibilitycondition (42) in the hypothesis. On the other hand, integrate by parts in II to arrive at II := (cid:90) S div ( a ∇ a ) dS = (cid:90) G |∇ a | dx + (cid:90) a ∆ a dx = (cid:90) G |∇ a | dx − λ (cid:90) G | a | dx, where the Helmholtz equation (53) has being used. Therefore, one arrives at (cid:61) (cid:32)(cid:90) S a + (cid:18) ∂a∂η (cid:19) + dS (cid:33) = (cid:61) (cid:18)(cid:90) G |∇ a | dx − λ (cid:90) G | a | dx (cid:19) = 0 , and consequently a = 0 in Ω and u solves the inhomogeneous Beltrami equation.Before proving the boundary condition and the decay and radiation properties, let us show that a alsovanishes in the interior domain. On the one hand, a solves the homogeneous Helmholtz equation in suchdomain and it also satisfies the interior homogeneous Dirichlet conditions in S since a − = a + on S and a = 0 in Ω. Moreover, λ is prevented from being a Dirichlet eigenvalue of the Laplacian in the interiordomain, so a also vanishes in G . In particular, the jumps relations (54) yields λg + w · η + div S ξ ≡ . (58)Furthermore, since u is now a solution to the next inhomogeneous Beltrami equation,curl u − λu = w, x ∈ Ω , taking trace values at S one gets η · (curl u ) + − λη · u + = w · η. Now, one can write the first term in an intrinsic way through the properties in Proposition A.1, specifically η · (curl u ) + = − div S ( η × u + ) = − div S ξ, and, consequently, the above formula can be restated as η × u + + w · η + div S ξ ≡ . (59)Then, comparing (58) and (59) entails the boundary condition η × u + = g .Finally, let us show the decay and radiation conditions on u . First, sinceΓ λ ( x ) , ∇ Γ λ ( x ) = O (cid:0) | x | − (cid:1) , when | x | → + ∞ , and w has compact support, then u enjoys the optimal decay u = O (cid:0) | x | − (cid:1) when | x | → + ∞ accordingto Theorem 2.5. Second, as u is again a sum of single and volume layer potential associated with theHelmholtz equation along with some partial derivatives, then u satisfies Sommerfeld radiation conditioncomponentwise thanks to Propositions 2.3 and 2.4. Therefore, one can show that u verifies SMH condi-tions (26) and (27) thanks to Remark 2.9. Since curl u − λu = w and w is compactly supported, then u actually satisfies SMB radiation condition (31) and this finishes the proof. (cid:3) Well-posedness of the boundary integral equation.
One should also notice that, in addition tothe uniqueness result proved in Theorem 2.16, we will also need a study of the regularity of the solution,which is obviously in C (Ω , C ) by the decomposition (46). We will prove in this next subsection that theregularity assumptions on the data w and g actually leads to C k +1 ,α (Ω , C ) regularity on u , estimatingits C k +1 ,α (Ω , C ) norm in terms of the natural norms of the data w and g . Some necessary potentialtheoretic estimates have been relegated to Section 6 to streamline the exposition.Let us start by studying the well-posedness of (43) using the Riesz–Fredholm theory for compactoperators, which follows easily from our previous estimates: Proposition 2.17.
The linear operator T λ : X k +1 ,α ( S ) −→ X k +1 ,α ( S ) is compact.Proof. The gain of regularity proved in Theorem 6.11 implies that T λ defines a continuous linear operator T λ : X k,α ( S ) −→ X k +1 ,α ( S ) . Since the embedding X k +1 ,α ( S ) (cid:44) → X k,α ( S ) is compact by the Ascoli–Arzel`a theorem, the propositionfollows. (cid:3) The proposition ensures that it is possible to apply Riesz–Fredholm theory to the operator I − T λ .In particular, I − T λ is one to one if, and only if, it is onto, i.e.,Ker (cid:18) I − T λ (cid:19) = 0 ⇐⇒ Im (cid:18) I − T λ (cid:19) = X k +1 ,α ( S ) . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 23
As it is hard to show explicitly that such operator is onto, let us equivalently show that it is one to one.This will be easier thanks to the uniqueness Lemma 2.15 for the Beltrami equation and the existenceTheorem 2.16.
Proposition 2.18.
The bounded linear operator I − T λ on X k +1 ,α ( S ) is one to one and onto. Conse-quently, the boundary integral equation (43) has a unique solution ξ ∈ X k +1 ,α ( S ) for any µ ∈ X k +1 ,α ( S ) .Proof. According to the preceding argument, we only have to show that Ker (cid:0) I − T λ (cid:1) = 0. To this end,let us consider an arbitrary ξ ∈ Ker (cid:0) I − T λ (cid:1) and show that ξ ≡
0. By definition, ξ ∈ X k +1 ,α ( S ) solvesthe boundary integral equation 12 ξ − T λ ξ = 0 , on S. Define u ( x ) := curl A ( x ) + λA ( x ), where A is the vector potential A ( x ) := (cid:90) S Γ λ ( x − y ) ξ ( y ) d y S. Thus, Theorem 2.16 for w ≡ g ≡ u ∈ C (Ω , C ) to the homogeneous Beltramiequation in Ω (cid:26) curl u = λu, x ∈ Ω ,η · u + = g = 0 , x ∈ S, that satisfies the Dirichlet boundary condition η × u + = ξ on S and the SMB radiation condition.We would like to show that this boundary value problem has a unique solution, but this does notfollow directly from Lemma 2.15. However, since η · u = 0 on S , then u + = − η × ( η × u + ) on S andwe have the following relation between the curl operator on S , curl S , and the curl operator on R (seeProposition A.1 in Appendix A):curl S u + = curl S ( − η × ( η × u + )) = η · curl u + = λ η · u + = 0 . As S is homeomorphic to a sphere, Poincar´e’s lemma shows that u has an associated potential ψ ∈ C ( S )on the surface, i.e., u = ∇ S ψ on S , where ∇ S stands for the Riemannian connection on the surface S (see Appendix A). Consequently, (cid:61) (cid:18)(cid:90) S u · ( η × u ) dS (cid:19) = (cid:61) (cid:18)(cid:90) S ∇ S ψ · ( η × ∇ S ψ ) dS (cid:19) = −(cid:61) (cid:18)(cid:90) S curl S ( ∇ S ψ ) ψ dS (cid:19) = 0 . The identity follows from an integration by parts on S and the classical property curl S ( ∇ S ψ ) = 0.Therefore, Lemma 2.15 yields the desired result. (cid:3) Remark 2.19.
The importance of the above result lies on the following facts. (1)
First, the existence part of the above result ensures that it is possible to choose some ξ solving(43). Obviously, it is essential to rigurously establish the existence Theorem 2.16. (2) Second, the uniqueness result shows that since ξ can uniquely be chosen, then (49) has a uniquesolution too. (3) Finally, it provides a very useful estimate for the subsequent result. Since I − T λ is linear,continuous and bijective, then (cid:0) I − T λ (cid:1) − is continuous by virtue of the Banach isomorphismtheorem. Consequently, there exists a positive constant c (which depends on G and λ ) such that c (cid:107) ξ (cid:107) C k +1 ,α ( S ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) I − T λ (cid:19) ξ (cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( S ) , (60) for any ξ ∈ X k +1 ,α ( S ) . We conclude by proving the following regularity result for the solution u of (49) according to Theorem2.16. It is an immediate consequence of the decomposition (46), the estimates for the volume and singlelayer potentials in Section 6 (Lemmas 6.10 and 6.1) and the estimate (60). Corollary 2.20.
Assume that the hypothesis in Theorem 2.16 are satisfied, fix any
R > such that G ⊆ B R (0) and assume that the closure of Ω R := B R (0) \ G contains the support of w . Then, there existssome nonnegative constant C = C ( k, α, G, R, λ ) such that the next estimate (cid:107) u (cid:107) C k +1 ,α (Ω) ≤ C (cid:8) (cid:107) w (cid:107) C k,α (Ω) + (cid:107) div w (cid:107) C k,α (Ω) + (cid:107) g (cid:107) C k +1 ,α ( S ) (cid:9) . (61) holds. In particular, not only does u belong to C (Ω , C ) , but also to C k +1 ,α (Ω , C ) . Optimal fall-off in exterior domains.
Before passing to the next section, it is worth discussingthe differences between the optimal fall-off | x | − of the solutions to inhomogeneous Beltrami equationand that of the solutions of the div-curl problem. First, it is well known that the exterior Neumannboundary value problem associated with the div-curl system curl u = w, x ∈ Ω , div u = f, x ∈ Ω ,u · η = g, x ∈ S,u = O ( | x | − ρ ) , x ∈ Ω , (62)where w, f = O ( | x | − ρ ) and ρ ∈ (1 , w has zero flux in the exterior domain. Moreover, the solution inherits theoptimal fall-off | x | − when w and f are assumed to have compact support. In particular, any harmonicfield ( w = 0 , f = 0) so obtained decays at infinity as | x | − . Such result is an easy consequence of theHelmholtz–Hodge representation formula in [38, Theorem 4.1] and the natural fall-off of the fundamentalsolution of the Laplace equation, Γ ( x ).In our case, the exterior Neumann boundary value problem associated with the inhomogeneous Bel-trami equation (49) has an associated representation formula of Helmholtz–Hodge type (46) that transfersthe “optimal fall-off” | x | − to the solution in Theorem 2.16 when w is assumed to have compact support.Let us show that it is indeed the optimal decay rate. To this end, assume that u solves the equationcurl u − λu = w, x ∈ Ω , (not necessarily fulfilling neither (32) nor (29)) for some divergence-free vector field w . Then, the solution u is divergence-free too. Hence, taking curl in the inhomogeneous Beltrami equation, we are led to thevector-valued Helmholtz equation − (∆ u + λ u ) = λw + curl w, x ∈ Ω . Consider K := supp w ⊆ Ω and notice that λw + curl w is also compactly supported in K . Imagine that u decayed as | x | − (1+ ε ) for some small ε >
0. Hence, a straightforward computation leads tolim R → + ∞ (cid:90) ∂B R (0) | u ( x ) | = 0 . Consequently, Rellich’s Lemma [10, Lemma 2.11] would show that u vanishes outside some sufficientlylarge ball centered at the origin and containing K . Then, the unique continuation principle of theHelmholtz equation allow proving that u is also compactly supported in K (see [31] for the study of suchproperty in many other linear PDEs with constants coefficients). In particular, g would vanish outside K ∩ S . In an equivalent way, the next result holds. Corollary 2.21.
Let u ∈ C k +1 ,α (Ω , R ) be a solution to curl u − λu = w, x ∈ Ω , for a divergence-free compactly supported w and some λ ∈ R \{ } . If u is transverse to S at some pointoutside the support of w , then u cannot decay faster than | x | − at infinity, i.e., there exists no ε > suchthat u = O ( | x | − (1+ ε ) ) , when | x | → + ∞ . The above Corollary can be interpreted in two different ways. First, it establishes the optimal fall-offof a “transverse” strong Beltrami field ( w = 0). Second, it also deals with some kind of “transverse”generalized Beltrami fields in exterior domains ( w = ϕu ) that will be of a great interest in our work. Werestrict to the second result since it contains the first one as a particular case. Corollary 2.22.
Let u ∈ C k +1 ,α (Ω , R ) be a generalized Beltrami field, i.e., (cid:26) curl u − f u = 0 , x ∈ Ω , div u = 0 , x ∈ Ω , whose proportionality factor is a compactly supported perturbation of a constant proportionality factor λ ∈ R \{ } , i.e., f = λ + ϕ for some ϕ ∈ C k,αc (Ω) . If u is transverse to S at some point outside thesupport of the perturbation ϕ , then u cannot decay faster than | x | − at infinity, i.e., there exists no ε > such that u = O ( | x | − (1+ ε ) ) , when | x | → + ∞ . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 25
Remark 2.23.
In particular, the above result leads to the natural counterpart for exterior domain of theLiouville theorem in [7, 36] about the fall-off of entire generalized Beltrami fields. Such theorem statesthat there is no globally defined generalized Beltrami field decaying faster than | x | − at infinity. As manyothers Liouville type results, it strongly depends on the solution being defined in the whole R . In our casewe remove this hypothesis but, in return, we need to argue with generalized Beltrami fields with constantproportionality factor outside a compact set enjoying some trasversality condition on the boundary surfaceof the exterior domain. An iterative scheme for strong Beltrami fields
Our objective in this section is to set the iterative scheme that we will use to establish the partialstability of strong Beltrami fields that will yield the existence of almost global Beltrami fields with anon-constant factor and complex vortex structures.3.1.
Further notation and preliminaries.
We devote a few lines to introduce some notation that willbe in continuous use in the rest of the paper. Although most of the results are classical [23], others areinspired in [27], where they have been used in the electromagnetic framework.On the differentiable surface S , we will consider local charts of the same regularity as S (that is, maps µ covering open subsets Σ ⊆ S of the form µ : D −→ R s (cid:55)−→ µ ( s ) , where µ ( D ) = Σ and D is a disk in the plane). Without any loss of generality, we will assume µ to bea local parametrization up to the boundary so that µ can be homeomorphically extended to the closure D , Σ = µ ( D ).We will also consider the corresponding C k and C k,α spaces of functions defined on a coordinateneighborhood Σ of S provided with a local chart µ . Up to the degree of smoothness of the surface, bycompactness they are known to be independent of the choice of the chart, so one can write C k (Σ) := { f ∈ C k (Σ) : f ◦ µ ∈ C k ( D ) } ,C k,α (Σ) := { f ∈ C k (Σ) : f ◦ µ ∈ C k,α ( D ) } and similarly for spaces on Σ. These spaces can be respectively endowed with the complete norms (cid:107) f (cid:107) C k (Σ) := (cid:107) f ◦ µ (cid:107) C k ( D ) , (cid:107) f (cid:107) C k,α (Σ) := (cid:107) f ◦ µ (cid:107) C k,α ( D ) . Let us consider a C k,α surface. An useful result is Calder´on’s extension theorem for C k,α functions,see e.g. [23, Lemma 6.37]: Proposition 3.1.
Let O ⊆ R be a C k,α domain with bounded boundary ∂O , and let O (cid:48) be any opensubset such that O ⊆ O (cid:48) . Then, there exists a linear operator P : C k,α ( O ) −→ C k,α ( O (cid:48) ) f (cid:55)−→ P ( f ) ≡ f , such that (1) P is an extension operator, i.e., P ( f ) | O = f, ∀ f ∈ C k,α ( O ) . (2) The support of P ( f ) is contained in the open subset O (cid:48) for evey f ∈ C k,α (Ω) . (3) P is continuous in the C k,α topology, i.e., (cid:107)P ( f ) (cid:107) C k,α ( O (cid:48) ) ≤ C P (cid:107) f (cid:107) C k,α ( O ) , ∀ f ∈ C k,λ ( O ) . (4) P is also continuous in the C m topology for any ≤ m ≤ k , i.e., (cid:107)P ( f ) (cid:107) C m ( O (cid:48) ) ≤ C P (cid:107) f (cid:107) C m ( O ) , ∀ f ∈ C k,α ( O ) . In the above inequalities, C P stands for a constant which depends on k, O and O (cid:48) . To describe the stream lines and tubes associated with a velocity field u ∈ C k +1 ,α (Ω , R ) in presenceof a boundary surface which u is not tangent to, it is convenient to consider an extension of the field totrivially obtain the following characterization from the classical Picard–Lindel¨of theorem for ODEs onH¨older spaces: G ΩΣ T (Σ , u ) b b µ ( s ) X ( T ; 0 , µ ( s )) Figure 3.
Stream lines and tubes of the velocity field u . Proposition 3.2.
Let O ⊆ R be a C k +1 ,α bounded domain, where k ≥ and < α ≤ . Considerany vector field u ∈ C k +1 ,α ( O, R ) , its associated extension u = P ( u ) ∈ C k +1 ,α ( R , R ) according toProposition 3.1, any point x ∈ R and an initial time t ∈ R . Consider the associated characteristicsystem (cid:40) dXdt = u ( X ) , t ∈ R ,X ( t ) = x . (63) Then, such problem is uniquely and globally (in time) solvable, its solution will be denoted X ( t ; t , x ) , X ( t ; t , · ) is a C k +1 global diffeomorphism of the Euclidean space for every t, t ∈ R and its inverse is X ( t ; t, · ) . The solutions to these problems are the stream lines of the extended velocity field u .Consider any x ∈ O and let T ( x ) ≥ be the greatest time for which the stream line X ( t ; 0 , x ) , t > remains inside the open subset O , i.e., T ( x ) := sup { T > X ( t ; 0 , x ) ∈ O ∀ t ∈ (0 , T ) } . Then, X ( t ; 0 , x ) , < t < T ( x ) is a stream line of u , or equivalently, it solves the ODE (cid:40) dXdt = u ( X ) , < t < T ( x ) ,X (0) = x . Notice that when X ( t ; 0 , x ) / ∈ O, ∀ t ∈ (0 , T ) for some T > , then T ( x ) = 0 , i.e., the correspondingstream line of u does not originally enter the region O . We will also consider stream tubes of a velocity field which emanate from an open subset of the surface S . Consider any vector field u ∈ C k +1 ,α (Ω , R ), u = P ( u ) ∈ C k +1 ,α ( R , R ) its extension according toCalder´on’s extension theorem, X ( t ; t , x ) its associated flux mapping through Proposition 3.2 and anopen subset Σ ⊆ S together with a local chart µ : D −→ S . The stream tube of u which emanates fromΣ is the collection of all stream lines of u radiating from the points in the open subset Σ, i.e., T (Σ , u ) := { X ( t ; 0 , µ ( s )) : s ∈ D, < t < T ( µ ( s )) } . It is also useful to consider bounded stream lines with “height”
T > T (Σ , u, T ) := { X ( t ; 0 , µ ( s )) : s ∈ D, < t < min { T, T ( µ ( s )) }} . Notice that in order for a stream line of u to be well defined, it is necessary that the velocity field pointstowards the exterior domain. The same condition leads to well defined stream tubes emanating from Σ. ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 27
The regularity in the preceding result follows from Peano’s differentiability theorem. The same regularityresult may be used in order to derive the regularity in the stream tubes parametrization.
Proposition 3.3.
Consider G, Σ , and µ verying the hypothesis (7), u ∈ C k +1 ,α (Ω , R ) be a velocity fieldin the exterior domain, and assume that the vector field u points towards the exterior domain at anypoint of Σ , i.e., there exits a positive ρ > such that u · η ≥ ρ on Σ . Then, a well defined stream lineof u emanates from each point of Σ and they smoothly foliate the whole stream tube T (Σ , u ) . To makethis statement more precise, let us define D (Σ , u ) := { ( t, s ) : s ∈ D, < t < T ( µ ( s )) } , and the mapping φ : D (Σ , u ) −→ T (Σ , u )( t, s ) (cid:55)−→ φ ( t, s ) := X ( t ; 0 , µ ( s )) . Then, (1) T ( µ ( s )) > , for each s ∈ D . (2) φ is bijective. (3) φ is a C k +1 diffeomorphism. (4) Jac( φ ) and Jac( φ ) − belongs to C k,α locally in t , i.e., there exists a function κ : R +0 × R +0 −→ R +0 which is increasing with respect to each variable, such that if one defines D (Σ , u, T ) := { ( t, s ) : s ∈ D, < t < min { T, T ( µ ( s )) }} and the mapping φ |D (Σ ,u,T ) : D (Σ , u, T ) −→ T (Σ , u, T ) , then, (cid:107) Jac( φ ) (cid:107) C k,α ( D (Σ ,µ,T )) , (cid:107) Jac( φ ) − (cid:107) C k,α ( T (Σ ,µ,T )) ≤ κ (cid:16) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:17) , for every positive number T .Proof. We sketch the proof of this result for the reader’s convenience (see [27, Lemma 5.1] for k = 0 andin [40, Proposici´on 2.1.7] for arbitrary k ). The first assertion is apparent: since u points outwards atany point in Σ, then the stream line of u arising from µ ( s ) points towards Ω at t = 0. Hence, a smallpiece of such stream line must stay in Ω. Regarding the second assertion, φ is clearly onto by virtue ofthe definition of T (Σ , u ). To check that φ is one to one, note that different stream lines cannot touchbecause of the uniqueness part in Proposition 3.2, and that the streamlines of u emerging from Σ cannotbe closed loops because u points outwards at Σ.The C k +1 regularity of φ is clear because so is X ( t ; t , x ) by Peano’s differentiability theorem as statedin Proposition 3.2. Let us show that its Jacobian matrix is regular at any point in D (Σ , u ) to obtain thesame regularity of φ − through the inverse mapping theorem. This matrix takes the formJac( φ )( t, s ) = (cid:18) ∂φ∂t ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂s ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂s ( t, s ) (cid:19) . For fixed s ∈ D , each column vector is a solution to the linear ODE˙ x ( t ) = Jac( u )( φ ( t, s )) x ( t ) . Thus, Jac( φ )( · , s ) is a solution matrix to such linear ODE, whose determinant at t = 0 equalsdet(Jac( φ )(0 , s )) = (cid:12)(cid:12)(cid:12)(cid:12) ∂µ∂s ( s ) × ∂µ∂s ( s ) (cid:12)(cid:12)(cid:12)(cid:12) u ( µ ( s )) · η ( µ ( s )) ≥ ρ ρ > . (64)Here ρ stands for any positive uniform lower bound of the first factor. Thus, Jac( φ )( t, s ) is regular forall t by the Jacobi–Liouville formula . In particular, the derivatives of Jac( φ ) and Jac( φ ) − up to order k can be continuously extended to D (Σ , u, T ) by the analogous properties of u and µ .Let us finally recursively show that all of them are bounded and the k -th order ones are α -H¨oldercontinuous indeed. First, notice that (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂t ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) u (cid:107) C (Ω) , (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂s i ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂µ∂s i ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) t (cid:107) Jac( u ) (cid:107) C (Ω) (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂s i ( τ, s ) (cid:12)(cid:12)(cid:12)(cid:12) dτ, for every ( t, s ) ∈ D (Σ , u, T ). As a consequence, Gronwall’s lemma amounts to the upper bound (cid:107) Jac( φ ) (cid:107) C ( D (Σ ,u,T )) ≤ (cid:107) u (cid:107) C (Ω) + (cid:107) µ (cid:107) C ( D, R ) e T (cid:107) Jac( u ) (cid:107) C ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) , for some function κ which is separately increasing.Assume now that the analogous estimate (cid:107) Jac( φ ) (cid:107) C m ( D (Σ ,u,T )) ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) , (65)holds true for some n such that 0 < n ≤ k and all m with 0 ≤ m < n and let us prove it for m = n . Fixany multi-index γ such that | γ | = n and take derivatives on the characteristic system (63) to arrive at D γ (cid:18) ∂φ i ∂t (cid:19) = D γ ( u i ( φ ( t, s ))) = γ ! (cid:88) ( l,β,δ ) ∈D ( γ ) ( D δ u i )( φ ( t, s )) l (cid:89) r =1 δ r ! (cid:18) β r ! D β r φ ( t, s ) (cid:19) δ r . The above formula is nothing but a chain rule for high order partial derivatives of a composition function.Here, D ( γ ) stands for the set of all the possible decompositions of γγ = l (cid:88) r =1 | δ r | β r , where δ r , β r are multi-indices, δ := (cid:80) lr =1 δ r and for every r = 1 , . . . , l − i r ∈ { , , } such that ( β r ) i = ( β r +1 ) i for every i (cid:54) = i r and ( β r ) i r < ( β r +1 ) i r . Similarly ∂∂t D γ (cid:18) ∂φ i ∂s j (cid:19) = l (cid:88) q =1 (cid:88) ρ ≤ γ (cid:88) ( l,β,δ ) ∈D ( ρ ) (cid:18) γρ (cid:19) ρ ! (cid:18) D δ ∂u i ∂x q (cid:19) ( φ ( t, s )) l (cid:89) r =1 δ r ! (cid:18) β r ! D β r φ ( t, s ) (cid:19) δ r D γ − ρ ∂φ k ∂s j ( t, s ) . Notice that the first derivative formula only involves derivatives of φ ( t, s ) and u up to order n . Regardingthe second formula, the only term involving a derivative of φ ( t, s ) of order n + 1 is the associated withthe multi-index ρ = 0. Hence, the next estimates hold true by virtue of (65) (cid:12)(cid:12)(cid:12)(cid:12) D γ (cid:18) ∂φ i ∂t (cid:19) ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) , (cid:12)(cid:12)(cid:12)(cid:12) D γ (cid:18) ∂φ i ∂s j (cid:19) ( t, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) (cid:88) q =1 (cid:18) (cid:90) t (cid:12)(cid:12)(cid:12)(cid:12) D γ (cid:18) ∂φ q ∂s j (cid:19) ( τ, s ) (cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:19) , for every ( t, s ) ∈ D (Σ , u, T ). Again, Gronwall’s lemma shows that (65) holds true when m = n .Finally, let us obtain the aforementioned α -H¨older estimate of the higher order derivatives of Jac( φ ).To this end, take any column vector x j ( t, s ) of the Jacobian matrix Jac( φ )( t, s ) and note that when γ = ( γ , γ , γ ) is a multi-index of the highest order k , then all the preceding derivative formulas can beadded up to obtain the PDE ∂∂t D γ x ji ( t, s ) = (cid:88) q =1 A i,jq ( γ ) ∂u i ∂x q ( φ ( t, s )) D γ x jq ( t, s ) + F i ( t, s )+ (cid:88) β ∈ Γ γ (cid:88) ( j ,...,j k +1 ) ∈ J γ (cid:88) ( i ,...,i k +1 ) ∈ I γ B j ,...,j k +1 i ,...,i k +1 ( β )( D β u i )( φ ( t, s )) x j i ( φ ( t, s )) · · · x j k +1 i k +1 ( φ ( t, s )) . (66)Here A i,jq ( γ ) and B j ,...,j k +1 i ,...,i k +1 ( β ) denote nonnegative constant coefficients and F i ( t, s ) consists of finitelymany sums and products of both derivatives of u up to order k and derivatives of φ up to order k .Furthermore, Γ γ is a set of 3-multi-indices with order k + 1 depending on γ and I γ , J γ are sets of ( k + 1)-multi-indices also depending on γ .Let us first prove the α -H¨older continuity in the variable s using the integral version of the aboveequation. Specifically, take s , s ∈ D , t ∈ (0 , T ) and notice that D γ x ji ( t, s ) − D γ x ji ( t, s ) = I + II + III + IV, where I := D γ x ji (0 , s ) − D γ x ji (0 , s ) ,II := (cid:90) t ( F i ( τ, s ) − F i ( τ, s )) dτ,III := (cid:88) q =1 A i,jq ( γ ) (cid:90) t (cid:18) ∂u i ∂x q ( φ ( τ, s )) D γ x jq ( τ, s ) − ∂u i ∂x q ( φ ( τ, s )) D γ x jq ( τ, s ) (cid:19) dτ, ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 29 IV := (cid:88) β ∈ Γ γ ( i ,...,i k +1 ) ∈ I γ ( j ,...,j k +1 ) ∈ J γ B j ,...,j k +1 i ,...,i k +1 ( β ) (cid:90) t ( D β u i )( φ ( τ, s )) x j i ( φ ( τ, s )) · · · x j k +1 i k +1 ( φ ( τ, s )) (cid:12)(cid:12)(cid:12) s s dτ. Regarding the terms
I, II , one can easily see that I ≤ κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) | s − s | α , II ≤ T κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) | s − s | α . In the first case, the estimate obviously follows from the regularity of µ in the particular case when D γ x ji involves no derivative of φ ( t, s ) with respect to t . A straightforward recursive argument on the order ofthe derivatives with respect to t yields the general assertion. The second case is obvious by the definitionof F i ( t, s ) and the mean value theorem. Furthermore, adding and subtracting crossed terms in III , it isclear that it can be bounded by the mean value theorem as
III ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) | s − s | α + κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:88) q =1 (cid:90) t | D γ x jq ( τ, s ) − D γ x jq ( τ, s ) | dτ. So far, only low order derivatives of u have being involved, and therefore the mean value theorem hassufficed to obtain Lipschitz conditions of such derivatives (terms I, II and
III ). In contrast, IV containsthe derivatives of u of the highest order, k + 1. Since they cannot be handled again by the mean valuetheorem, then the α -H¨older continuity of D k +1 u must be used. By appropriately adding and subtractingcrossed terms, using the above-mentioned H¨older continuity of D β u i on the first factor and the meanvalue theorem on the second one, one easily obtains the upper bound IV ≤ (cid:88) β ∈ Γ γ (cid:88) ( j ,...,j k +1 ) ∈ J γ (cid:88) ( i ,...,i k +1 ) ∈ I γ B j ,...,j k +1 i ,...,i k +1 ( β ) × T (cid:0) κ ( (cid:107) u (cid:107) C k +1 ,α (Ω , T ) α + k +1 [ D β u i ] α, Ω | s − s | α + κ ( (cid:107) u (cid:107) C k +1 ,α (Ω , T ) k +2 (cid:107) D β u i (cid:107) C (Ω) | s − s | (cid:1) . To conclude, let us combine all the above estimates and use Gronwall’s lemma to arrive at | D γ x ji ( t, s ) − D γ x ji ( t, s ) | ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω , T ) | s − s | α , for an appropriately function κ . Regarding the α -H¨older condition in the variable t , one only needs tonote that ∂ t D γ x ji ( t, s ) is uniformly bounded by virtue of (66).Finally, note thatJac( φ ) − = 1det(Jac( φ )) (cid:18) ∂φ∂s × ∂φ∂s (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂s × ∂φ∂t (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂t × ∂φ∂s (cid:19) (cid:124) , and that the Jacobi–Liouville formula along with the lower bound in (64) yield a uniform lower boundfor the Jacobian determinant:det(Jac( φ )( t, s )) = det(Jac( φ )(0 , s )) exp (cid:18)(cid:90) t Tr(Jac( u )( φ ( τ, s ))) dτ (cid:19) ≥ ρ ρ exp (cid:0) − T (cid:107) Jac( u ) (cid:107) C (Ω) (cid:1) . Hence, the C k +1 ,α ( D (Σ , u, T )) estimate for Jac( φ ) − easily follows from that of Jac( φ ). (cid:3) The analysis in the next sections requires stream tubes of u that are bounded and have both endson S . These structures were considered (although its existence was not proved) in [27]. In our setting,we will say that the stream tube of u arising from Σ is a ( ρ , T, δ )-stream tube of u when the previoustwo conditions hold, i.e., • u · η ≥ ρ on Σ. • For every s ∈ D there exists two associated positive numbers 0 < T ( s ) , T δ ( s ) < T such that X ( T ( s ); 0 , µ ( s )) ∈ S and X ( T δ ( s ); 0 , µ ( s )) ∈ S δ .Here ρ , T, δ are positive constants which measure the initial angle of the streams lines over Σ, the timeat which the whole tube has returned to the surface and the depth that the stream lines achieve into theinterior domain G , while S δ stands for the boundary of the subdomain of G made of the points in G atdistance at least δ from S , i.e., G δ := { x ∈ G : dist( x, S ) > δ } (see Figure 4.)Since a stream tube consists of integral curves, it is elementary that the diameter of a ( ρ , T, δ )-streamtube is bounded in terms of the sup norm of the vector field, the flow time T and the diameter at time 0as diam( T (Σ , u )) ≤ T (cid:107) u (cid:107) C (Ω) + diam(Σ) . (67)(A detailed proof of this can be found in [27, Lemma 4.6]). In a similar way, [27, Lemma 4.7] providesa criterion to obtain “almost” ( ρ , T, δ )-stream tubes for velocity fields which are “close enough” to any T (Σ , u ) µ ( s ) X ( T δ ( s ); 0 , µ ( s )) X ( T ( s ); 0 , µ ( s )) GS δ S Figure 4. ( ρ , T, δ )-stream tube of u .other given velocity field enjoying this kind of stream tubes. This merely asserts that, as is well known, a C -small perturbation of the initial vector field will not prevent the integral curves of the perturbed fieldfrom intersecting a surface to which the initial flow was transverse. This can be quantitatively writtenas follows: Lemma 3.4.
Let G, Σ , µ verify (7) and consider u , u ∈ C k +1 ,α (Ω , R ) . Define T i := T (Σ , u i ) its streamtubes emanating from Σ and assume that T is a ( ρ , T, δ ) -stream tube of u and (1) u · η = u · η on Σ . (2) u and u are “close enough” in C (Ω) norm. Specifically, assume (cid:107) u − u (cid:107) C (Ω) < − θ ) δC P T e − C P T (cid:107) u (cid:107) C , for some < θ < .Then, T is also a ( ρ , T, θδ ) -stream tube of u . Iterative scheme.
In this section we discuss the iterative scheme that is typically used in theliterature to obtain nonlinear force-free fields in the magnetohydrodynamical setting as small perturba-tions of harmonic fields: the
Grad–Rubin iterative method (see, the review [44]). It goes back to 1958,when it was originally proposed by H. Grad and H. Rubin in connection with applications to plasmaphysics. A numerical implementation in the context of coronal magnetic fields is introduced in [1], wherethe Grad–Rubin method was obtained through the decomposition of the Beltrami equation with smallproportionality factor f into a hyperbolic part, which transports the proportionality factor f along themagnetic field lines, and an elliptic one, to correct the magnetic field step by step using Ampere’s law.From a mathematical point of view, this method was used in [5] to obtain small perturbations ofharmonic fields in bounded domains, leading to generalized Beltrami fields with small non-constantproportionality factors.Let us assume that G, Σ , µ satisfies the hypotheses (7) and consider any initial harmonic vector field u in the exterior domain Ω, i.e., a solution to the equation curl u = 0 , x ∈ Ω , div u = 0 , x ∈ Ω , | u ( x ) | ≤ C | x | ρ − , x ∈ Ω , (68)where ρ ∈ (1 ,
3) is a parameter which controls the decay at infinity of the vector field u . Dynamically,we will assume that u has a tube that starts and ends on the boundary, i.e., that the field u has a( ρ , T, δ )-stream tube. ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 31
We want to known whether there exists a Beltrami field u with “small” proportionality factor f suchthat u (resp. f ) is close to u (resp. 0) and satisfies the exterior Neumann problem curl u = f u, x ∈ Ω , div u = 0 , x ∈ Ω ,u · η = u · η, x ∈ S, | u ( x ) | ≤ C | x | ρ − , x ∈ Ω . (69)To solve this problem, the Grad–Rubin method analyzes the following iterative scheme: curl u n +1 = f n u n , x ∈ Ω , div u n +1 = 0 , x ∈ Ω ,u n +1 · η = u · η, x ∈ S, | u n +1 ( x ) | ≤ C | x | ρ − , x ∈ Ω , (cid:26) ∇ f n · u n = 0 , x ∈ Ω ,f n = f , x ∈ Σ . (70)The strong convergence to a solution of (69) was rigorously proved in [27] for small enough prescriptions f of the proportionality factor on Σ and under the assumption that the field u has a ( ρ , T, δ )-streamtube by providing the a priori bounds that were missing in [5]. The way to go is the following. First, it iseasy to solve the steady transport equation in the right hand side when the stream tube of u n emanatingfrom the open subset Σ is a ( ρ , T, δ )-stream tube and the prescription f of the proportionality factorhas compact support inside of Σ. The transportation of f along this stream tube leads to the existenceand uniqueness of f n in C ,α (Ω) whenever u n lies in C ,α (Ω , R ). Second, in order to solve the exteriorinhomogeneous Neumann problem for the div-curl equation, a boundary integral equation method forharmonic fields was studied in [38]. Roughly speaking, the existence of solution to the Neumann problemwas solved by means of the Helmholtz–Hodge’s decomposition theorem for vector fields with suitable decayat infinity, splitting them into harmonic volume and single layer potentials depending on its divergence,its curl and its normal and tangential component. Although the tangential component η × u + is notprescribed in the boundary data, it can be obtained from a boundary integral equation which can bededuced from jump relations for the derivatives of harmonic single layer potentials (see [32, Theorem14.IV], [33, Teorema 2.II] or [9, Theorem 2.17] for a proof of jump relation and [38, 43] for a detailedstudy of the associated boundary integral equation). H¨older estimates for the solution follow from acareful study of the harmonic volume and single layer potentials in the decomposition [9, 25, 32, 33].Although only C ,α proportionality factors and C ,α magnetic fields were considered [27], a C k,α theorycan be obtained after studying higher order derivates of potentials. The only hypotheses that the datamust satisfy to ensure the well-posedness of the inhomogeneous div-curl problems are: • The inhomogeneous term f n u n in the equation for curl u n +1 must have “zero flux” in Ω. Specif-ically, the flux of f n u n across any closed surface in Ω must vanish or equivalently f n u n must bedivergence-free in Ω and its flux across the surface S has to be zero. This hypothesis can bededuced in each step from the corresponding hypothesis in the preceding step. As it is verified inthe step n = 0 because u is divergence-free and f verifies the steady transport equation, thesehypothesis are satisfied for every step n ∈ N in the iteration. • The right hand sides of the equations for div u n +1 and curl u n +1 must decay at infinity as | x | − ρ .This is easily checked since f n has compact support.A natural question is to ascertain whether these results can be adapted to get perturbations of strongBeltrami fields with any constant proportionality factor λ (cid:54) = 0. We devote now some lines to explain whya direct application of the same Grad–Rubin method cannot work. Assume that u is a strong Beltramifield with constant proportionality factor in the exterior domain Ω. We will restrict ourselves to strongBeltrami fields u with optimal decay at infinity, say | x | − (see [18, 36] and compare with the | x | − decayin the harmonic case). curl u = λu , x ∈ Ω , div u = 0 , x ∈ Ω , | u ( x ) | ≤ C | x | , x ∈ Ω . (71)Now, we would like to solve the next problem curl u = ( λ + ϕ ) u, x ∈ Ω , div u = 0 , x ∈ Ω ,u · η = u · η, x ∈ S, | u ( x ) | ≤ C | x | , x ∈ Ω , (72) where ϕ is a “small” perturbation of the constant proportionality factor λ . The same ideas as above leadto the next modification of the classical Grad–Rubin iterative method curl u n +1 = ( λ + ϕ n ) u n , x ∈ Ω , div u n +1 = 0 , x ∈ Ω ,u n +1 · η = u · η, x ∈ S, | u n +1 ( x ) | ≤ C | x | , x ∈ Ω , (cid:26) ∇ ϕ n · u n = 0 , x ∈ Ω ,ϕ n = ϕ , x ∈ Σ . Although the steady transport system in the right hand side can be solved in exactly the same way, wecannot ensure now the previous hypothesis leading to the well-posedness of the exterior inhomogeneousdiv-curl problems. On the one hand, the vanishing flux hypothesis follows from the same argument asabove. On the other hand, ( λ + ϕ ) u should decay as | x | − in order for the solution u to decay as | x | − .Unfortunately, it is not possible when λ (cid:54) = 0 because u has optimal decay | x | − .To solve this problem, we move the term λu in the equation for curl u from the inhomogeneous side,to the homogeneous one arriving at curl u n +1 − λu n +1 = ϕ n u n , x ∈ Ω , div u n +1 = 0 , x ∈ Ω ,u n +1 · η = u · η, x ∈ S, | u n +1 ( x ) | ≤ C | x | , x ∈ Ω , (cid:26) ∇ ϕ n · u n = 0 , x ∈ Ω ,ϕ n = ϕ , x ∈ Σ . However, as it will be shown in the next section, the exterior inhomogeneous Beltrami equation fordivergence-free vector fields is an overdetermined system in general. Notice that when one computes thedivergence in the first equation and assumes λ (cid:54) = 0, one recovers div u n +1 from the first equationdiv u n +1 = − λ div( ϕ n u n ) . Therefore, it is an easy task to check that as soon as u is divergence-free and ϕ n is a fist integral of u n ,then u n +1 is also divergence-free in each step of the iteration. Consequently, we can simplify the previousoverdetermined systems by removing the divergence-free conditions. curl u n +1 − λu n +1 = ϕ n u n , x ∈ Ω ,u n +1 · η = u · η, x ∈ S, | u n +1 ( x ) | ≤ C | x | , x ∈ Ω , (cid:26) ∇ ϕ n · u n = 0 , x ∈ Ω ,ϕ n = ϕ , x ∈ Σ . (73)The stationary problem along a ( ρ , T, δ )-stream tube of u n in the right hand side of (73) will be studiedin the C k +1 ,α setting in the next section. The inhomogeneous Beltrami equations in the left hand sidewas studied in the preceding Section 2 through the analysis of complex-valued solutions satisfying boththe L decay condition (3) and the L SMB radiation condition (4). Consequently, we arrive at the modified Grad–Rubin iterative method discussed in the Introduction (Equation (6)).3.3.
Linear transport problem.
We begin with the steady transport equations along ( ρ , T, δ )-streamtubes in the right hand side of (6). The main idea to find a solution is to transport ϕ along the foliatedstream tube and to check that this definition leads to regular enough factors f n of u n due to the regularityof the tube. Theorem 3.5.
Let G, Σ , µ satisfy the hypotheses (7), consider any u ∈ C k +1 ,α (Ω , R ) such that T (Σ , u ) is a ( ρ , T, δ ) -stream tube of such a velocity field and assume that ϕ ∈ C k +1 ,αc (Σ) . Consider the firstintegral equation associated with u (cid:26) u · ∇ ϕ = 0 in Ω ϕ = ϕ on Σ . (74) Then, there exists an unique solution ϕ along T (Σ , u ) , its support lies in the closure of T (Σ , u ) and it canbe extended to a global solution in Ω with zero value outside T (Σ , u ) . Moreover, it belongs to C k +1 ,α (Ω) and the estimate (cid:107) ϕ (cid:107) C k +1 ,α (Ω) ≤ (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) holds, for some continuous function κ : R +0 × R +0 −→ R +0 .Proof. Let us now sketch the proof of this result, which can be found in [27, Lemmas 4.8, 4.9 and 5.2] for k = 0. Define the Calder´on extension of u , u := P ( u ), according to Proposition 3.1 and denote its fluxmapping by X ( t ; t , x ). First, let us prove the uniqueness part of our assertion. Notice that as long as ϕ is a smooth first integral of u , then ddt ϕ ( X ( t ; 0 , µ ( s ))) = ( u · ∇ ϕ )( X ( t ; 0 , µ ( s ))) = ( u · ∇ ϕ )( X ( t ; 0 , µ ( s ))) = 0 , ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 33 for every ( t, s ) ∈ D (Σ , u ). Therefore, ϕ ( x ) = ϕ ( µ ( s ( x ))) for every x ∈ T (Σ , u ), where ( t ( x ) , s ( x )) = φ − ( x ). Second, regarding the existence assertion, the previous formula for ϕ defines a smooth functionin T (Σ , u ) (by virtue of the bijectivity and regularity of the parametrization φ in Proposition 3.2) whichobviously solves (74) along the stream tube. Furthermore, with the exception of the endpoints, it iscompactly supported in the interior of the tube. The extension of ϕ by zero outside the tube yields aglobal smooth solution of (74) in Ω.To show the bound for (cid:107) ϕ (cid:107) C k +1 ,α (Ω) (equivalently for (cid:107) ϕ (cid:107) C k +1 ,α ( T (Σ ,u )) ), let us fix any multi-index γ = ( γ , γ , γ ) such that | γ | ≤ k + 1 and note that D γ ϕ ( x ) = γ ! (cid:88) ( l,β,δ ) ∈D ( γ ) ( D δ ( ϕ ◦ µ ))( s ( x )) l (cid:89) r =1 δ r ! (cid:18) β r ! D β r s ( x ) (cid:19) δ r . for every x ∈ T (Σ , u ). First of all, it is necessary to know how to handle D β r s ( x ). To this end, note thatJac( φ − )( x ) = Jac( φ ) − ( φ − ( x )), so D ρ (Jac( φ − ) i,j )( x ) = n ρ (cid:88) n =1 (cid:89) β ∈ Γ n ≤ p,q ≤ A i,jn,p,q ( ρ, β )( D β (Jac( φ ) − p,q ))( φ − ( x )) , for every multi-index ρ such that | ρ | ≤ k . Here, A i,jn,p,q ( ρ, β ) stand for constant coefficients and Γ n is aset of 3-multi-indices of order at most | ρ | ≤ k . Expanding the products of sums by distributivity, eachterm in D γ ϕ takes the form( D δ ( ϕ ◦ µ ))( s ( x )) (cid:89) β ∈ Γ1 ≤ p,q ≤ B i,jp,q ( γ, β )( D β (Jac( φ ) − p,q ))( φ − ( x )) , where Γ is a set of multi-indices with degree at most k . The first factor can be bounded by (cid:107) ϕ (cid:107) C k +1 ,α (Σ) whilst the terms in the second factor are bounded by κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) as stated in Proposition 3.2.Hence, it is clear that (cid:107) ϕ (cid:107) C k +1 (Ω) ≤ (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) . Finally, for any multi-index with maximum order k +1, the α -H¨older seminorm of D γ ϕ can be estimatedas follows. Take x , x ∈ T (Σ , u ) and appropriately add and subtract the crossed terms. Since D δ ( ϕ ◦ µ )is bounded by (cid:107) ϕ (cid:107) C k +1 ,α (Σ) and D β (Jac( φ ) − pq ) is bounded by κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ), then it only remainsto obtain estimates for I : = ( D δ ( ϕ ◦ µ ))( s ( x )) (cid:12)(cid:12) x x ,II : = ( D β (Jac( φ ) − p,q ))( φ − ( x )) (cid:12)(cid:12) x x . First, we distinguish the cases | δ | < k + 1 and | δ | = k + 1. In the former case, the mean value theorem,the estimates in Proposition 3.2 for Jac( φ ) − and the estimate (67) of the diameter of the stream tube T (Σ , u ) yield the upper bound I ≤ (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) | x − x |≤ (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ ( (cid:107) u (cid:107) C k +1 ,α )( T (cid:107) u (cid:107) C (Ω) + diam(Σ)) − α | x − x | α . In the later case, the α -H¨older continuity of D δ ( ϕ ◦ µ ) gives rise to an analogous estimate I ≤ (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) α | x − x | α . Second, note that D β (Jac( φ ) − p,q ) is α -H¨older continuous with H¨older’s constant that can be boundedabove by κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) by virtue of Proposition 3.2. Thus, II ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) | φ − ( x ) − φ − ( x ) | α . The mean value theorem then leads to the desired upper estimate | D γ ϕ ( x ) − D γ ϕ ( x ) | ≤ κ ( (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) | x − x | α , appropriately modifying the separately increasing function κ . (cid:3) In addition to the existence and uniqueness results of (74), in order to take limits in (6) we will need acompactness result for { ϕ n } n ∈ N . Once we know that the sequence { u n } n ∈ N converges in C k +1 ,α (Ω , R ),a result of stability for the problem (74) leads to the convergence of the sequence { ϕ n } n ∈ N in C k,α (Ω).This stability result was proved in [27, Lemma 5.3] in the C ,α framework and can be easily extended to C k +1 ,α using the same lines as in Theorem 3.5 (the details, which are straightforward, can be found in[40, Corollary 2.4.4]): Corollary 3.6.
Let G , Σ , µ satisfy the properties (7). Consider any couple of vector fields u , u ∈ C k +1 ,α (Ω , R ) , and denote as T := T (Σ , u ) and T := T (Σ , u ) the associated stream tubes whichemanate from Σ . Assume that T i is a ( ρ , T, δ i ) -stream tube of u i . Consider any boundary data ϕ ∈ C k +1 ,αc (Σ) and the solutions ϕ and ϕ (according to Theorem 3.5) to each transport problem associatedwith u and u respectively: (cid:26) ∇ ϕ · u = 0 , x ∈ Ω ,ϕ = ϕ , x ∈ Σ , (cid:26) ∇ ϕ · u = 0 , x ∈ Ω ,ϕ = ϕ , x ∈ Σ . Then, (cid:107) ϕ − ϕ (cid:107) C k,α (Ω) ≤ (cid:107) ϕ (cid:107) C k +1 ,α (Σ) · κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) · κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:107) u − u (cid:107) C k +1 ,α (Ω) , where κ : R +0 × R +0 −→ R +0 is a continuous and separately increasing function which does not depend on u i , ϕ or T . Limit of the approximate solutions.
The existence and uniqueness results in Theorems 3.5 and2.16 together with the stability result for the transport problem in Corollary 3.6 now allow us to take thelimit as n → + ∞ in the modified Grad–Rubin iterative scheme (6). Therefore, we obtain a generalizedBeltrami field which is close to the initial strong Beltrami field and whose proportionality factor is anon-constant small enough perturbation of the initial constant proportionality factor λ : Theorem 3.7.
Let G, Σ , µ satisfy the hypotheses (7) and assume that (cid:54) = λ ∈ R is not a Dirichleteigenvalue of Laplace operator in the interior domain G . Consider any complex-valued strong Beltramifield v ∈ C k +1 ,α (Ω , C ) which satisfy the L SMB radiation condition (29) and the L decay property(32) in the exterior domain. Consider its real part u := (cid:60) v , and assume that T (Σ , u ) is a ( ρ , T, δ ) -stream tube of the velocity field u . Let ε be a positive number. Then, there exists a nonnegativeconstant δ for which the real parts u n +1 of the solutions v n +1 ∈ C k +1 ,α (Ω , C ) together with the solutions ϕ n ∈ C k +1 ,α (Ω) of the coupled problems in the modified Grad–Rubin iterative scheme (6) (Theorems 3.5and 2.16) have a limit vector field u ∈ C k +1 ,α (Ω , R ) and a limit perturbation of the proportionality factor ϕ ∈ C k,α (Ω) such that u n → u in C k +1 ,α (Ω , R ) , ϕ n → ϕ in C k,α (Ω) , as n → + ∞ , for any ϕ ∈ C k +1 ,αc (Σ) with (cid:107) ϕ (cid:107) C k +1 ,α (Σ) < δ . Also, ( u, λ + ϕ ) solves the followingboundary value problem curl u = ( λ + ϕ ) u, x ∈ Ω , div u = 0 , x ∈ Ω ,u · η = u · η, x ∈ S,ϕ = ϕ , x ∈ Σ . Furthermore, u = O ( | x | − ) as | x | → + ∞ , T (Σ , u ) is a ( ρ , T, δ/ -stream tube of u , ϕ has compactsupport inside the closure of such stream tube and u is close enough to u , specifically (cid:107) u − u (cid:107) C k +1 ,α (Ω) ≤ ε (cid:107) u (cid:107) C k +1 ,α (Ω) . Proof.
For simplicity of notation, we will denote the stream tubes associated with each vector field u n which emanates from Σ by T n := T (Σ , u n ). First of all, it is necessary to check whether the hypothesisof Theorems 3.5 and 2.16 hold and they can be deduced in each step from the corresponding hypothesesin the previous step in the iteration. Let us begin with the step n = 0: (cid:26) ∇ ϕ · u = 0 , x ∈ Ω ,ϕ = ϕ , x ∈ Σ , curl v − λv = ϕ u , x ∈ Ω ,v · η = u · η, x ∈ S, + L Decay property (3) , + L SBM radiation condition (4) . The hypotheses imply that T is a ( ρ , T, δ )-stream tube of u and ϕ ∈ C k +1 ,αc (Σ). Hence, thereexists a global solution ϕ to the transport equation (Theorem 3.5). Moreover, ϕ u ∈ C k +1 ,αc (Ω , R ) ⊆ C k,αc (Ω , R ) and its compact support is contained in the stream tube T . In particular, the estimate (67)ensures that supp( ϕ u ) ⊆ T ⊆ Ω R , where Ω R := B R (0) \ G and R := 2 T (cid:107) u (cid:107) C k +1 ,α (Ω) + diam(Σ). Onthe other hand, as S is regular enough, so η is and, consequently, u · η ∈ C k +1 ,α ( S ). An integration byparts leads to the following expression (cid:90) S ( λu · η + ϕ u · η ) dS = λ (cid:90) S u · η dS + (cid:90) S ϕ u · η dS ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 35 = λ (cid:90) S u · η dS + (cid:90) ∂B R (cid:48) (0) ϕ u · η dS − (cid:90) Ω R (cid:48) div( ϕ u ) dx For R (cid:48) > R , the second term vanishes as a consequence of the previous estimate for the diameter of theinitial stream tube. Regarding the third term, notice that the same argument as above leads todiv( ϕ u ) = ∇ ϕ · u + ϕ div u = 0 . Finally, in Appendix A we show that the Beltrami equation for u allows us to write u · η = − λ div S ( η × u ) . The divergence theorem then concludes that the first term vanishes too. Therefore, the hypothesesof Theorem 2.16 are satisfied, so there is a unique solution v to the corresponding complex-valuedinhomogeneous Beltrami equation in the right hand side of the step n = 0.Let us prove an estimate for u − u that will be useful to prove the Cauchy condition in C k +1 ,α (Ω , R )for the sequence { u n } n ∈ N . This vector field is the real part of v − v , which satisfies the complex-valuedexterior Neumann problem (curl − λ )( v − v ) = ϕ u , x ∈ Ω , ( v − v ) · η = 0 , x ∈ S, + L decay condition (3) , + L SMB radiation conditon (4) . Therefore, the uniqueness of the solution to this problem (Proposition 2.18), the C k +1 ,α estimates ofsuch solutions (Corollary 2.20), and the C k,α estimates for the solution of the steady transport equation(Theorem 3.5) allow us to obtain the following estimate for v − v and, consequently, for u − u : (cid:107) u − u (cid:107) C k +1 ,α (Ω) = (cid:107)(cid:60) ( v − v ) (cid:107) C k +1 ,α (Ω) ≤ (cid:107) v − v (cid:107) C k +1 ,α (Ω) ≤ C (cid:107) ϕ u (cid:107) C k,α (Ω) . Here C > k, α, λ, G and R . The Leibniz rule for the derivative ofa product reads D γ ( ϕ u ) = (cid:88) β ≤ γ (cid:18) γβ (cid:19) D β ϕ D γ − β u , for any multi-index γ .Therefore, the estimates in Theorem 3.5 for the derivatives up to order k of ϕ and the combinationof the mean value theorem and the Calder´on’s extension theorem (Proposition 3.1) to estimate the C ,α -norm of the derivatives of u up to order k allow us to arrive at the inequality (cid:107) D γ ( ϕ u ) (cid:107) C (Ω) ≤ C k (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:107) u (cid:107) C k +1 ,α (Ω) , for every multi-index γ with | γ | ≤ k , and (cid:107) D γ ( ϕ u ) (cid:107) C ,α (Ω) = (cid:107) D γ ( ϕ u ) (cid:107) C ,α ( T ) ≤ C k C P (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ ( (cid:107) u (cid:107) C k +1 ,α , T ) (cid:107) u (cid:107) C k +1 ,α (Ω) ( T (cid:107) u (cid:107) C k,α (Ω) + diamΣ) − α , for every multi-index γ so that | γ | = k and a nonnegative constant C k depending on k . To derive thelast estimate, we have used that | D γ − β u ( x ) − D γ − β u ( y ) | ≤ (cid:107) D γ − β u (cid:107) C ( R ) | x − y | ≤ C P (cid:107) u (cid:107) C k +1 ,α (Ω) | x − y | α (diam T ) − α , for every x, y ∈ T and the estimate (67) for the diameter of the ( ρ , T, δ )-stream tube of T . Hence thefollowing inequality (cid:107) u − u (cid:107) C k +1 ,α (Ω) ≤ K (cid:8) T (cid:107) u (cid:107) C k +1 ,α (Ω) + diamΣ) − α (cid:9) (cid:107) ϕ (cid:107) C k +1 ,α (Σ) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:107) u (cid:107) C k +1 ,α (Ω) holds, with a constant K = K ( k, α, λ, G, R ). Now, we can fix the small parameter δ such that it satisfies K (cid:8) T (cid:107) u (cid:107) C k +1 ,α (Ω) + diamΣ) − α (cid:9) ×× (cid:110) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) + (cid:107) u (cid:107) C k +1 ,α (Ω) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:111) δ <
12 min { ε , } .K (cid:8) T (cid:107) u (cid:107) C k +1 ,α (Ω) + diamΣ) − α (cid:9) ×× (cid:110) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) + (cid:107) u (cid:107) C k +1 ,α (Ω) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:111) (cid:107) u (cid:107) C k +1 ,α (Ω) δ <
14 2 δC P T e − C P (cid:107) u (cid:107) Ck +1 ,α (Ω) T . (75)Then we infer (cid:107) u − u (cid:107) C k +1 ,α (Ω) < min { ε , } (cid:107) u (cid:107) C k +1 ,α (Ω) , (cid:107) u − u (cid:107) C k +1 ,α (Ω) <
14 2 δC P T e − C P T (cid:107) u (cid:107) Ck +1 ,α (Ω) , (cid:107) u (cid:107) C k +1 ,α (Ω) ≤ (cid:107) u (cid:107) C k +1 ,α (Ω) . (76)To obtain similar estimates for the remaining terms of the iterative scheme we will use induction toshow that (cid:107) u n +1 − u n (cid:107) C k +1 ,α (Ω) ≤ n (cid:107) u − u (cid:107) C k +1 ,α (Ω) < min { ε , } n +1 (cid:107) u (cid:107) C k +1 ,α (Ω) , (cid:107) u n +1 − u n (cid:107) C k +1 ,α (Ω) <
12 12 n +1 δC P T e − C P T (cid:107) u (cid:107) Ck +1 ,α (Ω) , (cid:107) u n +1 − u (cid:107) C k +1 ,α (Ω) < min { ε , } n +1 (cid:88) i =1 i (cid:107) u (cid:107) C k +1 ,α (Ω) , (cid:107) u n +1 − u (cid:107) C k +1 ,α (Ω) < n +1 (cid:88) i =1 i δC P T e − C P (cid:107) u (cid:107) Ck +1 ,α (Ω) , (cid:107) u n +1 (cid:107) C k +1 ,α (Ω) < n +1 (cid:88) i =0 i (cid:107) u (cid:107) C k +1 ,α (Ω) . (77)This is true for n = 0 due to (76), so we can assume that the inductive hypotheses holds for all indicesless than n . Specifically, we assume that ϕ m , v m +1 are well defined, i.e., the corresponding problemshave a unique solution, that u m +1 are divergence-free and (77) hold for indices m < n .Let us now prove that the result is verified for the index m = n . The inductive hypotheses imply the ex-istence of a vector field v n ∈ C k +1 ,α (Ω , C ) and ϕ n − ∈ C k,α (Ω). Moreover, T n is a (cid:0) ρ , T, (cid:0) − (cid:80) ni =1 12 i (cid:1) δ (cid:1) )-stream tube of the real part u n = (cid:60) v n because of the third inequality in (77). Consequently, there exists aunique solution ϕ n ∈ C k,α (Ω) to the transport problem in the left hand side of (6) according to Theorem3.5. The last estimate in (77) along with (67) lead to T n ⊆ Ω R . Therefore, ϕ n is compactly supported inΩ R ⊆ Ω and the same argument as in the step n = 0 ensures the existence and uniqueness of a solution v n +1 ∈ C k +1 ,α (Ω , C ) to the complex-valued exterior Neumann problem for the inhomogeneous Beltramiequation in the right hand side of (6).Notice that the vanishing flux hypothesis in Theorem 2.16 is satisfied. To check it we repeat theprevious argument to get (cid:90) S ( λu · η + ϕ n u n · η ) dS = λ (cid:90) S u · η dS + (cid:90) ∂B R (cid:48) (0) ϕ n u n · η dS − (cid:90) Ω R (cid:48) div( ϕ n u n ) dx. The first term is zero as before, the second one also vanishes for a choice R (cid:48) > R and the last one is zerotoo because ϕ n is a first integral of u n and u n is divergence-free according to the induction hypothesis.Consequently, it is easy that u n +1 is also divergence-free.To conclude, let us prove the inductive hypothesis (77) for u n +1 − u n . Taking the difference of thecorresponding complex-valued exterior boundary value problems we have that v n +1 − v n solves (curl − λ )( v n +1 − v n ) = ϕ n u n − ϕ n − u n − , x ∈ Ω , ( v n +1 − v n ) · η = 0 , x ∈ S, + L decay conditions (3) , + L SMB radiation condition (4) . Again, thanks to the uniqueness property (Proposition 2.18), the C k +1 ,α estimates for these solutions(Corollary 2.20) and the C k,α estimates for the solution of the steady transport equation (Theorem 3.5), ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 37 we obtain the following estimate for v n +1 − v n and, consequently, for u n +1 − u n (cid:107) u n +1 − u n (cid:107) C k +1 ,α (Ω) = (cid:107)(cid:60) ( v n +1 − v n ) (cid:107) C k +1 ,α (Ω) ≤ (cid:107) v n +1 − v n (cid:107) C k +1 ,α (Ω) ≤ C (cid:107) ϕ n u n − ϕ n − u n − (cid:107) C k,α (Ω) . Now, ϕ n u n − ϕ n − u n − has compact support inside T n ∪ T n − ⊆ Ω R (see estimate (67) and the lastinequalities for the C k +1 ,α norms of u n and u n − in the inductive hypothesis). Thus, Theorem 6.10asserts that the constant C = C ( k, α, λ, G, R ) is the same as in the basic step because all the supportsof the inhomogeneous terms in the complex-valued exterior Neumann problems are contained in the samebounded subset Ω R of the exterior domain. This is a crucial fact because it prevents those constantsfrom depending on the iteration number n and avoids the blowup when n → + ∞ . Notice that (cid:107) ϕ n u n − ϕ n − u n − (cid:107) C k,α (Ω) ≤ (cid:107) ( ϕ n − ϕ n − ) u n (cid:107) C k,α (Ω) + (cid:107) ϕ n − ( u n − u n − ) (cid:107) C k,α (Ω) . Since T n is a (cid:0) ρ , T, (cid:0) − (cid:80) ni =1 12 i (cid:1) δ (cid:1) -stream tube of u n , T n − is a (cid:16) ρ , T, (cid:16) − (cid:80) n − i =1 12 i (cid:17) δ (cid:17) -streamtube of u n − and u n − · η = u · η = u n · η on S , we can apply both estimates in Theorem 3.5 andCorollary 3.6 to obtain the inequality (cid:107) ϕ n u n − ϕ n − u n − (cid:107) C k +1 ,α (Ω) ≤ K (cid:107) ϕ (cid:107) C k +1 ,α (Σ) (cid:8) T (cid:107) u (cid:107) C k +1 ,α (Ω) + diamΣ) − α (cid:9) × (cid:110) κ (2 (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) + (cid:107) u (cid:107) C k +1 ,α (Ω) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:111) (cid:107) u n − u n − (cid:107) C k +1 ,α (Ω) . Consequently, the estimate (cid:107) u n +1 − u n (cid:107) C k +1 ,α (Ω) ≤ K (cid:107) ϕ (cid:107) C k +1 ,α (Σ) (cid:8) T (cid:107) u (cid:107) C k +1 ,α (Ω) + diamΣ) − α (cid:9) × (cid:110) κ (2 (cid:107) u (cid:107) C k +1 ,α (Ω) , T ) + (cid:107) u (cid:107) C k +1 ,α (Ω) κ (cid:0) (cid:107) u (cid:107) C k +1 ,α (Ω) , T (cid:1) (cid:111) (cid:107) u n − u n − (cid:107) C k +1 ,α (Ω) holds, with a constant K independent of n . Since (cid:107) ϕ (cid:107) C k +1 ,α (Σ) < δ and δ is small enough to ensure(75), one has (cid:107) u n +1 − u n (cid:107) C k +1 ,α (Ω) < (cid:107) u n − u n − (cid:107) C k +1 ,α (Ω) , and the inductive hypothesis for indices less than n leads to the first two inequalities in (77).The last three estimates can be obtained as follows. Firstly, the preceding two estimates together withthe induction hypotheses lead to (cid:107) u n +1 − u (cid:107) C k +1 ,α (Ω) ≤ n (cid:88) i =0 (cid:107) u i +1 − u i (cid:107) C k +1 ,α (Ω) ≤ min { ε , } n (cid:88) i =0 i +1 (cid:107) u (cid:107) C k +1 ,α (Ω) = min { ε , } n +1 (cid:88) i =1 i (cid:107) u (cid:107) C k +1 ,α (Ω) . Similarly, we have (cid:107) u n +1 − u (cid:107) C k +1 ,α (Ω) ≤ n (cid:88) i =0 (cid:107) u i +1 − u i (cid:107) C k +1 ,α (Ω) ≤ n +1 (cid:88) i =1 i δC P T e − C P T (cid:107) u (cid:107) Ck +1 ,α (Ω) . The last inequality in (77) is obvious by the triangle inequality: (cid:107) u n +1 (cid:107) C k +1 ,α (Ω) ≤ (cid:107) u (cid:107) C k +1 ,α (Ω) + (cid:107) u n +1 − u (cid:107) C k +1 ,α (Ω) ≤ (cid:32) n +1 (cid:88) i =1 i (cid:33) (cid:107) u (cid:107) C k +1 ,α (Ω) = n +1 (cid:88) i =0 i (cid:107) u (cid:107) C k +1 ,α (Ω) . Using the above inequalities in (77) one can show that { u n } n ∈ N and { ϕ n } n ∈ N are Cauchy sequencesin C k +1 ,α (Ω , R ) and C k,α (Ω), respectively. On the one hand, we find (cid:107) u n + m − u n (cid:107) C k +1 ,α (Ω) ≤ n + m − (cid:88) i = n (cid:107) u i +1 − u i (cid:107) C k +1 ,α (Ω) < n + m − (cid:88) i = n i +1 (cid:107) u (cid:107) C k +1 ,α (Ω) ≤ + ∞ (cid:88) i = n +1 i (cid:107) u (cid:107) C k +1 ,α (Ω) = 12 n (cid:107) u (cid:107) C k +1 ,α (Ω) . Likewise, the third inequality in (77) along with the property u n · η = u · η on S, shows that T n are (cid:0) ρ , T, (cid:0) − (cid:80) ni =0 12 i (cid:1) δ (cid:1) -stream tubes of u n . Therefore, { ϕ n } n ∈ N also satisfies the Cauchy condition in C k,α (Ω) due to Corollary 3.6. Thus, it converges in C k,α to some ϕ ∈ C k,α (Ω). Let us now take the limit as n → + ∞ in the iterative scheme to deducediv u n +1 = 0 curl u n +1 − λu n +1 = ϕ n u n u n +1 · η = u · η ↓ ↓ ↓ ↓ ↓ ↓ div u = 0 curl u − λu = ϕu u · η = u · η. Moreover, the L SMB radiation condition (4) and the decay property (3) lead to complex-valuedsolutions v n to the exterior Neumann problem for the inhomogeneous Beltrami equations in the iterativescheme with the asymptotic behavior | v n ( x ) | ≤ C | x | , x ∈ Ω , for every n and a constant C independent of n . To check it, notice that Theorem 2.16 provides adecomposition of v n +1 into generalized volume and single layer potentials whose densities are u · η , ϕ n u n and the sequence ξ n of solutions to the boundary integral equations (43). The single layer potentials andits first order partial derivarives are dominated by the corresponding integral kernels Γ λ and ∇ Γ λ for x far enough from the surface S . This leads to an upper bound C | x | − where C depends on the C norm ofthe densities u · η and ξ n . Both quantities can be bounded above by (cid:107) u · η (cid:107) C k,α ( S ) and (cid:107) ϕ n u n (cid:107) C k,α (Ω) ,which are uniformly bounded with respect to n . Furthermore, the volume layer potentials and its firstorder partial derivatives can be bounded by C | x | − for an n -independent constant thanks to Theorem2.5 and the above argument. Consequently, we get the same asymptotic behavior at infinity for the limitvector field u .Let us show now that T (Σ , u ) is a ( ρ , T, δ/ u and that the support of ϕ lies in it.Since, by taking limits in the fourth inequality in (77), (cid:107) u − u (cid:107) C k +1 ,α (Ω) ≤
12 2 δC P T e − C P T (cid:107) u (cid:107) Ck +1 ,α (Ω) , Corollary 3.6 yields the first assertion. The second one is clear by taking into account that supp ϕ n ⊆ T n ,for every n ∈ N . Finally, to check that the limit solution is close to the initial strong Beltrami field u ,it suffices to take limits in the third inequality in (77) to get (cid:107) u − u (cid:107) C k +1 ,α (Ω) ≤ min { ε , } + ∞ (cid:88) i =1 i (cid:107) u (cid:107) C k +1 ,α (Ω) ≤ ε (cid:107) u (cid:107) C k +1 ,α (Ω) . (cid:3) Remark 3.8.
Notice that the generalized Beltrami field u ∈ C k +1 ,α (Ω , R ) obtained by means of thepreceding theorem solves the equation curl u − f u = 0 , x ∈ Ω , with proportionality factor f = λ + ϕ and a compactly supported perturbation ϕ ∈ C k,α (Ω) . Moreover, byconstruction it decays as | x | − at infinity. Let us show now why this is indeed the optimal decay. First,recall that div( ϕu ) = ∇ ϕ · u + ϕ div u = 0 , since u is divergence-free and ϕ is a first integral of u . Second, ϕ is compactly supported in T (Σ , u ) ,which is a ( ρ , T, δ/ -stream tube of u . Indeed, consider any open subset Σ (cid:48) ⊆ S such that supp ϕ ⊆ Σ (cid:48) ⊆ Σ (cid:48) ⊆ Σ . Then, the preceding proof actually shows that ϕ is compactly supported in T (Σ (cid:48) , u ) . Take any x ∈ Σ \ Σ (cid:48) and note that u ( x ) · η ( x ) ≥ ρ > . Hence, the transversality condition and Corollary 2.21 entail theoptimal decay | x | − of the vector field u . A related remark in the harmonic case ( λ = 0) is in order now. Remark 3.9.
Recall that a similar result to that in Theorem 3.7 was previously proved in [27] to obtaingeneralized Beltrami fields u ∈ C ,α (Ω , R ) (nonlinear force-free fields), i.e., solutions to curl u = f u, x ∈ Ω , with compactly supported small proportionality factors f ∈ C ,α (Ω) .On the one hand, the low regularity C ,α and C ,α is not a weakness in such result since despite notbeing directly considered in [27] , our results in Section 6 provide the necessary background to promote theexistence theorem in [27] to a high regularity setting. On the other hand, such generalized Beltrami fieldsdecay as | x | − at infinity. There is no contradiction neither with Corollary 2.22 (since it holds under ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 39 the assumption λ (cid:54) = 0 ) nor with the Liouville theorem in [7, 36] (since it just holds for globally definedgeneralized Beltrami fields).On the contrary, the latter can be used to show an interesting property of such generalized Beltramifields obtained as perturbations of harmonic fields. Specifically: they cannot be globally extended to thewhole space by virtue of the fall-off obstructions in [7, 36] . Nevertheless, the same cannot be directly saidfor generalized Beltrami fields obtained as perturbations of strong Beltrami fields. Knotted and linked stream lines and tubes in generalized Beltrami fields
Our objective in this section is to apply the convergence result for the modified Grad–Rubin method(6) that we established in the previous section (Theorem 3.7) to show the existence of almost globalBeltrami fields of class C k +1 ,α with a nonconstant factor that realize any given configuration of vortextubes and vortex lines, modulo a small diffeomorphism. Here k is an arbitrary integer.4.1. Knots and links in strong Beltrami fields.
The main result in [18] ensures the existence of strongBeltrami field with the sharp decay at infinity and exhibiting any finite collection of (possibly knottedand liked) vortex lines and thin vortex tubes. The vortex tubes can be though as (small deformationsof) metric neighborhoods of a smooth knotted loop in R . Specifically, for a closed curve Γ ⊆ R of anyknot type the associated tube of thickness ε > T ε (Γ) := { x ∈ R : dist( x, Γ) < ε } . With this notation, the main result of [18] reads as follows. In the statement, let us agree to saythat a vortex tube T of a field u is structurally stable if any divergence-free field u (cid:48) that is close enoughto u in C ,α ( V ) has an “invariant tube” (i.e., invariant torus) of the form Ψ( T ), where V is any fixedneighborhood of T and Ψ is a diffeomorphism of R that is close to the identity in C α . Theorem 4.1.
Let Γ , . . . , Γ n be n pairwise disjoint (possibly knotted and linked) closed curves in R .For any small enough ε , we can transform the collection of pairwise disjoint thin tubes T ε (Γ ) , . . . , T ε (Γ n ) by a diffeomorphism Φ of R , arbitrarily close to the identity in any C m norm, so that Φ( T ε (Γ )) , . . . , Φ( T ε (Γ n )) are vortex tubes of a strong Beltrami field u , which satisfies the equation curl u = λu in R forsome non-zero constant λ and decays at infinity as | x | − . Furthermore, these vortex tubes are structurallystable. For the benefit of the reader, let us briefly discuss the main ideas of the proof. The gist is to constructa “local” Beltrami field realizing the desired configuration of vortex tubes so that they are structurallystable and they approximate the local Beltrami field by a Beltrami field with the same constant that isglobal, that is, defined everywhere in R .Hence, the first step is to show the existence of Beltrami field that possess the desired collection ofvortex tubes and which is defined in a neigborhood of the boundary tori. Since the tubes can be chosenanalytic without loss of generality (by slightly deformating the initial vortex tube configuration), onecould try to do that using a kind of Cauchy–Kowalewski theorem that was proved in [20]. However, thiswould lead to a local Beltrami field defined in a region whose complement is not connected, and thisshould prevent us from applying any kind of approximation theorem (this restriction is already presentin the classical theorem of Runge, and appears in all the approximations theorems known to date). Thisleads to move from the above-mentioned Cauchy problem to the following boundary value problem ofNeumann type in the interior of the above tori (cid:26) curl (cid:101) u = λ (cid:101) u , x ∈ ∪ ni =1 T ε (Γ i ) , (cid:101) u · ν = 0 , x ∈ ∪ ni =1 ∂ T ε (Γ i ) , where ν stands for the unit outward normal vector to the boundary of the tube. The tangency boundarycondition ensures that any solution (cid:101) u has the invariant tori ∂ T ε (Γ ) , . . . , ∂ T ε (Γ n ). To get nontrivialsolutions, one can prescribe the L projection of (cid:101) u into the space of tangential harmonic fields (calledthe harmonic part of (cid:101) u ). Therefore, the above Neumann boundary value problem for a vector field (cid:101) u with fixed harmonic part is uniquely solvable by means of a variational approach as long as λ does notbelong to the spectrum of certain operator with a compact inverse (in particular, it works if | λ | < C/ε ).The above argument ensures the existence of local Beltrami fields tangent to the tubes but it does notsay anything about the structural stability of the tubes. This is obtained by applying a KAM theorem.What makes the proof subtle is that the applicability of the KAM theorem involves a combination ofdelicate PDE and dynamical systems estimates. Indeed, the above existence theorem applies to mostdomains and λ ’s, while the structural stability hinges on the smallness of λ . The point is to derive fineasymptotics for the field (cid:101) u for small ε and use this information to show that one can effectively use KAM theory on an associated Poincar´e map. The reason for which the estimates are subtle is that, fromthe point of view of KAM theory, the situation is very degenerate because the twist condition is barelysatisfied, and in turn this has a bearing on the power of the PDE estimates that are needed to make theKAM argument go through.What is of a greater direct interest for our purpose here is the way that we pass from the local solution (cid:101) u to a global Beltrami field u solving the same Beltrami equation in the whole space and exhibitingthe desired decay properties at infinity. Especially, in the next section we will need to use that the globalBeltrami field u is of the form u = curl(curl + λ )2 λ (cid:101) u , (78)where (cid:101) u is a finite Fourier–Bessel series of the form (cid:101) u ( x ) = L (cid:88) l =0 l (cid:88) m = − l c ml j l ( λ | x | ) Y ml (cid:18) x | x | (cid:19) , (79)where c ml are constant vectors in C , j l stands for the spherical Bessel function of first kind and l -th orderand Y ml is the m -th spherical harmonic of l -th order. Obviously (cid:101) u is real-valued.4.2. Knots and links in almost global generalized Beltrami fields.
Our goal in this section is toshow that the partial stability result for almost global Beltrami fields allows us to conclude the existenceof Beltrami fields with a non-constant proportionality factor that are defined in all of R but, say, in thecomplement of an arbitrarily small ball, and which have a collection of vortex tubes and vortex lines ofarbitrary topology. More precisely, our objective is to prove the following result. Let us recall that in theIntroduction we defined that a vortex tube (invariant torus) of a divergence-free field u is structurallystable if any divergence-free field that is close enough to u in C ,α has an invariant torus given by a C ,α -small diffeomorphism of the initial tube. Although we shall not state these properties explicitly, just asin [18] the vortex tubes that we construct are accumulated on by a positive-measure set of invariant torion which the vortex lines are ergodic. Theorem 4.2.
Let G be an exterior domain satisfying the hypotheses (7) and consider any collectionof disjoint knotted and linked thin tubes T ε (Γ ) , . . . , T ε (Γ n ) whose closure is contained in the exteriordomain Ω . Then, for ε small enough and any k, α there exists a nonzero constant λ , an open subset Σ ⊆ S and some δ > with the following property: for any function ϕ ∈ C k +1 ,αc (Σ) with (cid:107) ϕ (cid:107) C k +1 ,α (Σ) < δ there is a Beltrami field u ∈ C k +1 ,α (Ω , R ) with factor λ + ϕ , where ϕ is a function in C k,α (Ω) satisfying ϕ | Σ = ϕ : (cid:26) curl u = ( λ + ϕ ) u, x ∈ Ω , div u = 0 , x ∈ Ω . Furthermore, u = O (cid:0) | x | − (cid:1) as | x | → + ∞ , the support of ϕ is compact and lies in the ( ρ , T, δ ) -stream tube T (Σ , u ) of u radiating from Σ (with the exception of the endpoints) and T ε (Γ ) , . . . , T ε (Γ n ) can be modifiedby a diffeomorphism Φ close enough to the identity in any C m norm into a collection of structurally stablevortex tubes of u , Φ( T ε (Γ )) , . . . , Φ( T ε (Γ n )) , (possibly) knotted and linked with T (Σ , u ) .Proof. Take a curve Γ intersecting S transversally and such that T ε (Γ ) ∩ Ω has only a connected com-ponent. We also assume that Γ does not intersect any of the other curves Γ j , so that the setup is then asdepicted in Figure 5). For ε > (cid:48) arbitrarily close to the identity map in any C m norm such that Φ (cid:48) ( T ε (Γ )) , . . . , Φ (cid:48) ( T ε (Γ n )) are vortextubes of a strong Beltrami field u which satisfies the equation curl u = λu in R for some non-zeroconstant λ (of order ε ). By construction, these tubes are structurally stable and Φ (cid:48) can be assumedto be arbitrarily close to the identity in any C m norm, so the new thin tubes enjoy the same geometricfeatures as we had assumed on the initial ones. Let us then take the point x ∈ S ∩ Φ (cid:48) (Γ ) where u pointsoutwards and consider any open and connected neighborhood Σ of x in S such that Σ ⊆ S ∩ Φ (cid:48) ( T ε (Γ )).We recorded in Equations (78)-(79) that u is of the form u = curl(curl + λ )2 λ L (cid:88) l =0 l (cid:88) m = − l c ml j l ( λ | x | ) Y ml (cid:18) x | x | (cid:19) . Since u is obviously real-valued, it is the real part of the vector field v = curl(curl + λ )2 λ L (cid:88) l =0 l (cid:88) m = − l c ml h (1) l ( λ | x | ) Y ml (cid:18) x | x | (cid:19) , ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 41
A)B) C)
Figure 5.
A) Collection of knotted and linked vortex tubes of the strong Beltramifield u , { Φ (cid:48) ( T ε (Γ )) , Φ (cid:48) ( T ε (Γ )) } , respectively homeomorphic to the unknot and to thetrefoil. B) Transverse intersection of the vortex tube Φ (cid:48) ( T ε (Γ )) and the interior domain G . Here we have zoomed in the squared region on the left side of the above figure,showing the smaller outward pointing ( ρ , T, δ )-stream tube of u that emerges from Σ.The perturbation ϕ of the constant proportionality factor λ will be supported there. C)Zoom of the vortex tube Φ (cid:48) ( T ε (Γ )) with trefoil knot. It shows the internal structure ofsuch vortex tube of u , which contains uncountably many nested tori and knotted vortexlines.where h (1) l := j l + iy l is the spherical Hankel function of l -th order and y l denotes the spherical Besselfunction of the second kind and l -th order. By construction, v satisfies the Beltrami equation (and inparticular is smooth) in R \{ } , while it diverges at the origin due to the presence of a Bessel functionof the second kind. In particular, it is a Beltrami field in Ω.The advantage of v is that, as the Hankel function h (1) l has been chosen to satisfy the scalar radiationcondition ( ∂ r − iλ ) h (1) l ( λr ) = o ( r − ) , it is straightforward to check that v ∈ C k +1 ,α (Ω , C ) is a complex-valued solution to the Beltramiequation in the exterior domain Ω which satisfies the L SMB radiation condition (29) and the weak L decay property (32) (see [9, Equation 2.41] along with Remark 2.10 and Figure 1). It is also apparentthat T (Σ , u ) ⊆ Φ (cid:48) ( T ε (Γ )) is a ( ρ , T, δ )-stream tube of u by construction (see Figure 5), and that λ ∼ ε can be prevented from being a Dirichlet eigenvalue of the Laplace operator in the interior domain G as long as ε is taken small enough. Then, we are ready to apply the convergence Theorem 3.7 forthe modified Grad–Rubin method starting up with the strong Beltrami field u . This result ensures theexistence of δ > (cid:107) ϕ (cid:107) C k +1 ,α (Σ) ≤ δ , then there exists a generalized Beltrami field u ∈ C k +1 ,α (Ω , R ) and a perturbation ϕ ∈ C k,α (Ω) solving the exterior boundary value problem curl u = ( λ + ϕ ) u, x ∈ Ω , div u = 0 , x ∈ Ω ,u · η = u · η, x ∈ S,ϕ = ϕ , x ∈ Σ . Furthermore, u = O (cid:0) | x | − (cid:1) as | x | → + ∞ , T (Σ , u ) is a ( ρ , T, δ/
2) stream tube of u , ϕ is compactlysupported in the closure of such stream tube and (cid:107) u − u (cid:107) C k +1 ,α (Ω) can be made arbitrarily small. Inview of the structural stability of the vortex tubes of u , the theorem follows. (cid:3) Local stability of generalized Beltrami fields
Our objective in this section is to show that, in fact, any generalized Beltrami field possesses a localpartial stability property which can be essentially regarded as a local version of Theorem 3.7. We recallthat, in view of the results in [20], one cannot prove a full stability result even in arbitrarily smallopen sets, so we regard this partial stability (where partial is understood in a very precise sense) as asatisfactory counterpart to the results in this paper.5.1.
A local stability theorem.
We shall next present the local stability result that constitutes thecore of this section. The philosophy of this result is that, as one is able to perturb strong Beltrami fields,one should also be able to perturb generalized Beltrami fields in small domains, since in a small regiona C k,α function behaves as a constant plus a small perturbation. Somehow, this reduces our effort toestimates similar to the ones that we have already obtained, so our presentation of the proof of this resultwill be a little sketchier than before. The gist will be to show that, although the strong convergence of themodified Grad–Rubin iterative scheme cannot be granted in C k +1 ,α for u n and C k,α for f n , we can passto the limit in C ,α and C ,α provided that both the domain and the perturbation of the proportionalityfactor are small enough. Elliptic regularity will then yield the desired high order regularity by a bootstrapargument.In order to support our argument, let us first sketch the effect of the size of the domain on the solutionsof the next Neumann boundary value problem associated with the inhomogeneous Beltrami equation insome open ball B R ( x ) (cid:26) curl u − λu = w, x ∈ B R ( x ) ,u · η = 0 , x ∈ ∂B R ( x ) , (80)where w ∈ C ,α ( B R ( x ) , R ) has zero flux. We will be interested in the case where R becomes very small.This problem has being carefully analyzed in [43] for bounded domains and in [27] for exterior un-bounded domains in the harmonic case ( λ = 0). The non-harmonic counterpart was studied in [29] andSection 2 for the inhomogeneous Beltrami equation in bounded and exterior domains respectively. In thebounded setting, λ has to be assumed “regular” (see [29]). To this end, notice that taking | λ | < c/R (for an appropriate universal constant c >
0) prevents λ from being an eigenvalue of the Laplacian in B R ( x ). Hence, | λ | < c/R is a sufficient condition ensuring the well-posedness of (80). All the aboveresults provide an estimate for the unique solution u to (80) in terms of w of the form (cid:107) u (cid:107) C ,α ( B R ( x )) ≤ C λ,R (cid:107) w (cid:107) C ,α ( B R ( x )) , where the dependence of the constant C λ.R on λ and R is not explicit. The next technical result aims toprovide some explicit R -dependent estimate for u in some space. Lemma 5.1.
Let u ∈ C ,α ( B R ( x ) , R ) be the unique solution to the Neumann boundary value problemassociated with the Beltrami equation (80) for | λ | < c/R and R ∈ (0 , . Then, (cid:107) u (cid:107) C ,α ( B R ( x )) ≤ CR − α (cid:107) w (cid:107) C ,α ( B R ( x )) , (81) for some positive constant C depending on α but not on u, w, x or R .Proof. To obtain an explicit R -dependent estimate of u in some space, let us perform the next change ofvariables y = x − x R . Then, one obtains the following vector fields in the unit ball centered at the origin: U ( y ) = u ( x ) , W ( y ) = w ( x ) , solving the next Neumann boundary value problem for the Beltrami equation in B (0): (cid:26) curl U − λR U = R W, y ∈ B (0) ,U · η = 0 , y ∈ ∂B (0) . Thus, the above-mentioned results yield the following estimate for some R -independent positive constant C (cid:107) U (cid:107) C ,α ( B (0)) ≤ CR (cid:107) W (cid:107) C ,α ( B (0)) , where the assumption | λ | < c/R has been used to avoid the λ -dependence of the constant C . Note thatby definition (cid:107) W (cid:107) C ,α ( B (0)) = (cid:107) w (cid:107) C ( B R ( x )) + R α [ w ] α,B R ( x ) , ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 43 b x Σ R T (Σ R , v, T R ) B R ( x ) B R ( x ) Ω Figure 6.
Flow box T (Σ R , v, T R ) covering the small ball B R ( x ). (cid:107) U (cid:107) C ,α ( B (0)) = (cid:107) u (cid:107) C ( B R ( x )) + R (cid:88) i =1 (cid:107) ∂ x i u (cid:107) C ( B R ( x )) + R α (cid:88) i =1 [ ∂ x i u ] α,B R ( x ) . Since R ∈ (0 , (cid:3) Another key ingredient is to show that C ,α vector fields near a non-equilibrium point verify a “struc-turally stable” flow box theorem, to be understood in the next precise sense. Lemma 5.2.
Let u ∈ C ,α (Ω , R ) be a (nontrivial) vector field and consider some x ∈ Ω such that u ( x ) (cid:54) = 0 . There exist R > and δ > such that B R ( x ) ⊆ Ω , u vanishes nowhere in the ball and forevery < R < R there exists some surface Σ R ⊆ ∂B R ( x ) and a positive function T R ∈ C (Σ R ) such thatfor every v ∈ C ,α ( B R ( x ) , R ) with (cid:107) u − v (cid:107) C ,α ( B R ( x )) < δ , then B R ( x ) ⊆ T (Σ R , v, T R ) ⊆ B R ( x ) . Here, the above stream tube reads T (Σ R , v, T R ) := { X v ( t ; 0 , x ) : x ∈ Σ R , t ∈ (0 , T R ( x )) } ,v is the Calder´on extension of v from B R ( x ) to B R ( x ) (Proposition 3.1) and the height T R of thestream tube is not constant but it continuously depends, stream line by stream line, on the base point x ∈ Σ R (see Figure 6). Furthermore, the parametrizations µ R of Σ R can be normalized by choosing µ R ( s ) = Rµ ( s ) , s ∈ D R , for some open subset D R ⊆ D (0) of the unit disc centered at 0, and some local parametrization of theunit sphere µ : D (0) −→ ∂B ( x ). Since the proof follows the same lines as Lemma 3.4 in Section 3, weskip it and pass to the central result of this section. Theorem 5.3.
Let u be a nontrivial generalized Beltrami field of class C k +1 ,α (Ω , R ) , where k ∈ N and α ∈ (0 , , and consider its (nonconstant) proportionality factor f ∈ C k,α (Ω) . Take some nonequilibriumpoint x ∈ Ω of u and fix some ε > . Then, for each small enough radius R > there is some surface Σ R ⊆ ∂B R ( x ) and some constant δ R > so that for every ϕ ∈ C k +1 ,α (Σ R ) with (cid:107) ϕ (cid:107) C k +1 ,α (Σ R ,µ R ) < δ R there exist ϕ ∈ C k,α ( B R ( x )) and u ∈ C k +1 ,α ( B R ( x ) , R ) such that ϕ = ϕ on Σ R and u is a strongBeltrami field with proportionality factor f + ϕ enjoying the same normal component as u in ∂B R ( x ) ,i.e., curl u = ( f + ϕ ) u, x ∈ B R ( x ) , div u = 0 , x ∈ B R ( x ) ,u · η = u · η, x ∈ ∂B R ( x ) . Furthermore, (cid:107) u − u (cid:107) C k +1 ,α ( B R ( x )) ≤ ε (cid:107) u (cid:107) C k +1 ,α ( B R ( x )) . Proof.
The proof has two steps. First, we will prove the theorem for low H¨older exponents and regularity(namely, α ∈ (0 , /
2) and k = 0). Second, we will show a bootstrap argument based on elliptic gain ofregularity that will raise the estimates in the first step to its full strength and will conclude the proof ofthe theorem for general regularity and H¨older exponents.Then, let us first assume that α ∈ (0 , / λ := f ( x ) and fix some radius R > B R ( x ) ⊆ Ω, u vanishes nowhere in B R ( x ) and the assertions in Lemma 5.2 fulfil. Without loss ofgenerality, we can assume that R < min { , c/ | λ |} . Moreover, note that the homogeneous generalizedBeltrami equation can be restated as an inhomogeneous Beltrami equation with constant proportionalityfactor and an inhomogeneous term taking the form of a small remainder, i.e.,curl u − λ u = R ( x − x ) u , x ∈ Ω , (82)where f ( x ) = λ + R ( x − x ) for every x ∈ B R ( x ), i.e., R ( z ) := (cid:18)(cid:90) ∇ f ( x + θz ) dθ (cid:19) · z, z ∈ B R (0) . Next, consider the following modified iterative scheme of Grad–Rubin type. It consists of a sequenceof transport equations (cid:26) ∇ ϕ n · u n = −∇ f · u n , x ∈ B R ( x ) ,ϕ n = ϕ , x ∈ Σ R ., (83)along with a sequence of boundary value problems associated with the inhomogeneous Beltrami equation (cid:26) curl u n +1 − λ u n +1 = R ( x − x ) u n + ϕ n u n , x ∈ B R ( x ) ,u n +1 · η = u · η, x ∈ ∂B R ( x ) . (84)Note that they have been chosen in a consistent way so that as long as { u n } n ∈ N and { ϕ n } n ∈ N have limits(in some sense), then the limits u and ϕ give rise to a generalized Beltrami field whose proportionalityfactor is a perturbation f + ϕ of the initial factor f . Without loss of generality, we can assume that λ (cid:54) = 0 (in the case λ = 0 would need the additional condition div u n +1 = 0).Let us show that for every n ∈ N both u n +1 ∈ C ,α ( B R ( x ) , R ) and f n ∈ C ,α ( B R ( x )) are welldefined and that (cid:107) u n +1 − u n (cid:107) C ,α ( B R ( x )) ≤ n (cid:107) u − u (cid:107) C ,α ( B R ( x )) < min { ε , } n +1 (cid:107) u (cid:107) C ,α ( B R ( x )) , (cid:107) u n +1 − u (cid:107) C ,α ( B R ( x )) ≤ min { ε , } n +1 (cid:88) i =1 i (cid:107) u (cid:107) C ,α ( B R ( x )) , (cid:107) u n +1 (cid:107) C ,α ( B R ( x )) ≤ min { ε , } n +1 (cid:88) i =0 (cid:107) u (cid:107) C ,α ( B R ( x )) . (85)Let us start with n = 0. On the one hand, the transport problem (83) with n = 0 can be solved in B R ( x ) as B R ( x ) ⊆ T (Σ R , u , T R ) ⊆ B R ( x ) by virtue of Lemma 5.2. Indeed, ϕ ( X u ( t ; 0 , x )) = ϕ ( x ) − (cid:90) t ( ∇ f · u )( X u ( τ, , x )) dτ, x ∈ Σ R , t ∈ (0 , T R ( x ))defines a solution in T (Σ R , u , T R ) and, in particular, in B R ( x ). Now, notice that (cid:90) ∂B R ( x ) ( R ( · − x ) u + ϕ u ) · η dS + λ (cid:90) ∂B R ( x ) u · η dS = (cid:90) B R ( x ) ( ∇ ( f + ϕ ) · u + ( f + ϕ ) div u ) dx = 0 , and λ is regular (see [29]) with respect to the inhomogeneous problem (84) with n = 0 because R < R
0. Then, the aboveestimate for u − u can be written as (cid:107) u − u (cid:107) C ,α ( B R ( x )) ≤ CR α (cid:0) (cid:107) ϕ (cid:107) C ,α (Σ R ,µ R ) + 2 R − α (cid:1) (cid:8) κ (cid:0) (cid:107) u (cid:107) C ,α ( B R ( x )) , C , (cid:107) µ (cid:107) C ,α ( D (0)) (cid:1)(cid:9) (cid:107) u (cid:107) C ,α ( B R ( x )) . Hereafter we will assume that C (cid:18) δ R R α + 2 R − α (cid:19) (cid:8) κ (cid:0) (cid:107) u (cid:107) C ,α ( B R ( x )) , C , (cid:107) µ (cid:107) C ,α ( D (0)) (cid:1) +2 κ (cid:0) (cid:107) u (cid:107) C ,α ( B R ( x )) , C , (cid:107) µ (cid:107) C ,α ( D (0)) (cid:1) (cid:107) u (cid:107) C ,α ( B R ( x )) (cid:111) < ε , (87)with ε ∈ (0 ,
1) small enough so that ε (cid:107) u (cid:107) C ,α ( B R ( x )) < δ . Since we are considering low H¨olderexponents α ∈ (0 , / R ∈ (0 , R ) and δ R > f m ∈ C ,α ( B R ( x )) and u m +1 ∈ C ,α ( B R ( x ) , R ) forevery m < n such that they verify (83)–(85) and u m = 0 is divergence-free for every index m < n . Toclose the inductive argument let us prove the result for m = n . First, the transport problem (83) can beuniquely solved in B R ( x ) by virtue of Lemma 5.2, the inductive hypothesis (85) and the assumption on ε since (cid:107) u n − u (cid:107) C ,α ( B R ( x )) ≤ ε (cid:107) u (cid:107) C ,α ( B R ( x )) < δ . Second, the boundary value problem (84) can also be uniquely solved since (cid:90) ∂B R ( x ) ( R ( · − x ) u n + ϕ n u n ) · η dS + λ (cid:90) ∂B R ( x ) u · η dS = (cid:90) B R ( x ) ( ∇ ( f + ϕ n ) · u n + ( f + ϕ n ) div u n ) dx = 0 , by the inductive hypothesis and λ is assumed to be a regular value. Furthermore, a similar argumentto that in the step n = 0 shows that u n +1 is divergence-free again. Let us finally obtain the desiredestimates for u n +1 − u n . To this end, note that u n +1 − u n solves the boundary value problem (cid:26) (curl − λ )( u n +1 − u n ) = R ( · − x )( u n − u n − ) + ( ϕ n − ϕ n − ) u n + ϕ n − ( u n − u n − ) , x ∈ B R ( x ) , ( u n +1 − u n ) · η = 0 , x ∈ ∂B R ( x ) . Hence, we arrive at the following bound (cid:107) u n +1 − u n (cid:107) C ,α ( B R ( x )) ≤ CR α ( (cid:107)R ( · − x ) (cid:107) C ,α ( B R ( x )) (cid:107) u n − u n − (cid:107) C ,α ( B R ( x )) + (cid:107) ϕ n − ϕ n − (cid:107) C ,α ( B R ( x )) (cid:107) u n (cid:107) C ,α ( B R ( x )) + (cid:107) ϕ n − (cid:107) C ,α ( B R ( x )) (cid:107) u n − u n − (cid:107) C ,α ( B R ( x )) ) . On the one hand, the remainder can be bounded above as in (86). On the other hand, (cid:107) ϕ n (cid:107) C ,α ( B R ( x )) and (cid:107) ϕ n − ϕ n − (cid:107) C ,α ( B R ( x )) can be estimated as (cid:107) ϕ n − (cid:107) C ,α ( B R ( x )) ≤ (cid:0) (cid:107) ϕ (cid:107) C ,α (Σ R ,µ R ) + R − α + (cid:107) T R (cid:107) C (Σ R ) (cid:1) × κ (cid:0) (cid:107) u n − (cid:107) C ,α ( B R ( x )) , (cid:107) T R (cid:107) C (Σ R ) , (cid:107) µ R (cid:107) C ,α ( B R ( x )) (cid:1) , (cid:107) ϕ n − ϕ n − (cid:107) C ,α ( B R ( x )) ≤ (cid:0) (cid:107) ϕ (cid:107) C ,α (Σ R ,µ R ) + R − α + (cid:107) T R (cid:107) C (Σ R ) (cid:1) × κ (cid:0) (cid:107) u n (cid:107) C ,α ( B R ( x )) , (cid:107) T R (cid:107) C (Σ R ) , (cid:107) µ R (cid:107) C ,α ( B R ( x )) (cid:1) × κ (cid:0) (cid:107) u n − (cid:107) C ,α ( B R ( x )) , (cid:107) T R (cid:107) C (Σ R ) , (cid:107) µ R (cid:107) C ,α ( B R ( x )) (cid:1) × (cid:107) u n − u n − (cid:107) C ,α ( B R ( x )) . Consequently, the inductive hypothesis along with our choice (87) leads to the first inequality in (85) andthe remaining two inequalities obviously follows from the first one by virtue of the triangle inequality.As in Section 3, the first inequality in (85) shows that { u n } n ∈ N is a Cauchy sequence in C ,α ( B R ( x ) , R ).By completeness, consider u ∈ C ,α ( B R ( x )) such that u n → u in C ,α ( B R ( x , R )) . Moreover, the same reasoning as above yields the estimate (cid:107) ϕ n − ϕ m (cid:107) C ,α ( B R ( x )) ≤ (cid:0) (cid:107) ϕ (cid:107) C ,α (Σ R ,µ R ) + R − α + (cid:107) T R (cid:107) C (Σ R ) (cid:1) × κ (cid:0) (cid:107) u n (cid:107) C ,α ( B R ( x )) , (cid:107) T R (cid:107) C (Σ R ) , (cid:107) µ R (cid:107) C ,α ( B R ( x )) (cid:1) × κ (cid:0) (cid:107) u m (cid:107) C ,α ( B R ( x )) , (cid:107) T R (cid:107) C (Σ R ) , (cid:107) µ R (cid:107) C ,α ( B R ( x )) (cid:1) × (cid:107) u n − u m (cid:107) C ,α ( B R ( x )) , for every indices n, m ∈ N . Then, there exists some constant K = K ( δ R , R, (cid:107) u (cid:107) C ,α ) > (cid:107) ϕ n − ϕ m (cid:107) C ,α ( B R ( x )) ≤ K (cid:107) u n − u m (cid:107) C ,α ( B R ( x )) . Hence, { ϕ n } n ∈ N is also a Cauchy sequence in C ,α ( B R ( x )) and one can consider ϕ ∈ C ,α ( B R ( x )) suchthat ϕ n → ϕ in C ,α ( B R ( x )) . Taking limits in (83)-(84) we are led to a generalized Beltrami field u ∈ C ,α ( B R ( x ) , R ) solving curl u = ( f + ϕ ) u, x ∈ B R ( x ) , div u = 0 , x ∈ B R ( x ) ,u · η = u · η, x ∈ ∂B R ( x ) , for a perturbation ϕ ∈ C ,α ( B R ( x )) of the factor such that ϕ = ϕ on Σ R .Let us finally show that u ∈ C k +1 ,α ( B R ( x ) , R ) and ϕ ∈ C k,α ( B R ( x )) by a bootstrap argumentbased on the elliptic gain of regularity. Recall that the vector-valued boundary problem associated withthe Laplacian with relative boundary conditions, ∆ w = F, x ∈ B R ( x ) ,u · η = G, x ∈ ∂B R ( x ) , curl u × η = H, x ∈ ∂B R ( x ) , is well known to satisfy the estimate (cid:107) w (cid:107) C l +1 ,α ( B R ( x )) ≤ C (cid:0) (cid:107) F (cid:107) C l − ,α ( B R ( x )) + (cid:107) G (cid:107) C l +1 ,α ( ∂B R ( x )) + (cid:107) H (cid:107) C l,α ( ∂B R ( x )) (cid:1) . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 47
The key observation now is that, by acting with the curl operator on the equation for u , it follows that ∆ u = − curl(( f + ϕ ) u ) , x ∈ B R ( x ) ,u · η = u · η, x ∈ ∂B R ( x ) , curl u × η = ( f + ϕ ) u × η, x ∈ ∂B R ( x ) . Then, the next hierarchy of inequalities hold for every l ≥ (cid:107) u (cid:107) C l +1 ,α ( B R ( x )) ≤ C ( (cid:107) ( f + ϕ ) u (cid:107) C l,α ( B R ( x )) + (cid:107) u · η (cid:107) C l +1 ,α ( ∂B R ( x )) + (cid:107) ( f + ϕ ) u × η (cid:107) C l,α ( B R ( x )) ) . We then get that the fact that u is of class C ,α implies that ϕ is of class C ,α . In turns, it ensures that u is in C ,α and, repeating the argument as many times as necessary (up to the regularity on ϕ and u ,i.e., C k +1 ,α ) we derive the desired gain of regularity. Indeed, the estimate (cid:107) u − u (cid:107) C ,α ( B R ( x )) ≤ ε (cid:107) u (cid:107) C ,α ( B R ( x )) , can be promoted to its C k +1 ,α version, i.e., (cid:107) u − u (cid:107) C k +1 ,α ( B R ( x )) ≤ ε (cid:107) u (cid:107) C k +1 ,α ( B R ( x )) . So far, we have only taken low H¨older exponents α ∈ (0 , / u ∈ C k +1 ,α (cid:48) (Ω , R )and f ∈ C k,α (cid:48) (Ω) for some α (cid:48) ∈ ( α, u ∈ C k +1 ,α ( B R ( x ) , R ) and ϕ ∈ C k,α ( B R ( x )).The above argument, yields a strong Beltrami field u ∈ C k +1 ,α ( B R ( x ) , R ) with proportionality factor f + ϕ for some perturbation ϕ ∈ C k,α ( B R ( x )) such that ϕ = ϕ on Σ R as long as R is small enoughand (cid:107) ϕ (cid:107) C k +1 ,α (cid:48) (Σ R ) < δ R . Since (cid:107) ϕ (cid:107) C k +1 ,α (Σ R ) = (cid:107) ϕ ◦ µ R (cid:107) C k +1 ,α ( D R ) ≤ (cid:107) ϕ ◦ µ R (cid:107) C k +1 ,α (cid:48) ( D R ) = (cid:107) ϕ (cid:107) C k +1 ,α (cid:48) (Σ R ) , then, the above smallness assumption on the C k +1 ,α (Σ R ) norm ϕ follows from the corresponding as-sumption on the C k +1 ,α (cid:48) (Σ R ) norm, i.e., (cid:107) ϕ (cid:107) C k +1 ,α (cid:48) (Σ R ) < δ R . Since ϕ solves (cid:26) ∇ ϕ · u = −∇ f · u, x ∈ B R ( x ) ,ϕ = ϕ , x ∈ Σ R , then, a similar result to that in Theorem 3.5 leads to ϕ ∈ C ,α ( B R ( x )) because so is u , f and ϕ .In particular ϕ ∈ C ,α (cid:48) ( B R ( x )) and u ∈ C ,α (cid:48) ( B R ( x ) , R ). Then, the above bootstrap in the Bel-trami equation yields ϕ ∈ C k,α (cid:48) ( B R ( x )) and u ∈ C k +1 ,α (cid:48) ( B R ( x )), thereby concluding the proof of thetheorem. (cid:3) Potential theory techniques for inhomogeneous integral kernels
Our goal in this section is to extend some results of classical potential theory to inhomogeneous kernelslike the fundamental solution of the Helmholtz equation Γ λ ( x ) (see e.g. [8, 13, 23, 24, 30, 32, 33, 41, 42] inthe case of homogeneous kernels). While there are some previous results concerning the inhomogeneouscase (see [9, 10, 37] for a study of Γ λ ( x ) with non-zero λ ), only low order H¨older estimates have beenobtained. Our approach roughly follows the treatment of [25, 38] for the harmonic case ( λ = 0), and wewill introduce nontrivial modifications to derive higher order H¨older estimates of generalized volume andsingle layer potentials in the inhomogeneous setting. These results were used in Section 2 and, of course,the main point throughout is to be able to consider exterior (unbounded) domains.6.1. Inhomogeneous volume and single layer potentials.
In our context, all the integral kernelsthat we need to consider come from the fundamental solution of the 3-dimensional Helmholtz equation(11) Γ λ ( z ) = e iλ | z | π | z | = 14 π (cid:18) cos( λ | z | ) | z | + i sin( λ | z | ) | z | (cid:19) , z ∈ R \{ } . For λ = 0 we recover the Newtonian potential associated to the Laplace equation in R , [23, 24, 32, 33].As it is not longer homogeneous, the classical theory cannot be directly applied. Fortunately, this kernel can be though to be “almost homogeneous” in the following sense. Let usconsider the functions φ λ ( r ) := e iλr πr , r > ,ψ λ ( r ) := φ λ ( r ) − πr ≡ e iλr − πr , r > . (88)From the definition one has the following splitting φ λ ( r ) = 14 πr + ψ λ ( r ) , (89)and consequently, the following decomposition of the fundamental solutionΓ λ ( z ) = φ λ ( | z | ) = 14 π | z | + ψ λ ( | z | ) =: Γ ( z ) + R λ ( z ) (90)holds. This amounts to a decomposition of the inhomogeneous kernel Γ λ ( z ) into the homogeneous partΓ ( z ) and an inhomogeneous remainder R λ ( z ) enjoying lower order singularities at the origin. This isthe main argument supporting our subsequent results: we do not need our whole kernel to be purelyhomogeneous, but only the principal (or more singular) part. While higher order derivatives of harmonicpotentials can be directly controlled through the harmonic kernel Γ ( z ) and the classical results in [23,24, 32, 33], it is also important to control the behavior of the higher order derivatives of R λ ( z ).Specifically, we can compute the first derivative of ψ λ ( r ) and write it by means of homogeneousfunctions and ψ λ ( r ) itself ψ (cid:48) λ ( r ) = iλ πr + (cid:18) iλ − r (cid:19) ψ λ ( r ) . As ψ λ ( r ) is locally bounded near r = 0 and decay as r − at infinity, it is globally bounded. Thus, | ψ (cid:48) λ ( r ) | ≤ C (cid:18) r (cid:19) , r > . A recursive reasoning leads to estimates for higher order derivatives of ψ λ ( r ) of the type | ψ ( m ) λ ( r ) | ≤ C (cid:18) r m (cid:19) , r > , (91)where C = C ( λ, m ) is a nonnegative constant. Consequently, we have the following bounds for R λ ( z ) = ψ λ ( | z | ) and its higher order derivatives | D γ R λ ( z ) | ≤ C (cid:18) | z | | γ | (cid:19) , (92)for every z ∈ R \{ } and each multi-index γ , in contrast with the analogous bounds for Γ ( z ): | D γ Γ ( z ) | ≤ C | z | | γ | +1 . (93)A basic fact is that the remainder R λ ( z ), which is not homogeneous, is one degree less singularthan Γ ( z ), so we will combine statements about singular integrals (such as D Γ ( z )) for which theCalderon–Zygmund theory essentially applies, with a treatment of weakly singular integral kernels (suchas D R λ ( z )) based on the Hardy–Littlewood–Sobolev theorem. See also [37] for a treatment of pseudo-homogeneous kernels.For the sake of completeness, we shall next introduce the kind of kernels that we will consider in thissection. Let us consider a bounded domain D ⊆ R N and a continuous function K ( x, z ) , x ∈ D, z ∈ R N \{ } . K is said to be a weakly singular kernels of exponent β if there exists a nonnegative constant C such that | K ( x, z ) | ≤ C | z | β , ∀ x ∈ D, ∀ y ∈ R N \{ } , for a given 0 ≤ β ≤ N −
1. The kind of singular integral kernel that arises in this paper are first orderpartial derivatives of positively homogeneous kernel or degree − ( N − ∂∂z i K ( x, z ) , x ∈ D, z ∈ R \{ } , where K ( x, z ) satisfies K ( x, λz ) = 1 λ N − K ( x, z ) , for all x ∈ D, z ∈ R N \{ } , λ > K ( x, σ ) is continuous for x ∈ D and σ ∈ ∂B (0). ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 49
A classical results about the boundedness of generalized volume and single layer potencial in H¨olderspaces allows us to bound the single layer potential associated with Γ λ ( z ) both in bounded and unboundeddomains (see [33, Teorema 2.I]): Theorem 6.1 (Generalized single layer potential) . Let G ⊆ R be a bounded domain with regularity C k +1 ,α , Ω := R \ G its outer domain and S = ∂G the boundary surface. Consider the generalized singlelayer potential associated with the Helmholtz equation and generated by a density ζ : S −→ R over theboundary, ( S λ ζ )( x ) := (cid:90) S Γ λ ( x − y ) ζ ( y ) d y S, x ∈ R \ S. Then, S λ ζ is well defined both in G and Ω for each ζ ∈ C k,α ( S ) , it belongs to C k +1 ,α ( G ) and C k +1 ,α (Ω) respectively and we have the associated bounded linear operators S − λ : C k,α ( S ) −→ C k +1 ,α ( G ) ,ζ (cid:55)−→ ( S λ ζ ) | G , S + λ : C k,α ( S ) −→ C k +1 ,α (Ω) ,ζ (cid:55)−→ ( S λ ζ ) | Ω , i.e., there exists a nonnegative constant K = K ( k, α, λ, G ) so that ζ ∈ C k,α ( S ) satisfies (cid:107)S − λ ζ (cid:107) C k +1 ,α ( G ) ≤ K (cid:107) ζ (cid:107) C k,α ( S ) , (cid:107)S + λ ζ (cid:107) C k +1 ,α (Ω) ≤ K (cid:107) ζ (cid:107) C k,α ( S ) . Differentiation under the integral sign leads to ∇ ( S − λ ζ )( x ) = (cid:90) S ∇ x Γ λ ( x − y ) ζ ( y ) d y S, x ∈ G, ∇ ( S + λ ζ )( x ) = (cid:90) S ∇ x Γ λ ( x − y ) ζ ( y ) d y S, x ∈ Ω . We omit the proof of this theorem since we are interested in a more singular regularity result thatgeneralizes this one. Specifically, we will study the regularity along the boundary surface S of thesegeneralized single layer potentials along with some other related potentials with inhomogeneous kernelsthat arose in previous sections, where we will use arguments as in [33, Teorema 2.I]. In the next results, weshow the regularity of generalized volume (or Newtonian) potentials with compactly supported densitiesboth for interior and exterior domains, which conclude with the derivation of the classical H¨older–Korn–Lichtenstein–Giraud inequality for high order estimates of H¨older type in the inhomogeneous case.
Lemma 6.2.
Let G ⊆ R be a bounded domain with regularity C k +1 ,α , Ω := R \ G its exterior domainand S = ∂G the boundary surface. Define the generalized volume potential on G associated with theHelmholtz equation and generated by a density in G , ζ : G −→ R ( N − λ ζ )( x ) = (cid:90) G Γ λ ( x − y ) ζ ( y ) dy, x ∈ G. Then, N − λ ζ ∈ C k +2 ,α ( G ) is well defined over G for every ζ ∈ C k,α ( G ) , and N − λ : C k,α ( G ) −→ C k +2 ,α ( G ) ,ζ (cid:55)−→ N − λ ζ, defines a bounded linear operator, i.e., there exists a nonnegative constant K = K ( k, α, λ, G ) so that (cid:107)N − λ ζ (cid:107) C k +2 ,α ( G ) ≤ K (cid:107) ζ (cid:107) C k,α ( G ) , for every density ζ ∈ C k,α ( G ) .Proof. The proof follows the lines of [33, Teorema 3.II] for the harmonic case λ = 0, that we extend tothe inhomogeneous case.Let us obtain first a C estimate of N − λ ζ . SinceΓ λ ( z ) = O ( | z | − ) and ∇ Γ λ ( z ) = O ( | z | − ) as | z | → ,G is bounded and ζ ∈ C ( G ), then one can take derivatives under the integral sign, i.e., ∂∂x i ( N − λ ζ )( x ) = (cid:90) G ∂∂x i Γ λ ( x − y ) ζ ( y ) dy, x ∈ G. Moreover, straightforward computations supported by the local integrability of both Γ λ ( z ) and ∇ Γ λ ( z )show that (cid:107)N − λ ζ (cid:107) C ( G ) ≤ C (cid:107) ζ (cid:107) C ( G ) ≤ C (cid:107) ζ (cid:107) C k,α ( G ) . Fix any multi-index γ with | γ | ≤ k and takes derivatives again under the integral sign to get D γ ∂∂x i ( N − λ ζ )( x ) = (cid:90) G D γx ∂∂x i Γ λ ( x − y ) ζ ( y ) dy. For any index 1 ≤ l ≤ e l ≤ γ , an integration by parts recasts the above identity as D γ ∂∂x i ( N − λ ζ )( x ) = − (cid:90) G D γ − e l x ∂∂x i ∂∂y l Γ λ ( x − y ) ζ ( y ) dy = (cid:90) G D γ − e l x ∂∂x i Γ λ ( x − y ) ∂ζ∂y l ( y ) dy − (cid:90) S D γ − e l x ∂∂x i Γ λ ( x − y ) ζ ( y ) η l ( y ) d y S, where η stands for the exterior unit normal vector field along S . A recursive reasoning leads to D γ ∂∂x i ( N − λ ζ )( x ) = − γ (cid:88) m =1 (cid:90) S D γ − m e x ∂∂x i Γ λ ( x − y ) D ( m − e ζ ( y ) η ( y ) d y S − γ (cid:88) m =1 (cid:90) S D γ − γ e − γ e x ∂∂x i Γ λ ( x − y ) D γ e +( m − e ζ ( y ) η ( y ) d y S − α (cid:88) m =1 (cid:90) S D γ − γ e − γ e − m e x ∂∂x i Γ λ ( x − y ) D γ e + γ e +( m − e ζ ( y ) η ( y ) d y S + (cid:90) G ∂∂x i Γ λ ( x − y ) D γ ζ ( y ) dy. Therefore, the same argument as above for the last volume integral along with Theorem 6.1 for theboundary integrals show the following upper bound for the derivatives up to order k + 1 of the generalizedvolume potential: (cid:13)(cid:13)(cid:13)(cid:13) D γ ∂∂x i ( N − λ ζ ) (cid:13)(cid:13)(cid:13)(cid:13) C ( G ) ≤ K (cid:107) ζ (cid:107) C k,α ( G ) . Here | γ | ≤ k and 1 ≤ i ≤ k + 2. Let us then consider another index1 ≤ j ≤ D γ ∂ ∂x i ∂x j ( N − λ ζ )( x ) = − γ (cid:88) m =1 (cid:90) S D γ − m e + e j x ∂∂x i Γ λ ( x − y ) D ( m − e ζ ( y ) η ( y ) d y S − γ (cid:88) m =1 (cid:90) S D γ − γ e − γ e + e j x ∂∂x i Γ λ ( x − y ) D γ e +( m − e ζ ( y ) η ( y ) d y S − α (cid:88) m =1 (cid:90) S D γ − γ e − γ e − m e + e j x ∂∂x i Γ λ ( x − y ) D γ e + γ e +( m − e ζ ( y ) η ( y ) d y S + (cid:90) G ∂ ∂x i ∂x j Γ λ ( x − y ) D γ ζ ( y ) dy. Similar estimates for the boundary terms can be obtained in C ,α ( G ) by virtue of Theorem 6.1, whilethe volume integral has to be studied separately carrying out an adaptation of the ideas in the harmoniccase [33, Teorema 3.II].We first split it into two parts and use again integration by parts in the second term (cid:90) G ∂ ∂x i ∂x j Γ λ ( x − y ) D γ ζ ( y ) dy = (cid:90) G ∂ ∂x i ∂x j Γ λ ( x − y )( D γ ζ ( y ) − D γ ζ ( x )) dy + D γ ζ ( x ) (cid:90) G ∂∂x j ∂∂x i Γ λ ( x − y ) dy = (cid:90) G ∂ ∂x i ∂x j Γ λ ( x − y )( D γ ζ ( y ) − D γ ζ ( x )) dy − D γ ζ ( x ) (cid:90) S ∂∂x i Γ λ ( x − y ) η j ( y ) d y S =: F ( x ) − H ( x ) . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 51
The idea behind such decomposition is apparent now since the second term, H ( x ) can be bounded in C ,α ( G ) according to Theorem 6.1 (cid:107) H (cid:107) C ,α ( G ) ≤ K (cid:107) η (cid:107) C ,α ( G ) (cid:107) ζ (cid:107) C k,α ( S ) and we have cancelled an α power of the singularity in the first term F ( x ): | F ( x ) | ≤ [ D γ ζ ] α,G (cid:90) G (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂x i ∂x j Γ λ ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12) | x − y | α dy. Bearing (90) in mind, we obtain the derivative formulas ∂∂z i Γ λ ( z ) = (cid:18) − π | z | + ψ (cid:48) λ ( | z | ) (cid:19) z i | z | ,∂∂z i ∂∂z j Γ λ ( z ) = (cid:18) π | z | + ψ (cid:48)(cid:48) λ ( | z | ) (cid:19) z i | z | z j | z | + (cid:18) − π | z | + ψ (cid:48) λ ( | z | ) (cid:19) δ ij | z | − z i z j | z | | z | . Thus, the estimates (92) and (93) amount to the following C estimate | F ( x ) | ≤ C (cid:107) ζ (cid:107) C k,α ( G ) (cid:90) G (cid:18) | x − y | (cid:19) | x − y | α dy ≤ C (cid:107) ζ (cid:107) C k,α ( G ) (cid:90) G dy | x − y | − α . The local integrability of | z | α − along with the boundedness of G lead again to the upper bound (cid:107) F (cid:107) C ( G ) ≤ K (cid:107) ζ (cid:107) C k,α ( G ) . Let us finally show the local α -H¨older property for F , i.e., | F ( x ) − F ( x ) | ≤ C | x − x | α , for every x , x ∈ G such that | x − x | < δ and any small δ >
0. To this end, consider a neighborhood U of x with B d ( x ) ⊆ U ⊆ B d ( x )so that, F ( x ) − F ( x ) = (cid:90) G ∂ Γ λ ( x − y ) ∂x i ∂x j ( D γ ζ ( y ) − D γ ζ ( x )) dy − (cid:90) G ∂ Γ λ ( x − y ) ∂x i ∂x j ( D γ ζ ( y ) − D γ ζ ( x )) dy = (cid:90) G ∩ B d ( x ) ∂ Γ λ ( x − y ) ∂x i ∂x j ( D γ ζ ( y ) − D γ ζ ( x )) dy − (cid:90) G ∩ B d ( x ) ∂ Γ λ ( x − y ) ∂x i ∂x j ( D γ ζ ( y ) − D γ ζ ( x )) dy + (cid:90) G \ B d ( x ) (cid:18) ∂ Γ λ ( x − y ) ∂x i ∂x j − ∂ G λ ( x − y ) ∂x i ∂x j (cid:19) ( D γ ζ ( y ) − D γ ζ ( x )) dy − ( D γ ζ ( x ) − D γ ζ ( x )) (cid:90) G \ B d ( x ) ∂ Γ λ ( x − y ) ∂x i ∂x j dy. Taking Euclidean norms, we finally arrive at | F ( x ) − F ( x ) | ≤ (cid:90) G ∩ B d ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) | ( D γ ζ ( y ) − D γ ζ ( x )) | dy + (cid:90) G ∩ B d ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) | ( D γ ζ ( y ) − D γ ζ ( x )) | dy + (cid:90) G \ B d ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j − ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) | D γ ζ ( y ) − D γ ζ ( x ) | dy + | D γ ζ ( x ) − D γ ζ ( x ) | (cid:90) G \ U (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) dy, (94)where in the last three terms we have respectively used that G ∩ B d ( x ) ⊆ G ∩ B d ( x ), G \ B d ( x ) ⊆ G \ B d ( x ) and G \ B d ( x ) ⊆ G \ U .The first term in (94) can be bounded by virtue of the α -H¨older property for D γ ζ and the fact that D Γ λ ( z ) = O (cid:0) | z | − (cid:1) near the origin: (cid:90) G ∩ B d ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) | ( D γ ζ ( y ) − D γ ζ ( x )) | dy ≤ C (cid:90) G ∩ B d ( x ) | x − y | − α dy b x x G B d ( x ) ≡ U B d ( x ) b Figure 7.
Choice of U = B d ( x ) in the first case. ≤ C (cid:90) B d ( x ) dy | x − y | − α = 4 πC (cid:90) d r − α dr = 4 πC α α | x − x | α . A similar bound follows for the second term. Regarding the third term in (94), we find (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ∂z i ∂z j ( x − y ) − ∂ Γ λ ∂z i ∂z j ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ddθ (cid:20) ∂ Γ λ ∂z i ∂z j ( θx + (1 − θ ) x − y ) (cid:21) dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∇ ∂ ∂z i ∂z j Γ λ (cid:19) ( θx + (1 − θ ) x − y ) · ( x − x ) (cid:12)(cid:12)(cid:12)(cid:12) dθ, ≤ C (cid:90) dθ | θx + (1 − θ ) x − y | | x − x | dθ. Since y ∈ G \ B d ( x ) in the third term of the decomposition (94) and 0 ≤ θ ≤
1, then1 | θx + (1 − θ ) x − y | = 1 | (1 − θ )( x − x ) + ( x − y ) | ≤ | x − y | − | x − x | ) ≤ | x − y | . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ∂z i ∂z j ( x − y ) − ∂ Γ λ ∂z i ∂z j ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − x || x − y | , ∀ y ∈ G \ B d ( x ) . The above estimate allows obtaining the desired estimate of α -H¨older type for the third term in (94) (cid:90) G \ B d ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j − ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) | D γ ζ ( y ) − D γ ζ ( x ) | dy ≤ C (cid:107) ζ (cid:107) C k,α ( G ) | x − x | (cid:90) R \ B d ( x ) dy | x − y | − α = 4 πC (cid:107) ζ (cid:107) C k,α ( G ) | x − x | (cid:90) + ∞ d r − α dr =4 πC (cid:107) ζ (cid:107) C k,α ( G ) | x − x | (cid:90) + ∞ d r − α ) dr = 4 πC − α − α (cid:107) ζ (cid:107) C k,α ( G ) | x − x || x − x | − α = C (cid:107) ζ (cid:107) C k,α ( G ) | x − x | α . Concerning the last term in (94), we are done as long as one notices that D γ ζ ∈ C ,α ( G ) and shows (cid:90) G \ U (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ( x − y ) ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) dy ≤ C, for some positive constant C depending on δ but not on d = | x − x | . For that, assume first that2 d ≤ dist( x , S ) and define U := B d ( x ). Then G \ U ≡ G \ B d ( x ). Note that ∂ ( G \ U ) = S ∪ ∂B d ( x )and integrate by parts to get (cid:90) G \ U ∂ Γ λ ( x − y ) ∂x i ∂x j dy = (cid:90) S ∂ Γ λ ( x − y ) ∂x i η j ( y ) d y S − (cid:90) ∂B d ( x ) ∂ Γ λ ( x − y ) ∂x i ( y − x ) j | y − x | d y S. ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 53 b x x G B d ( x ) ≡ U B d ( x ) b b e x U ≡ B d ( e x ) Figure 8.
Choice of U = B d ( (cid:101) x ) in the second case.Theorem 6.1 provides an upper bound of the first term. On the other hand, by definitionsup y ∈ ∂B d ( x ) | x − y | ≤ | x − x | and one has the asymptotic behavior (cid:12)(cid:12)(cid:12)(cid:12) ∂ Γ λ ∂z i ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) | z | (cid:19) , z ∈ R \{ } , so the second term is bounded as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂B d ( x ) ∂ Γ λ ( x − y ) ∂x i ( y − x ) j | y − x | d y S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:101) C | x − x | (cid:90) ∂B d ( x ) d y S = 4 π (cid:101) C ≡ C. Secondly, let us consider the opposite case 2 d > dist( x , S ). Now the configuration is slightly different.Let us fix some (cid:101) x ∈ S so that | x − (cid:101) x | = dist( x , S ) and define U := B d ( (cid:101) x ). By definition x ∈ B d ( (cid:101) x )and consequently, B d ( x ) ⊆ U ⊆ B d ( x ). Thus, U is as in (94). The last term there takes the form (cid:90) G \ U ∂ Γ λ ( x − y ) ∂x i ∂x j dy = (cid:90) S ∂ Γ λ ( x − y ) ∂x i η j ( y ) d y S − (cid:90) ∂ ( U ∩ G ) ∂ Γ λ ( x − y ) ∂x i ν j ( y ) d y S. Since the first term can be bounded through the same reasonings as above, we focus on the second term.It will be estimated following the idea in [33, Lemma 2.IV]. To this end, define some cut-off function ξ (cid:16) | y − (cid:101) x | d (cid:17) for ξ ∈ C ∞ c ( R +0 ) such that ξ ( r ) = 1 , r ∈ (cid:2) , (cid:3) ,ξ ( r ) ∈ (0 , , r ∈ (cid:0) , (cid:1) ,ξ ( r ) = 0 , r ≥ , and consider the splitting (cid:90) ∂ ( G ∩ B d ( (cid:101) x )) ∂ Γ λ ( x − y ) ∂x i ν j ( y ) d y S = (cid:90) G ∩ ∂B d ( (cid:101) x ) ∂ Γ λ ( x − y ) ∂x i ν j ( y ) d y S + (cid:90) S ∩ B d ( (cid:101) x ) ∂ Γ λ ( x − y ) ∂x i (cid:20) − ξ (cid:18) | y − (cid:101) x | d (cid:19)(cid:21) ν j ( y ) d y S + (cid:90) S ∩ B d ( (cid:101) x ) ∂ Γ λ ( x − y ) ∂x i ξ (cid:18) | y − (cid:101) x | d (cid:19) ν j ( y ) d y S. (95) Bear in mind again that x ∈ B d ( (cid:101) x ), so | y − x | ≥ | y − (cid:101) x | − d = 4 d − d = d, for each y ∈ G ∩ ∂B d ( (cid:101) x ) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) G ∩ ∂B d ( (cid:101) x ) ∂ Γ λ ( x − y ) ∂x i ν j ( y ) d y S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:101) C | x − x | | ∂B d ( (cid:101) x ) | ≤ π (cid:101) C. In the second term y ∈ S ∩ B d ( (cid:101) x ). Moreover, in order that y belongs to the support of the cut-offfunction, one has to assume | y − (cid:101) x | ≥ d . Thus, | y − x | ≥ | y − (cid:101) x | − d ≥ d − d = d , and consequently, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ∩ B d ( (cid:101) x ) ∂ Γ λ ( x − y ) ∂x i ν j ( y ) d y S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:101) C | x − x | | S ∩ B d ( (cid:101) x ) | . The upper bound for the second term is done once we note that | S ∩ B d ( (cid:101) x ) | ≤ Cd . (96)To prove the corresponding bound for the third term in (95), we consider the potential S ( x ) = (cid:90) S ∂ Γ λ ( x − y ) ∂x i ξ (cid:18) | y − (cid:101) x | d (cid:19) ν j ( y ) d y S, x ∈ G, whose C ,α estimate follows again from Lemma 6.1: (cid:107)S(cid:107) C ,α ( G ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) ξ (cid:18) | · − (cid:101) x | d (cid:19) ν j (cid:13)(cid:13)(cid:13)(cid:13) C ,α ( S ) ≤ C (cid:18) d α (cid:19) . Let us now fix δ > x − θη ( x ) ∈ G for every couple x ∈ S and 0 < θ < δ . Thus, S ( (cid:101) x − dη ( (cid:101) x )) = 0 and consequently |S ( x ) | = |S ( x ) − S ( (cid:101) x − dη ( (cid:101) x )) | ≤ C (cid:18) d α (cid:19) | x − (cid:101) x + 4 dη ( (cid:101) x ) | α ≤ C (cid:18) d α (cid:19) (3 d + 4 d ) α ≤ (cid:101) C. (cid:3) Remark 6.3.
Let us elaborate on the key inequality (96), which is a key regularity property intimatelyconnected with deep issues in harmonic analysis. Indeed, a surface S is said to be Alhfolrs–David regularwhen | S ∩ B R ( x ) | ≤ CR , for every couple x ∈ S , R > and some nonnegative constant C . These surfaces (originally curves)arise from the study of singular integrals along curves [13] , and had already appeared in [6, 8, 30] on L estimates for the Cauchy integral along Lipschitz curves. His results were improved in [13] to the moregeneral setting of Alhfors–David curves and it was generalized in [41] to the N -dimensional framework.Specifically, Ahlfors–David regularity was shown to control singular integral operators that are much moregeneral than the Cauchy integral. Of course, C k,α surfaces are Ahlfors–David regular. Lemma 6.4.
Let G ⊆ R be a bounded domain with regularity C k +1 ,α , Ω := R \ G its exterior domainand S = ∂G the boundary surface. Define the generalized volume potential on Ω associated with theHelmholtz equation and generated by a density in G , ζ : G −→ R ( N + λ ζ )( x ) = (cid:90) G Γ λ ( x − y ) ζ ( y ) dy, x ∈ Ω . Then, N + λ ζ ∈ C k +2 ,α (Ω) is well defined for every ζ ∈ C k,α ( G ) , and N + λ : C k,α ( G ) −→ C k +2 ,α (Ω) ,ζ (cid:55)−→ N + λ ζ, is a bounded linear operator, i.e., (cid:107)N + λ ζ (cid:107) C k +2 ,α (Ω) ≤ K (cid:107) ζ (cid:107) C k,α ( G ) , for every density ζ ∈ C k,α ( G ) . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 55
Proof.
Our argument is based on some ideas of [33, Teorema 3.II]. Consider
R > G ⊆ B R (0) and let us estimate (cid:107)N + λ ζ (cid:107) C k +2 ,α (Ω) in terms of (cid:107)N + λ ζ (cid:107) C k +2 ,α ( R \ B R (0)) and (cid:107)N + λ ζ (cid:107) C k +2 ,α (Ω R ) ,where Ω R stands for B R (0) \ G . Set d R := min {| x − y | : x ∈ R \ B R (0) , y ∈ G } > , and assume that d R < | D γx Γ λ ( x − y ) | ≤ (cid:101) C d | γ | +1 R , for every multi-index γ and every x ∈ R \ B R (0) and y ∈ G . One can then take derivatives under theintegral sign and obtain the desired estimate for the C k +2 ,α norm in R \ B R (0). On the other hand,consider ζ ∈ C k,α ( R ) through Proposition 3.1. Then,( N + λ ζ )( x ) = (cid:90) B R (0) Γ λ ( x − y ) ζ ( y ) dy − (cid:90) Ω R Γ λ ( x − y ) ζ ( y ) dy, for every x ∈ Ω R . Since both B R (0) and Ω R are C k +1 ,α bounded domains, then Lemma 6.2 leads to (cid:107)N + λ ζ (cid:107) C k +2 ,α (Ω R ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) B R (0) Γ λ ( x − y ) ζ ( y ) dy (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k +2 ,α ( B R (0)) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) Ω R Γ λ ( x − y ) ζ ( y ) dy (cid:13)(cid:13)(cid:13)(cid:13) C k +2 ,α (Ω R ) ≤ M (cid:107) ζ (cid:107) C k +2 ,α ( B R (0)) ≤ M C P (cid:107) ζ (cid:107) C k +2 ,α ( G ) . (cid:3) Now, we focus on similar estimates for singular and weakly singular kernels in the whole space R N .This results are classical in the homogeneous harmonic case, Γ ( z ), and can be found in [24, 32, 33].However, not only we will need harmonic potentials, but we will also deal with general singular andweakly singular kernels. To this end, we remind [25, Satz 3.4, Satz 5.4]. Theorem 6.5 (Weakly singular kernels) . Let us consider ≤ β ≤ N − , < α < and K ( x, z ) , x ∈ D, z ∈ R N \{ } a weakly singular integral kernel of exponent β satisfying the following three hypothesis: (1) For each x ∈ D K ( x, · ) ∈ C ( R N \{ } ) . (97)(2) For each x ∈ D and z ∈ R N \{ } |∇ z K ( x, z ) | ≤ C | z | β +1 . (98)(3) For all x , x ∈ D and z ∈ R N \{ } one has | K ( x , z ) − K ( x , z ) | ≤ C | x − x | α | z | β . (99) Then, for all
R > there exists a nonnegative constant M = M ( N, α, β, R ) so that the generalized volumepotential ( N K ζ )( x ) := (cid:90) R N K ( x, x − y ) ζ ( y ) dy, x ∈ D, with density ζ ∈ C ,αc ( B R (0)) belongs to C ,α ( D ) and (cid:107)N K ζ (cid:107) C ,α ( D ) ≤ M (cid:107) ζ (cid:107) C ,α ( R N ) . Theorem 6.6 (Singular kernels) . Consider < α < and K ( x, z ) , x ∈ D, z ∈ R N \{ } a kernelsatisfying the following hypotheses: (1) K ( x, z ) is positively homogeneous of degree − ( N − with respect to the second variable, i.e., forall x ∈ D, z ∈ R N \{ } and λ > K ( x, λz ) = 1 λ N − K ( x, z ) . (100) (2) K ( x, z ) has the following regularity properties for every x ∈ D and each index ≤ i, j ≤ N : K ∈ C ( D × ( R N \{ } )) , K ( x, · ) ∈ C ( R N \{ } ) ,∂K∂x i ∈ C ( D × ( R N \{ } )) , ∂K∂x i ( x, · ) ∈ C ( R N \{ } ) ,∂K∂z i ∈ C ( D × ( R N \{ } )) ,∂ K∂z i ∂x j ∈ C ( D × ( R N \{ } )) , ∂ K∂z i ∂z j ∈ C ( D × ( R N \{ } )) . (101)(3) The first derivatives of K ( x, z ) are H¨older-continuous with exponent α with respect to x in thesense that, for each x , x ∈ D, z ∈ R N \{ } and for all index ≤ i ≤ N , (cid:12)(cid:12)(cid:12)(cid:12) ∂K∂x i ( x , z ) − ∂K∂x i ( x , z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − x | α | z | N − , (cid:12)(cid:12)(cid:12)(cid:12) ∂K∂z i ( x , z ) − ∂K∂z i ( x , z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − x | α | z | N . (102) Then, for every
R > there exists a nonnegative constant M = M ( N, α, R ) so that the generalized volumepotential N K ζ with density ζ ∈ C ,αc ( B R (0)) belongs to C ,α ( D ) and satisfies the estimate (cid:107)N K ζ (cid:107) C ,α ( D ) ≤ M (cid:107) ζ (cid:107) C ,α ( R N ) . Moreover, its first partial derivatives can be computed as ∂∂x i ( N K ζ ) = N ∂K∂xi ζ + N ∂K∂zi ζ. Notice that the singular integral kernel ∂K∂z i has an associated singular integral operator N ∂K∂zi , wherethe integrals are understood in the sense of Cauchy principal values, i.e., (cid:16) N ∂K∂zi ζ (cid:17) ( x ) = PV (cid:90) R N ∂K∂z i ( x, x − y ) ζ ( y ) dy, by virtue of the cancelation properties arising from the homogeneity in z of the original kernel K ( x, z ).Another interesting observation, which explains some differences between volume potentials in the whole R N and volume potentials in a bounded domain, is the change of variables formula( N K ζ )( x ) = (cid:90) R N K ( x, x − y ) ζ ( y ) dy = (cid:90) R N K ( x, z ) ζ ( x − z ) dz, (103)which lets us take derivatives in any of the fwo factors. When the kernel is not sufficiently well behaved,we can put the derivatives on the density, or the other way round. However, for densities on G as inLemma 6.2, the previous change of variable is not allowed and the only way to transfer derivatives tothe densities is by means of the integration by parts argument in Lemma 6.2. This gives rise to a newboundary term appears that must be studied by estimating single layer potentials as in Theorem 6.1.As a consequence, one can prove the next two corollaries, where higher order derivatives of thesegeneralized volume potentials can be considered. Corollary 6.7.
Les us consider ≤ β ≤ N − , < α < , k, m ∈ N so that β + m ≤ N − and K ( x, z ) , x ∈ D, z ∈ R N \{ } , a weakly singular integral kernel of exponent β which satisfies the followinghypotheses: (1) For each γ , γ so that | γ | ≤ k and | γ | ≤ m , D γ + γ x K ( x, z ) is weakly singular with exponent β and D γ x D γ z K ( x, z ) is a finite sum of weakly singular integral kernels with exponent β + n where n = 0 , . . . , | γ | , i.e., (cid:12)(cid:12) D γ + γ x K ( x, z ) (cid:12)(cid:12) ≤ C | z | β , | D γ x D γ z K ( x, z ) | ≤ C | z | β + | γ | . (104)(2) For every x ∈ D and γ , γ such that | γ | ≤ k and | γ | ≤ m ( D γ + γ x K )( x, · ) , ( D γ x D γ z K )( x, · ) ∈ C ( R N \{ } ) . (105) ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 57 (3)
For all x ∈ D , z ∈ R N \{ } and γ , γ so that | γ | ≤ k and | γ | ≤ m (cid:12)(cid:12) ∇ z D γ + γ x K ( x, z ) (cid:12)(cid:12) ≤ C | z | β +1 , |∇ z D γ x D γ z K ( x, z ) | ≤ C | z | β + | γ | +1 . (106)(4) For any x , x ∈ D , z ∈ R N \{ } and γ , γ satisfying | γ | ≤ k and | γ | ≤ m | D γ + γ x K ( x , z ) − D γ + γ x K ( x , z ) | ≤ C | x − x | α | z | β , | D γ x D γ z K ( x , z ) − D γ x D γ z K ( x , z ) | ≤ C | x − x | α | z | β + | γ | . (107) Then, for every
R > there exists a nonnegative constant M = M ( N, α, β, k, m, R ) such that the gener-alized volume potential N K ζ generated by a density ζ ∈ C k,αc ( B R (0)) belongs to C k + m,α ( D ) and verifiesthe estimate (cid:107)N K ζ (cid:107) C k + m,α ( D ) ≤ M (cid:107) ζ (cid:107) C k,α ( R N ) . Moreover, for every multi-index γ = γ + γ so that | γ | ≤ k and | γ | ≤ mD γ ( N K ζ ) = (cid:88) δ ≤ γ (cid:18) γ δ (cid:19) (cid:16) N D δ + γ x K D γ − δ ζ + N D δx D γ z K D γ − δ ζ (cid:17) . Corollary 6.8.
Let < α < , k ∈ N , x ∈ D, z ∈ R N \{ } and K ( x, z ) be a weakly singular kernel,which has the following properties: (1) K ( x, z ) is positively homogeneous of degree − ( N − in the second variable, i.e., K ( x, λz ) = 1 λ N − K ( x, z ) . (108)(2) K ( x, z ) has the regularity properties D γx K ∈ C ( D × ( R N \{ } )) , ( D γx K )( x, · ) ∈ C ( R N \{ } ) ,∂∂x i D γx K ∈ C ( D × ( R N \{ } )) , (cid:18) ∂∂x i D γx K (cid:19) ( x, · ) ∈ C ( R N \{ } ) ,∂∂z i D γx K ∈ C ( D × ( R N \{ } )) ,∂ ∂z i ∂x j D γx K ∈ C ( D × ( R N \{ } )) , ∂ ∂z i ∂z j D γx K ∈ C ( D × ( R N \{ } )) , (109) for each index ≤ i, j ≤ N and each γ with | γ | ≤ k . (3) The derivatives of K ( x, z ) with respect to x up to order k are H¨older-continuous with exponent α in the sense that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂∂x i D γx K (cid:19) ( x , z ) − (cid:18) ∂∂x i D γx K (cid:19) ( x , z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − x | α | z | N − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂∂z i D γx K (cid:19) ( x , z ) − (cid:18) ∂∂z i D γx K (cid:19) ( x , z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − x | α | z | N (110) for each x , x ∈ D , z ∈ R N \{ } , each index ≤ i ≤ N and | γ | ≤ k .Then, for every R > there exists a nonnegative constant M = M ( N, α, k, R ) such that the generalizedvolume potential N K ζ generated by any density ζ ∈ C k,αc ( B R (0)) belongs to C k +1 ,α ( D ) and (cid:107)N K ζ (cid:107) C k +1 ,α ( D ) ≤ M (cid:107) ζ (cid:107) C k,α ( R N ) . Moreover, the partial derivatives of the volume potential are ∂∂x i D γx ( N K ζ ) = (cid:88) δ ≤ γ (cid:18) γδ (cid:19) (cid:16) N ∂∂xi D γx K D δ − γ ζ + N ∂∂zi D γx K D δ − γ ζ (cid:17) , for | γ | ≤ k and ≤ i ≤ N . When the constants C appearing in the statements of the above results do not depend on the chosenbounded domain D , the above estimates can be extended from H¨older estimates over D , to globalestimates in R N . This is the case for the integral kernels which do not depend on the variable x (e.g.,Γ ( z ), R λ ( z ) and Γ λ ( z )). In this way, we get the next result in the spirit of Lemmas 6.2 and 6.4. Lemma 6.9.
Consider the generalized volume potential with compactly supported density ζ : R −→ R associated with the Helmholtz equation ( N λ ζ )( x ) := (cid:90) R Γ λ ( x − y ) ζ ( y ) d y S, x ∈ R . Then, for every
R > there exists a nonnegative constant K = K ( k, α, λ, R ) such that N λ ζ is well definedfor every ζ ∈ C k,αc ( B R (0)) , belongs to C k +2 ,α ( R ) and the following estimate is verified (cid:107)N λ ζ (cid:107) C k +2 ,α ( R ) ≤ K (cid:107) ζ (cid:107) C k,α ( R ) . Combining the above results, we can estimate generalized volume potentials in Ω whose densitieshave compact support in Ω by means of an appropriate splitting. Using Calder´on’s extension theorem(Proposition 3.1), for every ζ ∈ C k,αc (Ω) there exists an extension ζ ∈ C k,αc ( R ), so N + λ ζ = (cid:0) N λ ζ (cid:1)(cid:12)(cid:12) Ω − N + λ (cid:0) ζ (cid:12)(cid:12) G (cid:1) in Ω . Then, Lemmas 6.4 and 6.9 lead to the following result:
Theorem 6.10 (Generalized volume potential) . Let G ⊆ R be a bounded domain with regularity C k +1 ,α , Ω := R \ G its exterior domain and S = ∂G the boundary surface. Define the generalized volume potentialin Ω with density ζ : Ω −→ R associated with the Helmholtz equation ( N + λ ζ )( x ) = (cid:90) Ω Γ λ ( x − y ) ζ ( y ) dy, x ∈ Ω . Let
R > be such that G ⊆ B R (0) , and define Ω R := B R (0) \ G . Then, N + λ ζ is well defined in Ω andbelongs to C k +2 ,α (Ω) , for every ζ ∈ C k,α (Ω) such that supp( ζ ) ⊆ Ω R . In addition, the bound (cid:107)N λ ζ (cid:107) C k +2 ,α (Ω) ≤ K (cid:107) ζ (cid:107) C k,α (Ω) holds for some K > depending on k, α, λ, G, R but not on ζ . Regularity of the boundary integral operator.
The next step is to analyze the regularityproperties of the boundary integral operator T λ (44) arising in the boundary integral equation associatedwith the boundary data η × u (43) in Theorem 2.16. Firstly, we split the operator T λ into simpler integraloperators. By inspection, T λ is given by T λ = M Tλ + λ S Tλ . M Tλ ζ is known as the magnetic dipole operator , which is the tangent component of the electric fieldgenerated by a dipole distribution with density ζ ∈ X ( S ), i.e.,( M Tλ ζ )( x ) := (cid:90) S η ( x ) × curl x (Γ λ ( x − y ) ζ ( y )) d y S, x ∈ S. S Tλ is the tangential component of the generalized single layer potential generated by ζ ,( S Tλ ζ )( x ) = (cid:90) S Γ λ ( x − y ) η ( x ) × ζ ( y ) d y S, x ∈ S. The integral kernel of S Tλ is weakly singular over S , so this integral is absolutely convergent undersuitable hypotheses for ζ . The integral kernel of M Tλ looks singular over S but, as we shall see below,this integral is just weakly singular when ζ is a tangent vector field on S . Thus, this integral is actuallyabsolutely convergent under minimal assumptions on ζ . In order to see why, notice that, given anytangent field along S , ζ ∈ X k,α ( S ), one has the decomposition η ( x ) × ( ∇ x Γ λ ( x − y ) × ζ ( y )) = η ( x ) · ζ ( y ) ∇ x Γ λ ( x − y ) − η ( x ) · ∇ x Γ λ ( x − y ) ζ ( y )= ( η ( x ) − η ( y )) · ζ ( y ) ∇ x Γ λ ( x − y ) − η ( x ) · ∇ x Γ λ ( x − y ) ζ ( y ) . Consequently, the j -th coordinates of the integrands read( η ( x ) × ( ∇ x Γ λ ( x − y ) × ζ ( y ))) j = ( η ( x ) − η ( y )) · ζ ( y ) ∂ x j Γ λ ( x − y ) − η ( x ) · ∇ x Γ λ ( x − y ) ζ j ( y )= (cid:88) i =1 ( η i ( x ) − η i ( y )) ζ i ( y ) ∂ x j Γ λ ( x − y ) − η ( x ) · ∇ x Γ λ ( x − y ) ζ j ( y ) , (Γ λ ( x − y ) η ( x ) × ζ ( y )) j = Γ λ ( x − y )( η ( x ) × ζ ( y )) · e j = Γ λ ( x − y )( e j × η ( x )) · ζ ( y ) = (cid:88) i =1 Γ λ ( x − y )( e i × e j ) · η ( x ) ζ i ( y ) . ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 59
Consider any extension (cid:101) η ∈ C k +4 ,αc ( R ) of the outward unit normal vector field η to the compact surface S and define the kernels K D λ ( x, z ) = (cid:101) η ( x ) · ∇ Γ λ ( z ) ,K i,jλ ( x, z ) = ( (cid:101) η i ( x ) − (cid:101) η i ( x − z )) ∂ z j Γ λ ( z ) , (111) (cid:101) K i,jλ ( x, z ) = ( e i × e j ) · (cid:101) η ( x ) Γ λ ( z ) . Then, we have the associated splitting of the operators M Tλ and S Tλ ( M Tλ ζ ) j ( x ) = (cid:88) i =1 T K i,jλ ζ i − T K D λ ζ j , ( S Tλ ζ ) j ( x ) = (cid:88) i =1 T (cid:101) K i,jλ ζ i , (112)where the integral operators in the above decomposition are( T K D λ ζ j )( x ) = (cid:90) S K D λ ( x, x − y ) ζ j ( y ) d y S, ( T K i,jλ ζ i )( x ) = (cid:90) S K i,jλ ( x, x − y ) ζ i ( y ) d y S, (113)( T (cid:101) K i,jλ ζ i )( x ) = (cid:90) S (cid:101) K i,jλ ( x, x − y ) ζ i ( y ) d y S. Since every C compact surface satisfies | η ( x ) · ( x − y ) | ≤ L | x − y | , | η ( x ) − η ( y ) | ≤ L | x − y | , for each x, y ∈ S , then all the preceding integral kernels are weakly singular. In particular, it preventsthese integrals from being considered in the Cauchy principal value sense.The study of H¨older estimates for all these potentials can be performed along the same lines as in [25,Satz 4.3, Satz 4.4]. In that work, the author dealt with the homogeneous harmonic case λ = 0, wherethe kernels have a simpler form. In our case λ (cid:54) = 0, we will decompose the 3-dimensional kernels intoa homogeneous part and an inhomogeneous but less singular part as in (90). Then, we will consider acoordinate system over S which allows transforming the integrals over S into integrals over planar domainsby means of a change of variables. The homogeneous and more singular parts will satisfy the hypothesisin Corollary 6.8 and the terms in the remainder will verify those in Corollary 6.7. Our regularity resultthen reads as follows: Theorem 6.11.
Let G be a bounded domain of class C k +5 , S = ∂G the boundary surface, η ∈ C k +4 ( S, R ) the outward unit normal vector field along S and any extension (cid:101) η ∈ C k +4 c ( R , R ) of η .Let K D λ ( x, z ) , K i,jλ ( x, z ) and (cid:101) K i,jλ ( x, z ) be the kernels given by (111) and T K D λ , T K i,jλ and T (cid:101) K i,jλ the as-sociated boundary integral operators given by (113). Then, these integral operators are bounded from C k,α ( S ) into C k +1 ,α ( S ) , i.e., the following linear operators are continuous: T K D λ : C k,α ( S ) −→ C k +1 ,α ( S ) ,T K i,jλ : C k,α ( S ) −→ C k +1 ,α ( S ) ,T (cid:101) K i,jλ : C k,α ( S ) −→ C k +1 ,α ( S ) . As a consequence, the linear operators M Tλ : X k,α ( S ) −→ X k +1 ,α ( S ) , S Tλ : X k,α ( S ) −→ X k +1 ,α ( S ) are bounded too. Remark 6.12.
The above regularity assumptions on the boundary surface will be discussed during theproof of the theorem. Roughly speaking, we will need C k +5 boundaries for the operators in (113) of firstand second type to be bounded from C k,α ( S ) to C k +1 ,α ( S ) whilst assuming C k +4 boundaries suffices toensure the corresponding result for the third kind of operators in (113). See [25, Satz 4.3, Satz 4.4] forthe homogeneous harmonic case λ = 0 . Let us recall that a similar formalism was introduced in [37] to deal with weakly singular operators whose homogeneous kernels are of Calder´on–Zygmund type aftera finite amount of derivatives is taken. The pseudo-homogeneous kernels are those that can be split into finitely many homogeneous weakly singular kernels of the preceding type and an arbitrarily regularremainder. Its associated integral operators gain m derivatives on any Sobolev space with finite exponent < p < ∞ , − m being the class of the pseudo-homogeneous kernel (see [37, Chapter 4, Section 3] formore details). In particular, [37, Example 4.3] shows that (cid:101) K i,jλ ( x, z ) is pseudo-homogeneous of class − and the aforementioned regularity result shows that S Tλ is bounded from W r,p ( S ) to W r +1 ,p ( S ) . Thisapproach involves splitting the exponential function in Γ λ ( z ) on its Taylor series, giving rise to finitelymany homogeneous kernels and a regular enough remainder. Since (cid:101) K i,jλ ( x, z ) has separated variables,it could be approached within this framework, but one cannot say the same about other of the kernelsin (111). On the contrary, the ideas in [25] work well for H¨older regularity using singular and weaklysingular kernels like those in Corollaries 6.7 and 6.8.Proof. Since the kernel (cid:101) K i,jλ ( x, z ) can be analyzed through a similar reasoning (as shown in [25] for thecase λ = 0), we will restrict out analysis to the kernels K i,jλ ( x, z ) and K D λ ( x, z ), which were not studiedin [25]. Let us then split these inhomogeneous kernels into a homogeneous part and some less singularpart (see the decomposition (90) and the functions φ λ and ψ λ in (88)). To this end, notice that K D λ ( x, z ) = φ (cid:48) λ ( | z | ) | z | (cid:101) η ( x ) · z,K i,jλ ( x, z ) = ( (cid:101) η i ( x ) − (cid:101) η i ( x − z )) φ (cid:48) λ ( | z | ) | z | z j , (114)Consequently, K i,jλ ( x, z ) = K i,jλ, + K i,jλ, ,K D λ ( x, z ) = K D λ, + K D λ, , (115)where, K i,jλ, ( x, z ) := − π ( (cid:101) η i ( x ) − (cid:101) η i ( x − z )) z j | z | , K i,jλ, ( x, z ) := ( (cid:101) η i ( x ) − (cid:101) η i ( x − z )) ψ (cid:48) λ ( | z | ) | z | z j ,K D λ, ( x, z ) := − π (cid:101) η ( x ) · z | z | , K D λ, ( x, z ) := (cid:101) η ( x ) · ψ (cid:48) λ ( | z | ) z | z | . (116)Notice that the associated integral operators only involve values x, y ∈ S . Define d S := 2 max x,y ∈ S | x − y | and let us take x ∈ S and z ∈ B d S (0), so in this case we have1 | z | β + 1 | z | β ≤ (cid:0) d M − mS (cid:1) | z | M , | z | β + | z | β ≤ (cid:0) d M − mS (cid:1) | z | m (117)for any couple of exponents β , β ≥ z ∈ B d S (0). Here m and M stand for the minimum andmaximum values i.e., m := min { β , β } , M := max { β , β } . Consider the function arising in (116), f λ ( r ) := ψ (cid:48) λ ( r ) r , r > , and note that (91) leads to | f ( m ) λ ( r ) | ≤ C (cid:18) r + 1 r m +2 (cid:19) , r > , | f ( m ) λ ( r ) | ≤ (cid:101) C r m +2 , r ∈ (0 , d S ) . (118)for some C > m and some (cid:101) C depending on m and d S .Let us study the boundedness of the integral operators associated with the integral kernels K i,jλ,n and K D λ,n for n = 0 ,
1. To this end, let us consider a finite covering of S by M coordinate neighborhoodsΣ , . . . , Σ M ⊆ S endowed with the associated local charts µ m : D m −→ Σ m belonging to C k +5 ( D m , R )and enjoying homeomorphic extensions up to the boundary of the planar disks D m ⊆ R . Also consider ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 61 the associated partition of unity of class C k +5 , i.e., { ϕ m } Mm =1 ⊆ C k +5 ( S ) such that supp ϕ m ⊆ Σ m foreach index m = 1 , . . . , M and 1 = M (cid:88) m =1 ϕ m ( x ) , for any x ∈ S. We will denote the Jacobian of each local chart µ m by J m ( s ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂µ m ∂s × ∂µ m ∂s (cid:19) ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:114) det (cid:16) g ijm ( s ) (cid:17) , s ∈ D m . Here g ijm ( s ) stands for the ( i, j ) component of the induced Euclidean metric on S with respect to the localchart µ m , i.e., (cid:0) g ijm ( s ) (cid:1) = (cid:18) E m ( s ) F m ( s ) F m ( s ) G m ( s ) (cid:19) , where E m , F m , G m stands for the coefficients of the first fundamental form, namely, E m ( s ) := ∂µ m ∂s ( s ) · ∂µ m ∂s ( s ) , F m ( s ) := ∂µ m ∂s ( s ) · ∂µ m ∂s ( s ) , G m ( s ) := ∂µ m ∂s ( s ) · ∂µ m ∂s ( s ) . Consequently,( T K i,jλ,n ζ )( µ m ( s )) = M (cid:88) m (cid:48) =1 (cid:90) D m (cid:48) K i,jλ,n ( µ m ( s ) , µ m ( s ) − µ m (cid:48) ( t )) ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt, (119)( T K D λ,n ζ )( µ m ( s )) = M (cid:88) m (cid:48) =1 (cid:90) D m (cid:48) K D λ,n ( µ m ( s ) , µ m ( s ) − µ m (cid:48) ( t )) ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt. (120)We will study the most singular case m (cid:48) = m and then show how the case m (cid:48) (cid:54) = m follows from it. Animportant fact is that we will extract the most singular homogeneous parts of K i,jλ, ( x, z ) and K D λ, ( x, z )by virtue of the splitting (115). However, the change of variables in the coordinate neighborhoods Σ m gives rise to new inhomogeneous planar kernels, K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( t )) and K D λ, ( µ m ( s ) , µ m ( s ) − µ m ( t )) . To solve this difficulty, we will decompose them again into the more sigular homogeneous part, whichstands for a planar homogeneous kernel of degree −
1, and some inhomogeneous but less singular term.Then, we will prove the corresponding regularity results for each term through Corollaries 6.7 and 6.8.Since both K i,jλ, ( x, z ) and K D λ, ( x, z ) can be studied by means of a similar reasoning, we will justanalyze one of them, e.g. K i,jλ, ( x, z ). In fact, K D λ, ( x, z ) stands for the integral kernel of the adjointoperator of the harmonic Neumann–Poincar´e operator and was studied in [25, Satz 4.4]. Inspired by [25,Lemma 4.2], let us expand µ m ( s ) − µ m ( t ) though the integral form of Taylor’s theorem up to secondorder, | µ m ( s ) − µ m ( t ) | = ( P m ( s, s − t ) + Q m ( s, s − t )) / , (121)where P m ( s, u ) := (cid:88) p,q =1 ∂µ m ∂s p ( s ) · ∂µ m ∂s q ( s ) u p u q = (cid:88) p,q =1 g pqm ( s ) u p u q = (( g pqm ( s )) u ) · u, (122) Q m ( s, u ) := − (cid:88) p,q,r =1 ∂µ m ∂s p ( s ) · (cid:18)(cid:90) (1 − θ ) ∂ µ m ∂s q ∂s r ( s − θu ) dθ (cid:19) u p u q u r + (cid:88) p,q,r,l =1 (cid:18)(cid:90) (1 − θ ) ∂ µ m ∂s p ∂s q ( s − θu ) dθ (cid:19) · (cid:18)(cid:90) (1 − θ ) ∂ µ m ∂s r ∂s l ( s − θu ) dθ (cid:19) u p u q u r u l . (123)Straightforward computations shows that P m ( s, u ) is positively homogeneous on u of degree 2, i.e., P m ( s, ρu ) = ρ P m ( s, u ) , (124) for all s ∈ D m , u ∈ R \{ } and ρ >
0. Moreover, the estimates1 C | u | ≤ | P m ( s, u ) | ≤ C | u | , | Q m ( s, u ) | ≤ C | u | , C | u | ≤ | P m ( s, u ) + Q m ( s, u ) | ≤ C | u | . | D γs P m ( s, u ) | ≤ C | u | , | D γs Q m ( s, u ) | ≤ C | u | , | D γs ( P ( s, u ) + Q ( s, u )) | ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u i D γs P m ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u i D γs Q m ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u i D γs ( P ( s, u ) + Q ( s, u )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂u i ∂u j D γs P m ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂u i ∂u j D γs Q m ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂u i ∂u j D γs ( P ( s, u ) + Q ( s, u )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | . (125)hold for each s ∈ D m , u ∈ R such that s − u ∈ D m and every multi-index with | γ | ≤ k . See [25, Satz4.2] for the details, which are starightforward.Our homogenization procedure follows from the next splitting, where ( (cid:101) η ◦ µ m ) i and ( µ m ) j are expandedagain by means of the integral form of Taylor’s theorem up to second order K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( t )) = H i,jλ, ( s, s − t ) + R i,jλ, ( s, s − t ) , where the homogeneous part H i,jλ, ( s, u ) and the remainder R i,jλ, ( s, u ) take the form H i,jλ, ( s, u ) := − π P m ( s, u ) − / (cid:88) p,q =1 ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) ∂ ( µ m ) j ∂s q ( s ) u p u q ,R i,jλ, ( s, u ) := (cid:101) R i,jλ, ( s, u ) + (cid:98) R i,jλ, ( s, u ) , and the remainder is split into (cid:101) R i,jλ, ( s, u ) := − π (cid:16) ( P m ( s, u ) + Q m ( s, u )) − / − P m ( s, u ) − / (cid:17) × (cid:32) (cid:88) p,q =1 ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) ∂ ( µ m ) j ∂s q ( s ) u p u q (cid:33) , (cid:98) R i,jλ, ( s, u ) := − π ( P m ( s, u ) + Q m ( s, u )) − / × (cid:40) − (cid:88) p,q,r =1 (cid:18)(cid:90) (1 − θ ) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ∂s q ( s − θu ) dθ (cid:19) ∂ ( µ m ) j ∂s r ( s ) u p u q u r − (cid:88) p,q,r =1 ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) (cid:18)(cid:90) (1 − θ ) ∂ ( µ m ) j ∂s q ∂s r ( s − θu ) dθ (cid:19) u p u q u r + (cid:88) p,q,r,l =1 (cid:18)(cid:90) (1 − θ ) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ∂s q ( s − θu ) dθ (cid:19) ×× (cid:18)(cid:90) (1 − θ ) ∂ ( µ m ) j ∂s r ∂s l ( s − θu ) dθ (cid:19) u p u q u r u l (cid:27) . Note again that only small values of u = s − t are involved here; specifically, s ∈ D m and u ∈ D d mS (0) for d mS := 2 max s,t ∈ D m | s − t | . Hence, one enjoy similar bounds to those in (117) with z replaced with u .Let us next analyze each term in the above decomposition for K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( t )). Firstly,since P m ( s, u ) is positively homogeneous on u with degree 2, then H i,jλ, ( s, u ) is positively homogeneous on u with degree −
1. The regularity properties in the second part in Corollary 6.8 can be straighforwardly
ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 63 checked. Let us then concentrate on the regularity properties in the third part of such corollary and, tothis end, let us compute the next partial derivative D γs H i,jλ, ( s, u ) = − π (cid:88) σ ≤ γ (cid:18) γσ (cid:19) D σs (cid:16) P m ( s, u ) − / (cid:17) (cid:34) (cid:88) p,q =1 D γ − σs (cid:18) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) ∂ ( µ m ) j ∂s q ( s ) (cid:19) u p u q (cid:35) . Define the homogeneous function h ( t ) := t − / , t > D σs (cid:16) P m ( s, u ) − / (cid:17) = (cid:88) ( l,β,δ ) ∈D ( σ ) ( D δ h )( P m ( s, u )) l (cid:89) r =1 δ r ! (cid:18) β r ! D β r s P m ( s, u ) (cid:19) δ r . Recall that ( l, β, δ ) ∈ D ( σ ) stands for the decompositions σ = l (cid:88) r =1 | δ r | β r , where β = ( β , . . . , β l ), δ = (cid:80) lr =1 δ r and for each r ∈ { , . . . , l − } there exists some i r ∈ { , } suchthat ( β r ) i r < ( β r +1 ) i r and ( β r ) i = ( β r +1 ) i for every i (cid:54) = i r .By virtue of (125), the derivatives with respect to s behave as (cid:12)(cid:12)(cid:12) D γs H i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12) ≤ C | u | . Let us take derivatives with respect to u and arrive at ∇ u D γs H i,jλ, ( s, u )= − π (cid:88) σ ≤ γ (cid:18) γσ (cid:19) ∇ u D ηs (cid:16) P m ( s, u ) − / (cid:17) (cid:34) (cid:88) p,q =1 D σ − γs (cid:18) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) ∂ ( µ m ) j ∂s q ( s ) (cid:19) u p u q (cid:35) − π (cid:88) σ ≤ γ (cid:18) γσ (cid:19) D σs (cid:16) P m ( s, u ) − / (cid:17) (cid:34) (cid:88) p,q =1 D σ − γs (cid:18) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) ∂ ( µ m ) j ∂s q ( s ) (cid:19) ∇ u ( u p u q ) (cid:35) , They can be similarly estimated by means of (125): (cid:12)(cid:12)(cid:12) D γs ∇ u H i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12) ≤ C | u | . Thus, H i,jλ, has the regularity properties required in Corollary 6.8, so (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) D m H i,jλ, ( s, s − t ) ϕ m ( µ m ( t )) ζ ( µ m ( t )) J m ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( D m ) ≤ M (cid:107) ζ (cid:107) C k,α (Σ m ) . Let us now move to the remainder R i,jλ, ( s, u ) and show that the hypoteses in Corollary 6.7 are satisfiedtoo. On the one hand, in the first term (cid:101) R i,jλ, ( s, u ) in R i,jλ, ( s, u ) one can rearranged terms as( P m ( s, u ) + Q m ( s, u )) − / − P m ( s, u ) − / = (cid:90) ddθ ( P m ( s, u ) + θQ m ( s, u )) − / dθ = − Q m ( s, u ) (cid:90) ( P m ( s, u ) + θQ m ( s, u )) − / dθ. Therefore, a D γs derivative of (cid:101) R i,jλ, ( s, u ) takes the form D γs (cid:101) R i,jλ, ( s, u ) = 14 π (cid:88) σ ≤ γ (cid:18) γσ (cid:19) D σs (cid:18) Q m ( s, u ) (cid:90) ( P m ( s, u ) + θQ m ( s, u )) − / dθ (cid:19) × (cid:88) p,q =1 D γ − σ (cid:18) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) ∂ ( µ m ) j ∂s q ( s ) (cid:19) u p u q . If we consider (cid:101) h ( t ) = t − / , a similar argument shows that D σs (cid:18) Q m ( s, u ) (cid:90) ( P m ( s, u ) + θQ m ( s, u )) − / dθ (cid:19) = (cid:88) ρ ≤ σ (cid:18) σρ (cid:19) D ρs ( Q m ( s, u )) (cid:90) D σ − ρs (cid:16) ( P m ( s, u ) + θQ m ( s, u )) − / (cid:17) dθ = (cid:88) ρ ≤ σ (cid:18) σρ (cid:19) D ρs ( Q m ( s, u )) (cid:90) (cid:88) ( l,β,δ ) ∈D ( σ − ρ ) ( D δ (cid:101) h )( P m ( s, u ) + θQ m ( s, u )) × l (cid:89) r =1 δ r ! (cid:18) β r ! D β r s ( P m ( s, u ) + θQ m ( s, u )) (cid:19) δ r dθ. Now, the estimates in (125) yields (cid:12)(cid:12)(cid:12) D γs (cid:101) R i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u l D γs (cid:101) R i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂u l ∂u l D γs (cid:101) R i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | . These estimates ensure that all the hypotheses in Corollary 6.7 are satisfied, so (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) D m (cid:101) R i,jλ, ( s, s − t ) ϕ m ( µ m ( t )) ζ ( µ m ( t )) J m ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( D m ) ≤ M (cid:107) ζ (cid:107) C k,α (Σ m ) . Regarding the second term (cid:98) R i,jλ, ( s, u ) of R i,jλ, ( s, u ) we can use a similar argument. First, the formulafor the D γs derivative D γs (cid:98) R i,jλ, ( s, u ) = 14 π (cid:88) σ ≤ γ (cid:18) γσ (cid:19) D σs (cid:16) ( P m ( s, u ) + Q m ( s, u )) − / (cid:17) × (cid:40) (cid:88) p,q,r =1 D γ − σs (cid:18)(cid:18)(cid:90) (1 − θ ) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ∂s q ( s − θu ) dθ (cid:19) ∂ ( µ m ) j ∂s r ( s ) (cid:19) u p u q u r + (cid:88) p,q,r =1 D γ − σs (cid:18) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ( s ) (cid:18)(cid:90) (1 − θ ) ∂ ( µ m ) j ∂s q ∂s r ( s − θu ) dθ (cid:19)(cid:19) u p u q u r − (cid:88) p,q,r,l =1 D γ − σs (cid:18)(cid:18)(cid:90) (1 − θ ) ∂ ( (cid:101) η ◦ µ m ) i ∂s p ∂s q ( s − θu ) dθ (cid:19) ×× (cid:18)(cid:90) (1 − θ ) ∂ ( µ m ) j ∂s r ∂s l ( s − θu ) dθ (cid:19)(cid:19) u p u q u r u l (cid:27) , and the chain rule for high order derivatives lead to D σs (cid:16) ( P m ( s, u ) + Q m ( s, u )) − / (cid:17) = (cid:88) ( l,β,δ ) ∈D ( σ ) ( D δ h )( P m ( s, u ) + Q m ( s, u )) l (cid:89) r =1 δ r ! (cid:18) β r ! D β r s ( P m ( s, u ) + Q m ( s, u )) (cid:19) δ r . Consequently, the estimates in (125) show that (cid:12)(cid:12)(cid:12) D γs (cid:98) R i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u l D γs (cid:98) R i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂u l ∂u l D γs (cid:98) R i,jλ, ( s, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , and Corollary 6.7 yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) D m (cid:98) R i,jλ, ( s, s − t ) ϕ m ( µ m ( t )) ζ ( µ m ( t )) J m ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( D m ) ≤ M (cid:107) ζ (cid:107) C k,α (Σ m )ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 65 Now we move to K i,jλ, ( x, z ) and expand (cid:101) η i ◦ µ m and ( µ m ) j through Taylor’s theorem in integral formup to first order K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( s − u ))= f λ ( | P m ( s, u ) + Q m ( s, u ) | / ) (cid:88) p,q =1 (cid:18)(cid:90) ∂ ( (cid:101) η i ◦ µ m ) ∂s q ( s − θu ) dθ (cid:19) (cid:18)(cid:90) ∂ ( µ m ) j ∂s q ( s − θu ) dθ (cid:19) u p u q . Then, the D γs derivative of K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( t )) takes the form D γs K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( s − u ))= (cid:88) σ ≤ γ (cid:18) γσ (cid:19) D σs (cid:16) f λ ( | P m ( s, u ) + Q m ( s, u ) | / ) (cid:17) × (cid:34) (cid:88) p,q =1 D γ − σs (cid:18)(cid:18)(cid:90) ∂ ( (cid:101) η i ◦ µ m ) ∂s q ( s − θu ) dθ (cid:19) (cid:18)(cid:90) ∂ ( µ m ) j ∂s q ( s − θu ) dθ (cid:19)(cid:19) u p u q (cid:35) . Again, by the chain derivative formula we arrive at D σs (cid:16) f λ (( P m ( s, u ) + Q m ( s, u )) / ) (cid:17) = (cid:88) ( l,β,δ ) ∈D ( σ ) D δ ( f λ ( · / )) (cid:12)(cid:12)(cid:12) P m ( s,u )+ Q m ( s,u ) l (cid:89) r =1 δ r ! (cid:18) β r ! D β r s ( P m ( s, u ) + Q m ( s, u )) (cid:19) δ r . Notice that (118) leads to (cid:12)(cid:12)(cid:12)(cid:12) d k dr k (cid:16) f λ ( r / ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:101) C r k +1 , ∀ r ∈ (0 , d mS ) . Consequently, (125) proves the upper bounds (cid:12)(cid:12)(cid:12) D γs K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( s − u )) (cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂∂u l D γs K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( s − u )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂u l ∂u l D γs K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( s − u )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | u | , so the hypotheses in Corollary 6.7 are satisfied and (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) D m K i,jλ, ( µ m ( s ) , µ m ( s ) − µ m ( t )) ϕ m ( µ m ( t )) ζ ( µ m ( t )) J m ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( D m ) ≤ M (cid:107) ζ (cid:107) C k,α (Σ m ) . In order to complete the proof of the theorem, let us show how to deal with the terms m (cid:48) (cid:54) = m in (119)and (120). The idea is to obtain estimates over Σ m ∩ Σ m (cid:48) and Σ m \ Σ m (cid:48) separately. First, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) D m (cid:48) K i,jλ ( µ m ( s ) , µ m ( s ) − µ m (cid:48) ( t )) ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( µ − m (Σ m ∩ Σ m (cid:48) )) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) D m (cid:48) K i,jλ ( µ m (cid:48) ( s ) , µ m (cid:48) ( s ) − µ m (cid:48) ( t )) ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k +1 ,α ( D m (cid:48) ) ≤ CM (cid:107) ζ (cid:107) C k,α (Σ m (cid:48) ) . Second, define C m (cid:48) := µ − m (cid:48) (supp ϕ m (cid:48) ) , K m (cid:48) := µ m (cid:48) ( C m (cid:48) ) and d m,m (cid:48) := dist (cid:0) Σ m \ Σ m (cid:48) , K m (cid:48) (cid:1) > , as showed in Figure 9. This avoids the singularity near z = 0 in the preceding kernels. Hence, we cantake derivatives under the integral sign, obtaining the formula D γs (cid:90) D m (cid:48) K i,jλ ( µ m ( s ) ,µ m ( s ) − µ m (cid:48) ( t )) ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt = (cid:88) ( l, ( δ ,δ ) ,β ) ∈D ( γ ) (cid:90) C m (cid:48) (cid:16) D δ x D δ z K i,jλ (cid:17) ( µ m ( s ) , µ m ( s ) − µ m (cid:48) ( t )) S Σ m = µ m ( D m )Σ m ′ = µ m ′ ( D m ′ ) K m ′ = µ m ′ ( C m ′ ) d m,m ′ Figure 9.
Overlapping coordinate neighborhoods Σ m and Σ m (cid:48) . × (cid:34) l (cid:89) r =1 δ r ! δ r ! (cid:18) β r ! D β r µ m ( s ) (cid:19) δ r (cid:18) β r ! D β r µ m ( s ) (cid:19) δ r (cid:35) × ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt. for each s ∈ µ − m (Σ m \ Σ m (cid:48) ). Since | D δ x D δ z K i,jλ ( x, z ) | ≤ (cid:101) C | z | | δ | , for every z ∈ B d m,m (cid:48) (0), then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D γs (cid:90) D m (cid:48) K i,jλ ( µ m ( s ) , µ m ( s ) − µ m (cid:48) ( t )) ϕ m (cid:48) ( µ m (cid:48) ( t )) ζ ( µ m (cid:48) ( t )) J m (cid:48) ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cd | γ | | C m (cid:48) |(cid:107) ζ (cid:107) C (Σ m (cid:48) ) . Here, 0 < d < d < d m,m (cid:48) for every m (cid:48) (cid:54) = m . Since one can take any | γ | ≤ k + 2 by theregularity of S , then we obtain the desired estimate for m (cid:48) (cid:54) = m and the result follows. (cid:3) Appendix A. Gradient, curl and divergence on surfaces
In this Appendix we record some well known formulas for the gradient, curl and divergence operatorson a compact surface S ⊆ R . These formulas have been useful in several sections to analyze boundaryintegrals. This is particularly true in the case of Lemma 2.15.Let us consider the vector spaces of smooth tangent vector fields along S and smooth 1-forms, i.e., X ( S )and Ω ( S ) respectively. It is well known that these vector spaces can be identified using the Riemannianmetric on S by virtue of the musical isomorphisms (cid:91) : X ( S ) −→ Ω ( S ) , (cid:93) : Ω ( S ) −→ X ( S ) X (cid:55)−→ X (cid:91) , α (cid:55)−→ α (cid:93) . These are defined as X (cid:91) ( Y ) = X · Y, α (cid:93) · X = α ( X ) . for any given X, Y ∈ X ( S ) and α ∈ Ω ( S ).The gradient vector field over S of any function f ∈ C ( S ) can be identified with the exterior differential1-form over S through the musical isomorphisms: ∇ S f := ( d S f ) (cid:93) . If f ∈ C ( R ) is any extension of f , it turns out that ∇ S f is the tangential component to the surface ofthe R gradient field ∇ f , that is, ∇ S f = − η × ( η × ∇ f ) on S. ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 67
Now, we focus on the divergence and curl of a tangent vector field X ∈ X ( S ). They can be distribu-tionally defined by the identity (cid:90) S div S ( X ) ϕ dS = − (cid:90) S X · ∇ S ϕ dS, ∀ ϕ ∈ C ∞ ( S ) , (cid:90) S curl S ( X ) ϕ dS = − (cid:90) S X · ( η × ∇ S ϕ ) dS, ∀ ϕ ∈ C ∞ ( S ) . Another way to provide a coordinate-free expression for div S and curl S is through the Hodge star operator ∗ and the codifferential δ S . Recall that ∗ acts on each k -forms space Ω k ( S ) as the bijection ∗ : Ω k ( S ) −→ Ω − k ( S ) , given by α ∧ ∗ β = α · β area S , where area S stands for the Riemannian area 2-form on S and α, β ∈ Ω ( S ). The dot symbol here is thepointwise inner product of k -forms induced by the musical isomorphisms. Its inverse can be computedthought the next classical formula ∗∗ = ( − k (2 − k ) I in Ω k ( S ) . Analogously, δ S : Ω k ( S ) −→ Ω k − ( S ) , acts on each k -forms space Ω k ( S ) as δ S α := ( − k − ( ∗ d S ∗ ) α. Recall that δ S is the adjoint of d S . Specifically, for any α ∈ Ω ( S ) and ϕ ∈ C ( S ) one has (cid:90) S ϕ δ S α dS = (cid:90) S d S ϕ · α dS, where the above pointwise inner product is the one induced by the Riemannian metric in S through themusical isomorphisms, i.e., (cid:90) S ϕ δ S α dS = (cid:90) S ( d S ϕ ) (cid:93) · α (cid:93) dS = (cid:90) S ∇ S ϕ · α (cid:93) dS. As a consequence, take any couple X ∈ X ( S ) and ϕ ∈ C ( S ) and note that (cid:90) S div S ( X ) ϕ dS = − (cid:90) S ∇ S ϕ · X, dS = − (cid:90) S δ S ( X (cid:91) ) ϕ dS. Consequently, div S X = − δ S ( X (cid:91) ) = ( ∗ d S ∗ )( X (cid:91) ) on S. With curl S we can also argue as above to arrive at the analogous formulacurl S X = ( ∗ d S )( X (cid:91) ) on S. To conclude, let us list a few useful identities that follow from the definition of ∇ S , div S and curl S : Proposition A.1. (1) curl S ( X ) = − div S ( η × X ) ∀ X ∈ X ( S ) . (2) curl S ( − η × ( η × F )) = η · curl F ∀ F ∈ C (Ω) . (3) curl S ( ∇ S f ) = 0 ∀ f ∈ C ( S ) . (4) div S ( η × ∇ S f ) = 0 ∀ f ∈ C ( S ) . (5) (Poincar´e’s lemma) Assume that S is simply connected and consider any tangent vector field X ∈ X ( S ) such that curl S ( X ) = 0 . Then, there exists some f ∈ C ( S ) such that X = ∇ S f . Appendix B. Obstructions to the existence of generalized Beltrami fields
In this Appendix we will review the main results on the non-existence of Beltrami fields with a non-constant factor proved in [20], as they are of direct interest for the theorems that we have presented inthis paper.Hence, let us consider in this Appendix a solution to the Beltrami field equation with a factor f :curl u = f u , div u = 0 . (126)We will not specify the domain of the solution as the results that we will review are mostly local. Thekey observation is that, as the divergence of u is zero, f is a first integral of u : u · ∇ f = 0 . Since this first integral condition is very restrictive, it stands to reason that Equation (126) shouldnot admit any nontrivial solutions for most functions f . Before we make this idea precise in the nextparagraphs, let us point out that the (well established) idea of constructing the iterations starting bydragging a function along the integral curves of a field, as we have done in the main body of this work,is fully consistent with the intuition that the first integral condition is the heart of the matter.The first obstruction to the existence of solutions to the Beltrami equation (126) is the following: Theorem B.1.
Let D ⊆ R be a domain and assume that the function f is nonconstant and of class C ,α .Suppose that the vector field u satisfies the Eq. (126) in D . Then there is a nonlinear partial differentialoperator P (cid:54) = 0 , which can be computed explicitly and involves derivatives of order at most , such that u ≡ unless P [ f ] is identically zero in D . In particular, u ≡ for all f in a set of infinite codimensionof C k,α ( U ) with any k ≥ . It should be noticed that Theorem B.1 is of a purely local nature, as it provides obstructions for theexistence of nontrivial Beltrami fields in any open set and most proportionality factors.A less powerful but more easily visualized obstruction is that if f has a regular level set homeomorphicto the sphere, then Equation (126) does not have any nontrivial solutions. In particular, there are noBeltrami fields whenever f has local extrema or is a radial function. This is related to the classicaltheorem of Cowling ensuring that there are no poloidal Beltrami fields with nonconstant factor and axialsymmetry [4]: Theorem B.2.
Suppose that the function f is of class C ,α in a domain D ⊆ R . If a regular level set f − ( c ) has a connected component in D homeomorphic to the sphere, then any solution to Equation (126) in D is identically zero. Although we will not repeat here the proof of these results, which can be found in [20], let us give a fewwords on the main idea. The proof of these theorems is based on formulating the Beltrami equation (126)as a constrained evolution problem. Indeed, one can show that Equation (126) is locally equivalent, in asense that can be made precise, to the assertion that there is a time-dependent 1-form β ( t ) on a surfaceΣ that satisfies the evolution equation ∂ t β = T ( t ) β (127)together with the differential constraint dβ = 0 . (128)Here T ( t ) is a time-dependent tensor field that depends on f and the exterior differential d is computedwith respect to the coordinates on the surface Σ, which, in turn, is a regular level set of f . It should bestressed that this formulation depends strongly on the choice of coordinates.This formulation lays bare the reason for which the Beltrami equation does not generally admit nonzerosolutions: the evolution (127) is not generally compatible with the constraint (128), and the resultingcompatibility conditions translate into equations that f and its derivatives must satisfy. In Theorems B.1and B.2 we have presented the first two of these compatibility conditions, but in fact the method of proofyields a whole hierarchy of explicitly computable obstructions (with increasingly cumbersome expressions)to the existence of solutions. To ascertain how many of these obstructions are actually independentremains an interesting open problem.Furthermore, the above formulation provides an appealing explanation, without even resorting tothe statement of the previous theorems, of the reason for which the attempts at constructing solutionsto (126) using variational techniques have failed: while the regularity of the equation is indeed determinedby an elliptic system, its existence is in fact controlled by a constrained evolution problem for which theexistence theory is ill posed. ENERALIZED BELTRAMI FIELDS AND VORTEX STRUCTURES IN THE EULER EQUATIONS 69
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