Capillary-gravity water waves with discontinuous vorticity: existence and regularity results
aa r X i v : . [ m a t h . A P ] N ov CAPILLARY-GRAVITY WATER WAVES WITH DISCONTINUOUSVORTICITY: EXISTENCE AND REGULARITY RESULTS
ANCA–VOICHITA MATIOC AND BOGDAN–VASILE MATIOC
Abstract.
In this paper we construct periodic capillarity-gravity water waves with an arbi-trary bounded vorticity distribution. This is achieved by reexpressing, in the height functionformulation of the water wave problem, the boundary condition obtained from Bernoulli’sprinciple as a nonlocal differential equation. This enables us to establish the existence of weaksolutions of the problem by using elliptic estimates and bifurcation theory. Secondly, we in-vestigate the a priori regularity of these weak solutions and prove that they are in fact strongsolutions of the problem, describing waves with a real-analytic free surface. Moreover, assumingmerely integrability of the vorticity function, we show that any weak solution corresponds toflows having real-analytic streamlines. Introduction
This paper is concerned with periodic capillary-gravity water waves traveling over a homoge-neous fluid and having an arbitrary bounded vorticity distribution. Our study is motivated bythe physical setting of wind generated waves which possess a thin layer of high vorticity [39],or even high vorticity regions beneath the wave crests [37]. On the other hand, in the near-bedregion there may exist strong tidal currents which interact with the water waves and contributeso to the transportation of sediments [38]. The plethora of phenomena resulting from the wave-current interactions makes the study of rotational water waves so interesting, cf. [5, 26, 42].Indeed, for irrotational waves in the absence of an underlying current the fluid velocity, thepressure, and the particle paths in the flow present very regular features that can be describedqualitatively even for waves of large amplitude (see [3, 4, 43]). However, already within thesetting of irrotational steady waves with an underlying uniform current one encounters newparticle path patterns, cf. [10, 22], while the behavior of the velocity field and of the pressureis considerably altered by an underlying current of constant non-zero vorticity. For rotationalwaves the most dramatic changes (in the form of critical layers) are triggered by the presenceof stagnation points in the flow but even in the absence of stagnation points significant changesoccur (see [41]). A discontinuous vorticity enhances these departures from features that holdwithin the irrotational regime, as indicated by the numerical simulations in [27, 28].On the basis of a rigorous theory, exact periodic gravity water waves with a discontinuousvorticity have been shown to exist in [11] by making use of a weak formulation of the waterwave problem. Subsequently, capillary-gravity water waves interacting with several verticallysuperposed and linearly sheared currents of different vorticities have been constructed in [31],by regarding the height function formulation of the hydrodynamical problem as a diffractionproblem. We develop herein a rigorous existence theory for capillary-gravity water waves witha bounded general vorticity function, some of the analysis in [31] serving as a preliminary step.
Mathematics Subject Classification.
Key words and phrases.
Local bifurcation; bounded vorticity; capillarity-gravity waves; real-analyticstreamlines.
The existence of exact capillary-gravity water waves was first established in the irrotationalsetting [23, 24, 25, 40], the existence theory for rotational waves being developed more recentlyin the setting of waves with constant vorticity, stagnation points, and possibly with overhangingprofiles [30] (see also [12, 32]), or for waves with a general Hölder continuous vorticity distribution[44]. Many papers are also dedicated to the study of the properties of capillary-gravity waterwaves and of the flow beneath them, such as the regularity of the wave profile and that of thestreamlines [19, 20, 2, 33, 45], or the description of the particle paths [18].The first goal of this paper is to establish the existence of two-dimensional capillary-gravitywater waves with an arbitrary bounded vorticity and without stagnation points. This is achievedby using the height function formulation of the water wave problem and by defining a suitablenotion of weak solution for this problem. Reexpressing the boundary condition obtained fromBernoulli’s law as a nonlocal boundary condition, we obtain a new equivalent formulation of theproblem which enables us to consider the existence problem of weak solutions in an abstractbifurcation setting. Using elliptic theory [17] and local bifurcation tools [13], we then establishthe existence of infinitely many bifurcation branches consisting of non-laminar weak solutions ofthe hydrodynamical problem. Our second goal is to determine the a priori regularity propertiesof the weak solutions in the case when the vorticity function is merely integrable. This problemis in the setting of rotational waves very recent [8, 14], but its implications are very importantwhen studying the symmetry properties of water waves. More precisely, in view of [34, Theorem3.1 and Remark 3.2] and [15, Corollary 1.2] the following statement holds true:
Within the set of all periodic gravity waves without stagnation points the symmetricwaves with one crest and trough per period are characterized by the property that all thestreamlines have a global minimum on the same vertical line.
We emphasize that the gravity waves with only one crest and trough per period are symmetricwaves [6, 7, 36]. The availability of Schauder estimates for the new formulation of the problemstands at the basis of our regularity result where we state that the streamlines and the waveprofiles corresponding to such weak solutions are real-analytic graphs. As a particular case, weestablish the real-analyticity of the streamlines also for pure capillary water waves, generalizingprevious results [21, 2]. Our regularity result could serve as a tool when studying the symmetryof waves with capillary effects, the symmetry problem being in this setting still open. Besides,the additional regularity properties of the weak solutions help us to prove that the weak solutionsthat we have found are in fact strong solutions, and even classical if the vorticity function iscontinuous.The outline of the paper is as follows: we start the Section 2 by presenting the mathematicalmodel and state at the end the main existence result Theorem 2.1. In Section 3 we derive anew formulation of the water wave problem, which we recast in Section 4 as a nonlinear andnonlocal problem. After proving the existence of local bifurcation branches of weak solutionsfor the latter problem in Theorem 4.6, we study in Section 5 the a priori regularity of the weaksolutions when assuming merely integrability of the vorticity function, cf. Theorem 5.1 andCorollary 5.2. We conclude the paper with the proof of Theorem 2.1.2.
The mathematical model and the existence result
The mathematical model.
We start by presenting three equivalent mathematical modelswhich describe the propagation of periodic water waves over a rotational, inviscid, and incom-pressible fluid, under the influence of gravity and capillary forces. The waves that we considerare two-dimensional and they travel at constant speed c > . In a reference frame which moves APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 3 in the same direction as the wave and with the same speed c the equations of motion are thesteady-state Euler equations ( u − c ) u x + vu y = − P x , ( u − c ) v x + vv y = − P y − g,u x + v y = 0 . in Ω η . (2.1a)We have assumed that the free surface of the wave is the graph y = η ( x ) , that the fluid hasconstant density, set to be , and that the fluid bed is located at y = − d . Hereby, d > is theaverage mean depth of the fluid, meaning that the fluid domain is Ω η := { ( x, y ) : x ∈ S and − d < y < η ( x ) } , whereby S is the unit circle. This notation is used to express the fact that the function η, thevelocity field ( u, v ) , and the pressure P are π -periodic in x. Since we incorporate the effect ofsurface tension in our problem, the equations (2.1a) are supplemented by the following boundaryconditions P = P − ση ′′ / (1 + η ′ ) / on y = η ( x ) ,v = ( u − c ) η ′ on y = η ( x ) ,v = 0 on y = − d, (2.1b)with P denoting the constant atmospheric pressure and σ > being the surface tension coeffi-cient. Moreover, the vorticity of the flow is the scalar function ω := u y − v x in Ω η . (2.1c)The goal of this paper is to prove the existence of solutions of the problem (2.1) in the class η ∈ C − ( S ) , u, v, P ∈ C − (Ω η ) , ω ∈ L ∞ (Ω η ) , (2.2)and to study their additional regularity properties. Hereby we may identify the spaces C k − ( S ) and C k − (Ω η ) , ≤ k ∈ N , which contain functions that have Lipschitz continuous derivatives oforder k − , with W k ∞ ( S ) and W k ∞ (Ω η ) , respectively, cf. [16].The problem (2.1) can also be formulated in terms of the stream function ψ : Ω η → R , whichis given by ψ ( x, y ) := − p + Z y − d ( u ( x, s ) − c ) ds for ( x, y ) ∈ Ω η . It follows readily from this formula that ψ ∈ C − (Ω η ) satisfies ∇ ψ = ( − v, u − c ) . Additionally,it can be shown that the problem (2.1) is equivalent to the following free boundary problem ∆ ψ = γ ( − ψ ) in Ω η , |∇ ψ | + 2 g ( y + d ) − σ η ′′ (1 + η ′ ) / = Q on y = η ( x ) ,ψ = 0 on y = η ( x ) ,ψ = − p on y = − d, (2.3)cf. [5, 9, 35]. We emphasize that the first boundary condition in (2.3) is obtained from Bernoulli’sprinciple which states that the total energy E := ( u − c ) + v g ( y + d ) + P − Z ψ γ ( − s ) ds is constant in Ω η . In (2.3), the constant p < represents the relative mass flux, Q ∈ R isrelated to the so-called total head, and the function γ is the vorticity function. The existence of A.–V. MATIOC AND B.–V. MATIOC the vorticity function is obtained under the additional assumption that the horizontal velocityof each fluid particle is less than the wave speed u − c < in Ω η . (2.4)Indeed, the relation (2.4) together with (2.1a) imply, cf. [9, 35], that there exists a function γ ∈ L ∞ (( p , such that ω ( x, y ) = γ ( − ψ ( x, y )) almost everywhere in Ω η .Assuming (2.4), the stream function formulation (2.3) can be reexpressed in terms of theso-called height function. Indeed, the assumption (2.4), ensures that the mapping Φ : Ω η → Ω given by Φ( x, y ) := ( q, p )( x, y ) := ( x, − ψ ( x, y )) for ( x, y ) ∈ Ω η , whereby Ω := S × ( p , , is a diffeomorphism of class C − . Consequently, the height function h : Ω → R defined by h ( q, p ) := y + d for ( q, p ) ∈ Ω belongs to C − (Ω) = W ∞ (Ω) and it solvesthe nonlinear boundary value problem (1 + h q ) h pp − h p h q h pq + h p h qq − γ ( p ) h p = 0 in Ω , h q + (2 gh − Q ) h p − σ h p h qq (1 + h q ) / = 0 on p = 0 ,h = 0 on p = p , (2.5)together with the condition min Ω h p > . (2.6)We stress at this point that the function h associates to each point ( q, p ) ∈ Ω the value of theheight of fluid particle ( x, y ) = Φ − ( q, p ) above the flat bed. Particularly, the wave profile isparametrized by the map η = h ( · , − d, implying that h ( · , ∈ C − ( S ) . With this observation,the boundary condition of (2.5) on p = 0 is also meaningful. In fact, each streamline of thesteady flow corresponds to a level curve of ψ and is therefore parametrized by the function h ( · , p ) − d , whereby p ∈ [ p , is uniquely determined by the streamline. Moreover, as a directconsequence of (2.6), that the first equation of (2.5) is uniformly elliptic. The equivalence ofthe problems (2.1), (2.3), and (2.5) under the assumption (2.4) (or equivalently (2.6)) in the W ∞ − setting follows easily from previous contributions [5, 14, 35] (see also [11]).Our main existence result is the following theorem. Theorem 2.1 (Existence result) . Let γ ∈ L ∞ (( p , be given. Then, there exists a positiveinteger n and connected curves C k , k ∈ N \ { } , consisting only of solutions of the problem (2.5) - (2.6) with the property that each solution h belonging to one of the curves satisfies ( i ) h ∈ W ∞ (Ω) , ( ii ) h ( · , p ) is a real-analytic map for all p ∈ [ p , . Each curve C k contains a laminar flow (all the streamlines being parallel to the flat bed) and allthe other points on the curve correspond to solutions that have minimal period π/ ( kn ) , onlyone crest and trough per period, and are symmetric with respect to the crest line.Remark . The integer n in Theorem 2.1 may be chosen to be n = 1 provided that thecondition (4.14) is satisfied.We emphasize that the regularity property ( ii ) of the solutions found in Theorem 2.1 guaran-tees that the wave surface and all the streamlines of the flows are real-analytic graphs. Additionalregularity properties of the solutions found in Theorem 2.1 are derived in Section 5, cf. Theorem5.1. APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 5 A fourth equivalent formulation for the water wave problem
The main difficulty in proving the existe Γ( p ) := Z p γ ( s ) ds for p ≤ p ≤ , (3.1)nce of solutions of problem (2.5) is due to the fact that we have to deal with a quasilinear ellipticequation in a W ∞ − setting. A further impediment is raised by the nonlinear boundary conditionon p = 0 which contains second order derivatives of the unknown. Therefore, we cannot attack(2.5) directly. Instead, we consider a weak formulation of (2.5) and establish first the existenceof weak solutions to (2.5) that satisfy (2.6). Later on, we improve the regularity of these weaksolutions and show finally that they are the strong solutions found in Theorem 2.1. To this end,we introduce the anti-derivative Γ : [ p , → R of γ by the relation Γ( p ) := Z p γ ( s ) ds for p ≤ p ≤ , (3.2)and observe that the first equation of (2.5) can be recast in the equivalent form (cid:18) h q h p (cid:19) q − Γ + 1 + h q h p ! p = 0 in Ω .This relation motivates us to introduce the following notion of weak solution of (2.5). Definition 3.1.
A function h ∈ C (Ω) is called a weak solution of (2.5) if ( i ) h ( · , ∈ C ( S ) ; ( ii ) h satisfies both boundary conditions of (2.5) ; ( iii ) h satisfies the following integral equation Z Ω h q h p φ q − Γ + 1 + h q h p ! φ p d ( q, p ) = 0 for all φ ∈ C (Ω) . (3.3)We have denoted by C (Ω) the space containing continuously differentiable functions withcompact support in Ω . It is easy to see that any classical solution of the problem (2.5)-(2.6),cf. [44], is also a weak solution of this problem. The disadvantage of this definition is that oneneeds to require more regularity from h on the boundary p = 0 , fact which makes it difficult toconsider a suitable functional analytic setting for this concept of weak solutions. Fortunately, wecan recast the second order boundary condition of (2.5), which is obtained from to Bernoulli’sprinciple, as a nonlinear and nonlocal equation. This operation has the benefit of transformingthe boundary condition from a differential equation of order two–we lose two derivatives due tothe curvature term–into a nonlocal equation of order zero.In the following tr will denote the trace operator with respect to boundary p = 0 , that is tr v = v ( · , for all v ∈ C (Ω) . Let α ∈ (0 , be fixed for the remainder of the paper. Lemma 3.2.
Let (1 − ∂ q ) − ∈ L ( C α ( S ) , C α ( S )) denote the inverse of the linear operator − ∂ q : C α ( S ) → C α ( S ) . ( i ) Assume that h ∈ C α (Ω) is a weak solution of (2.5) that satisfies additionally thecondition (2.6) . Then, h also satisfies the following equation h + (1 − ∂ q ) − tr (cid:0) h q + (2 gh − Q ) h p (cid:1) (1 + h q ) / σh p − h ! = 0 on p = 0 . (3.4) A.–V. MATIOC AND B.–V. MATIOC ( ii ) Assume that h ∈ C α (Ω) verifies the condition (2.6) and that h is a weak solution ofthe problem (1 + h q ) h pp − h p h q h pq + h p h qq − γ ( p ) h p = 0 in Ω ,h + (1 − ∂ q ) − tr (cid:0) h q + (2 gh − Q ) h p (cid:1) (1 + h q ) / σh p − h ! = 0 on p = 0 ,h = 0 on p = p , (3.5) that is h satisfies the last two equations of (3.5) pointwise and the first equation in theweak sense defined in Definition 3.1 ( iii ) . Then, h is a weak solution of (2.5) .Proof. It is easy to see that ( ii ) follows from ( i ) . On the other hand, if we assume that ( ii ) issatisfied, we only need to show that tr h ∈ C α ( S ) and that h satisfies the second boundarycondition of (2.5). Noticing that the second equation of (3.5) implies tr h = − (1 − ∂ q ) − tr (cid:0) h q + (2 gh − Q ) h p (cid:1) (1 + h q ) / σh p − h ! we deduce that tr h ∈ C α ( S ) , and, applying the operator (1 − ∂ q ) to the latter equation, weobtain the desired conclusion. (cid:3) The advantage of the formulation (3.5) of the water wave problem is that all its equations arewell-defined for functions h ∈ C α (Ω) . This allows us to introduce a functional analytic settingand to recast (3.5) as a bifurcation problem. Then, using the theorem on local bifurcation fromsimple eigenvalues due to Crandall and Rabinowitz [13] we determine weak solutions of (3.5)and (2.6) which are located on real-analytic curves.4. Local bifurcation of weak solutions
We now introduce a parameter λ into the problem (3.5) which is used to describe the trivialsolutions of (3.5). These are laminar flows, with a flat surface and parallel streamlines, and aredenoted by H. Indeed, if H ∈ C α (Ω) is a weak solution of (3.5) and (2.6) which is independentof q , then H ( p ) = 0 , H (0) + (1 − ∂ q ) − gH (0) − Q ) H p (0)2 σH p (0) − H (0) ! = 0 , and Z Ω (cid:18) Γ + 12 H p (cid:19) φ p d ( q, p ) = 0 for all φ ∈ C (Ω) .This last relation ensures that Γ + 1 / (2 H p ) is a constant function. Taking into account that (1 − ∂ q ) − c = c for all c ∈ R , we find a constant λ > [ p , Γ such that we have H ( p ) := H ( p ; λ ) := Z pp p λ − s ) ds, p ∈ [ p , . (4.1)Requiring that H solves also the boundary condition on p = 0 , we determine the head Q as afunction of the parameter λQ := Q ( λ ) := λ + 2 g Z p p λ − p ) dp. (4.2) APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 7
Let us observe that in fact H ∈ C − ([ p , , and that the constant λ is related to the horizontalspeed at the top of the laminar flow by the relation √ λ = 1 H p (0) = ( c − u ) (cid:12)(cid:12) y =0 . We now present an abstract functional analytic setting which allows us to recast the problem(3.5) as an operator equation. We choose therefore an integer n ∈ N with n ≥ (which will befixed later on) and define the Banach spaces: X := n h ∈ C α π/n (Ω) : h is even in q and h (cid:12)(cid:12) p = p = 0 o ,Y := { f ∈ D ′ (Ω) : f = ∂ q φ + ∂ p φ for φ , φ ∈ C α π/n (Ω) with φ odd and φ even in q } ,Y := { ϕ ∈ C α π/n ( S ) : ϕ is even } , whereby we have identified, when defining Y , the unit circle S with the line p = 0 . The subscript π/n means that we consider functions which are π/n –periodic only. The space Y is a Banachspace with the norm k f k Y := inf {k φ k α + k φ k α : f = ∂ q φ + ∂ p φ } . Moreover, we introduce the operator F := ( F , F ) : (2 max [ p , Γ , ∞ ) × X → Y := Y × Y bythe relations F ( λ, h ) := (cid:18) h q H p + h p (cid:19) q − Γ + 1 + h q H p + h p ) ! p , F ( λ, h ) := tr h + (1 − ∂ q ) − tr (cid:0) h q + (2 g ( H + h ) − Q )( H p + h p ) (cid:1) (1 + h q ) / σ ( H p + h p ) − h ! for ( λ, h ) ∈ (2 max [ p , Γ , ∞ ) × X, whereby H = H ( · ; λ ) and Q = Q ( λ ) are given by (4.1)and (4.2), respectively. Let us observe that the function F is well-defined and it depends real-analytically on its arguments, that is F ∈ C ω ((2 max [ p , Γ , ∞ ) × X, Y ) . (4.3)Whence, the problem (3.5) is equivalent to the following abstract equation F ( λ, h ) = 0 in Y , (4.4)the laminar flow solutions of (3.5) corresponding to the trivial solutions of FF ( λ,
0) = 0 for all λ ∈ (2 max [ p , Γ , ∞ ) . (4.5)We emphasize that if ( λ, h ) is a solution of (4.4), then the function h + H ( · ; λ ) is a weak solutionof (3.5), when Q = Q ( λ ) , and it also satisfies the condition (2.6) if h is sufficiently small.In order to prove the existence of branches of solutions of (4.4) bifurcating from the laminarflows h = 0 , we need to determine particular λ for which ∂ h F ( λ, ∈ L ( X, Y ) is a Fred-holm operator of index zero with a one-dimensional kernel. We first prove that ∂ h F ( λ, is aFredholm operator of index zero for every value of λ ∈ (2 max [ p , Γ , ∞ ) . To this end, given
A.–V. MATIOC AND B.–V. MATIOC λ ∈ (2 max [ p , Γ , ∞ ) , we note that the Fréchet derivative ∂ h F ( λ, is the linear operator ( L, T ) ∈ L ( X, Y ) given by Lw := (cid:18) w q H p (cid:19) q + (cid:18) w p H p (cid:19) p ,T w := tr w + (1 − ∂ q ) − tr gw − λ / w p σ − w ! for w ∈ X. (4.6) Lemma 4.1.
Given λ ∈ (2 max [ p , Γ , ∞ ) , the Fréchet derivative ∂ h F ( λ, ∈ L ( X, Y ) is aFredholm operator of index zero.Proof. Given w ∈ X, we note that ( L, T ) w = ( L, tr ) w + , (1 − ∂ q ) − tr gw − λ / w p σ − w !! . Recalling that (1 − ∂ q ) − ∈ L ( C α ( S ) , C α ( S )) , the operator X ∋ w , (1 − ∂ q ) − tr gw − λ / w p σ − w !! ∈ Y is compact, so that our conclusion is immediate if ( L, tr ) : X → Y is an isomorphism. How-ever, the latter property follows readily from the existence and uniqueness result stated in [17,Theorem 8.34]. (cid:3) The kernel of the Fréchet derivative.
We now identify certain λ for which the Fréchetderivative ∂ h F ( λ,
0) = (
L, T ) has a one-dimensional kernel. To this end, let w ∈ X be a vectorin the kernel of ( L, T ) and define for each k ∈ N the Fourier coefficients w k ( p ) := h w ( · , p ) | cos( kn · ) i L := Z π w ( q, p ) cos( knq ) dq for p ∈ [ p , .Clearly, we have w k ∈ C α ([ p , for all k ∈ N . Given ψ ∈ C (( p , , we define the function φ ( q, p ) := ψ ( p ) cos( knq ) for ( q, p ) ∈ Ω , and observe that φ ∈ C (Ω) . Whence, in virtue of Lw = 0 , integration by parts gives Z p (cid:18) w ′ k H p ψ ′ + ( kn ) w k H p ψ (cid:19) dp = 0 . This relation being true for all ψ ∈ C (( p , and since H p ∈ C − ([ p , W ∞ (( p , , weconclude that w k ∈ H (( p , is a strong solution of the equation (cid:18) w ′ k H p (cid:19) ′ − ( kn ) w k H p = 0 in L (( p , . Moreover w ∈ X implies that w k ( p ) = 0 , while multiplying the relation T w = 0 by cos( knq ) and making use of the symmetry of the operator (1 − ∂ q ) − , that is h f | (1 − ∂ q ) − g i L = h (1 − ∂ q ) − f | g i L for all f, g ∈ C α ( S ) , (4.7)we determine a third relation ( g + σ ( kn ) ) w k (0) = λ / w ′ k (0) . APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 9
Summarizing, the Fourier coefficient w k ∈ H (( p , solves the problem ( a w ′ ) ′ − µaw = 0 in L (( p , , ( g + σµ ) w (0) = λ / w ′ (0) ,w ( p ) = 0 , (4.8)when µ = ( kn ) . Hereby, we use the shorthand a := 1 /H p ∈ C − ([ p , . Thus, if we wish that ∂ h F ( λ, has a one-dimensional kernel, we need to impose conditions on λ which guarantee thatthe system (4.8) has non-trivial solutions–which form a one-dimensional subspace of H (( p , –for only one constant µ ∈ { ( kn ) : k ∈ N } . This motivates us to study, for each ( λ, µ ) ∈ (2 max [ p , Γ , ∞ ) × [0 , ∞ ) , the Sturm-Liouville operator R λ,µ : H → L (( p , × R , whereby H := { w ∈ H (( p , w ( p ) = 0 } and R λ,µ w := (cid:18) ( a w ′ ) ′ − µaw ( g + σµ ) w (0) − λ / w ′ (0) (cid:19) for w ∈ H. Therefore, given ( λ, µ ) ∈ (2 max [ p , Γ , ∞ ) × [0 , ∞ ) , we define the functions v i ∈ C − ([ p , ,with v i := v i ( · ; λ, µ ) , as being the solutions of the initial value problems ( ( a v ′ ) ′ − µav = 0 in L (( p , ,v ( p ) = 0 , v ′ ( p ) = 1 , ( ( a v ′ ) ′ − µav = 0 in L (( p , ,v (0) = λ / , v ′ (0) = g + σµ. (4.9)These problems can be seen as system of first order linear ordinary differential equations, andtherefore the existence and uniqueness of v i follows from the classical theory, cf. [1]. Proposition 4.2.
For every ( λ, µ ) ∈ (2 max [ p , Γ , ∞ ) × [0 , ∞ ) , the operator R λ,µ is a Fredholmoperator of index zero and its kernel is at most one-dimensional. Furthermore, the kernel of theoperator R λ,µ is non-trivial exactly when the functions v i , i = 1 , , given by (4.9) , are linearlydependent. In this case we have Ker R λ,µ = span { v } . Proof.
Observe first that the operator R λ,µ can be writen as a sum R λ,µ = R I + R c , with R I w := (cid:18) ( a w ′ ) ′ − µaw − λ / w ′ (0) (cid:19) and R c w := (cid:18) g + σµ ) w (0) (cid:19) for all w ∈ H, R c being a compact operator. Furthermore, if the equation R I w = ( f, A ) , with ( f, A ) ∈ L (( p , × R , has a solution w ∈ H, then Z p (cid:0) a w ′ ϕ ′ + µawϕ (cid:1) dp = − Aϕ (0) − Z p f ϕ dp (4.10)for all ϕ ∈ H ∗ := { w ∈ H (( p , w ( p ) = 0 } . Noticing that the right-hand side of (4.10)defines a linear functional in L ( H ∗ , R ) , and that the left-hand side corresponds to a boundedbilinear and coercive functional in H ∗ × H ∗ , the existence and uniqueness of a solution w ∈ H ∗ follows from the Lax-Milgram theorem, cf. [17, Theorem 5.8]. This solution is actually in H and therefore R I is an isomorphism. This proves the Fredholm property of R λ,µ .In order to see that the kernel of R λ,µ is at most one-dimensional, we consider two solutions w , w ∈ H (( p , of the equation ( a w ′ ) ′ − µaw = 0 in L (( p , . Multiplying the equationsatisfied by w with w and that satisfied by w with w , we obtain, after subtracting the newidentities, that a ( w w ′ − w w ′ ) = const. in [ p , . (4.11)Thus, if additionally w , w ∈ H, then the constant is zero and, in view of a > , w and w are linearly dependent. Finally, it is not difficult to see that if the functions v and v , given by (4.9), are linearly dependent, then they both belong to Ker R λ,µ . On the other hand, if = v ∈ Ker R λ,µ , using the relation (4.11), we get that v is colinear with v and v . This provesthe claim. (cid:3)
Thus, we need to determine for which ( λ, µ ) the Wronskian W ( p ; λ, µ ) := (cid:12)(cid:12)(cid:12)(cid:12) v v v ′ v ′ (cid:12)(cid:12)(cid:12)(cid:12) vanishes on the whole interval [ p , . Recalling (4.11), the Wronskian vanishes on [ p , ifand only if it vanishes at p = 0 . Summarizing, R λ,µ has a one-dimensional kernel exactlywhen ( λ, µ ) is a solution of the equation W (0; λ, µ ) = 0 . Taking into account that all theequations of (4.9) depend real-analytically on the variable ( λ, µ ) , we deduce that the function W (0; · , · ) : (2 max [ p , Γ , ∞ ) × [0 , ∞ ) → R , defined by W (0; λ, µ ) := λ / v ′ (0; λ, µ ) − ( g + σµ ) v (0; λ, µ ) , (4.12)is real-analytic. Determining the zeros of W (0; · , · ) when µ = 0 is rather easy. Indeed, for µ = 0 , we can determine v explicitly v ( p ; λ,
0) = Z pp a ( p ) a ( s ) ds, p ∈ [ p , . Consequently, W (0; λ,
0) = 0 if and only if λ solves the equation g = Z p a ( p ) dp. (4.13)The right-hand side of (4.13) is a strictly decreasing function of λ , Z p a ( p ) dp −→ λ →∞ and Z p a ( p ) dp −→ λ → [ p , Γ ∞ . Consequently, there exists a unique λ ∈ (2 max [ p , Γ , ∞ ) which satisfies (4.13). When µ > , there are in general no explicit formula for v , and the problem of determining the zeros of W (0; · , · ) is more intriguing. However, the arguments used in [31] can be adapted to our contextto prove the following statement. Proposition 4.3.
Given λ > λ , there exists a unique solution µ ( λ ) ∈ (0 , ∞ ) of the equation W (0; λ, µ ) = 0 . Moreover, we have W (0; λ , · ) − { } = { , µ ( λ ) } whereby µ ( λ ) = 0 if Z p a ( p ) (cid:18)Z pp a ( s ) ds (cid:19) dp < σg . (4.14) The function µ : [ λ , ∞ ) → [0 , ∞ ) is real-analytic in ( λ , ∞ ) , strictly increasing, and lim λ →∞ µ ( λ ) = ∞ . (4.15) Proof.
The proof is similar to that of the Lemmas 4.3 - 4.7 in [31], the restriction γ ∈ L ∞ (( p , leading only to minor modifications. Therefore, we omit it. (cid:3) In virtue of Proposition 4.3, there exists a smallest positive integer n ( n = 1 if (4.14) issatisfied) such that for all k ∈ N \ { } , there exists a unique constant λ k ∈ ( λ , ∞ ) with theproperty that µ ( λ k ) := ( kn ) . (4.16) APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 11
Because µ is strictly increasing and recalling (4.15), we deduce that λ k ր ∞ . Summarizing, if k ≥ , we have that W (0; λ k , ( ln ) ) = 0 , with l ∈ N , if and only if l = k. Consequently, R λ k , ( ln ) has a non-trivial kernel if and only if l = k. We have thus shown that, for all k ≥ , the kernelof the Fréchet derivative ∂ h F ( λ k , is one-dimensional. More precisely, we have Ker ∂ h F ( λ k ,
0) = span { w ∗ } , (4.17)whereby w ∗ ( q, p ) := v ( p ) cos( knq ) and v is the solution of the first system of (4.9) when setting µ = ( kn ) . That w ∗ is an element of X follows easily from v ∈ C − ([ p , . The transversality condition.
In order to apply the theorem on bifurcation from simpleeigenvalues due to Crandall and Rabinowitz [13] to the operator equation (4.4) we still need toprove that ∂ λh F ( λ k , w ∗ ] / ∈ Im ∂ h F ( λ k , . (4.18)To this end, we need to characterize the range Im ∂ h F ( λ k , . Lemma 4.4.
Given k ∈ N with k ≥ , the pair ( f, ϕ ) ∈ Y , with f := ∂ q φ + ∂ p φ , belongs to Im ∂ h F ( λ k , if and only if we have Z Ω φ w ∗ q + φ w ∗ p d ( q, p ) − Z S ×{ } φ w ∗ dq − σ (1 + ( kn ) ) Z S ×{ } ϕw ∗ dq = 0 . (4.19) Proof.
Let us presuppose that there exists w ∈ X such that ( L, T ) w = ( f, ϕ ) . For every positiveinteger m , we define the function ψ m ∈ H (( p , by the relation ψ m ( p ) := for p + 1 /m ≤ x ≤ − /m,m ( x − p ) for p ≤ x ≤ p + 1 /m, − mx for − /m ≤ x ≤ . Then ψ m w ∗ ∈ H (Ω) = C (Ω) k·k H , and, since Lw = f in Y , a density argument leads us tothe following relation Z Ω w q H p ψ m ∂ q w ∗ + w p H p ∂ p ( ψ m w ∗ ) d ( q, p ) = Z Ω φ ψ m ∂ q w ∗ + φ ∂ p ( ψ m w ∗ ) d ( q, p ) . Letting m → ∞ , it is easy to see that Z Ω w q w ∗ q H p + w p w ∗ p H p d ( q, p ) − Z S ×{ } w p w ∗ H p dq = Z Ω φ w ∗ q + φ w ∗ p d ( q, p ) − Z S ×{ } φ w ∗ dq. (4.20)On the other hand, if we multiply the relation T w = ϕ by w ∗ and integrate it over a period,gives, after exploiting the relation (1 − ∂ q ) − (tr w ∗ ) = tr w ∗ kn ) (4.21)and the symmetry of the operator (1 − ∂ q ) − , the following integral relation ( g + σ ( kn ) ) Z S ×{ } ww ∗ dq − Z S ×{ } w p w ∗ H p dq = σ (1 + ( kn ) ) Z S ×{ } ϕw ∗ dq. (4.22) Subtracting (4.22) from (4.20), we find that Z Ω φ w ∗ q + φ w ∗ p d ( q, p ) − Z S ×{ } φ w ∗ dq − σ (1 + ( kn ) ) Z S ×{ } ϕw ∗ dq = Z Ω w q w ∗ q H p + w p w ∗ p H p d ( q, p ) − ( g + σ ( kn ) ) Z S ×{ } ww ∗ dq. (4.23)However, recalling that ( L, T ) w ∗ = 0 , similar arguments to those presented above show that theright-hand side of (4.23) is zero, and we obtain the desired relation (4.19). To finish the proof,let us observe that the relation (4.19) defines a closed subspace of Y that has codimension oneand contains the range Im ∂ h F ( λ k , . Since the range also has codimension one, we concludethat every pair ( f, ϕ ) that satisfies (4.19) belongs Im ∂ h F ( λ k , . (cid:3) Lemma 4.5.
The transversality condition (4.18) is satisfied for all k ∈ N with k ≥ .Proof. Differentiating (4.6) with respect to λ we obtain, in virtue of ∂ λ H p = − / (2 H p ) , that ∂ λh F ( λ k , w ∗ ] = (cid:18) H p w ∗ q (cid:19) q + (cid:18) w ∗ p H p (cid:19) p , − λ / k σ (1 − ∂ q ) − tr w ∗ p ! . We only need to check that the relation (4.19) is not satisfied by ∂ λh F ( λ k , w ∗ ] ∈ Y . To thisend, we set φ := H p w ∗ q , φ := 3 w ∗ p H p , ϕ := − λ / k σ (1 − ∂ q ) − tr w ∗ p , and, recalling (4.7) and (4.21), we conclude that Z Ω φ w ∗ q + φ w ∗ p d ( q, p ) − Z S ×{ } φ w ∗ dq − σ (1 + ( kn ) ) Z S ×{ } ϕw ∗ dq = Z Ω H p w ∗ q w ∗ p H p d ( q, p ) > . (cid:3) Gathering (4.3), (4.5), (4.17), Proposition 4.3 and the Lemmas 4.1 and 4.5, the theorem onbifurcation from simple eigenvalues due to Crandall and Rabinowitz [13] yields the followingresult for the bifurcation problem (4.4).
Theorem 4.6 (Local bifurcation) . Let γ ∈ L ∞ (( p , be given. Then, there exists a positiveinteger n and, for each k ∈ N \ { } , there exists ε k > and a real-analytic curve ( λ k , h k ) : ( λ k − ε k , λ k + ε k ) → (2 max [ p , Γ , ∞ ) × X, consisting only of solutions of the problem (4.4) . Moreover, we have that λ k ( s ) = λ k + O ( s ) ,h k ( s ) = sw ∗ + O ( s ) , as s → ,whereby w ∗ ∈ X is given by w ∗ ( q, p ) := v ( p ) cos( knq ) and v denotes the solution of the firstsystem of (4.9) when µ = ( kn ) . Moreover, in a neighborhood of ( λ k , , the solutions of (4.4) are either trivial or are located on the local curve ( λ k , h k ) . If the condition (4.14) is satisfied,then n = 1 . APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 13
The points on the curves ( λ k , h k ) , k ≥ correspond to weak solutions of the problem (3.5)and (2.6). The next lemma shows, under an additional regularity assumption, that all weaksolutions h ∈ C α (Ω) of (3.5) and (2.6) are in fact strong solutions (even classical solutionsif γ ∈ C ([ p , ). This additional regularity assumption is shown later on, cf. Proposition 5.4,to be a priori satisfied by the weak solutions h ∈ C α (Ω) of (3.5) and (2.6) even when thevorticity function is merely integrable. Lemma 4.7.
Assume that h ∈ C α (Ω) is a weak solution of (3.5) and (2.6) correspondingto a vorticity function γ ∈ L ∞ (( p , (resp. γ ∈ C ([ p , ) Additionally, we assume that h q ∈ C α (Ω) . Then, h ∈ W ∞ (Ω) (resp. h ∈ C (Ω) ) and h satisfies the first equation of (2.5) almost everywhere in Ω .Proof. Because ∂ p ( h q ) ∈ C α (Ω) , we deduce that h p is differentiable with respect to q and that ∂ q h p = ∂ p h q ∈ C α (Ω) . Therefore, we have that ∂ p Γ + 1 + h q h p ! = ∂ q (cid:18) h q h p (cid:19) ∈ C α (Ω) and ∂ q Γ + 1 + h q h p ! ∈ C α (Ω) , the first relation being understood in the sense of distributions. Particularly, we deduce that Γ + 1 + h q h p ∈ C − (Ω) . But, since h p satisfies (2.6) and it is also bounded, this implies h p ∈ C − (Ω) . Summarizing, wehave shown that h ∈ C − (Ω) = W ∞ (Ω) . The same arguments lead us h ∈ C (Ω) if γ ∈ C ([ p , . The final part of the claim follows now directly from (3.3), cf. [17, Lemma 7.5]. (cid:3) Regularity of weak solutions
We consider now an arbitrary non-laminar weak solution h ∈ C α (Ω) of the water waveproblem (3.5), when requiring merely integrability of the vorticity function. Assuming that h satisfies also the condition (2.6), we establish additional regularity properties for this weaksolution. The main result of this section is the following theorem. Theorem 5.1 (Regularity result) . Assume that γ ∈ L (( p , and let α ∈ (0 , be given.Given a weak solution h ∈ C α (Ω) of (3.5) that satisfies (2.6) , we have that ∂ mq h ∈ C α (Ω) for all m ∈ N . Moreover, there exists a constant
L > with the property that k ∂ mq h k α ≤ L m − ( m − (5.1) for all integers m ≥ . An immediate consequence of Theorem 5.1 is the following corollary.
Corollary 5.2.
Let h satisfy the assumptions of Theorem 5.1. Then, the wave surface and allthe other streamlines are real-analytic graphs.Proof. In view of Theorem 5.1, the function h ( · , p ) is real-analytic for all p ∈ [ p , . Since thestreamlines of the flow coincide with the graphs [ q h ( q, p ) − d ] , the conclusion is obvious. (cid:3) Remark . The result established in Theorem 5.1 is true also for pure capillary water waves.Indeed, neglecting gravity corresponds to putting g = 0 in (3.5), modification which does notinfluence the proof of Theorem 5.1. Particularly, the streamlines and the wave profile of capil-lary water waves with a merely integrable vorticity function are real-analytic. This generalizesprevious results [2, 21]. We first prove that the distributional derivative ∂ mq h , m ∈ N and m ≥ , is a weak solution ofa linear elliptic equation satisfying certain nonlocal boundary conditions. This property appearsas a consequence of the invariance of the problem (3.5) with respect to horizontal translations. Proposition 5.4.
Let γ ∈ L (( p , be given and assume that h ∈ C α (Ω) is a weak solutionof (3.5) which satisfies additionally (2.6) . Given m ∈ N with m ≥ , the derivative ∂ mq h belongsto C α (Ω) and it is a weak solution of the elliptic boundary value problem (cid:16) h p ∂ q w (cid:17) q − (cid:16) h q h p ∂ p w (cid:17) q − (cid:16) h q h p ∂ q w (cid:17) p + (cid:16) h q h p ∂ p w (cid:17) p = ( f m ) q + ( g m ) p in Ω ,w − (1 − ∂ q ) − tr ( w + a w q + a w p ) = P i =1 ϕ im on p = 0 ,w = 0 on p = p , (5.2) whereby f m , g m , ϕ m ∈ C α (Ω) are given by f m := m − X k =1 (cid:18) m − k (cid:19) (cid:20) − ∂ kq (cid:18) h p (cid:19) ∂ q ( ∂ m − kq h ) + ∂ kq (cid:18) h q h p (cid:19) ∂ p ( ∂ m − kq h ) (cid:21) ,g m := m − X k =1 (cid:18) m − k (cid:19) " ∂ kq (cid:18) h q h p (cid:19) ∂ q ( ∂ m − kq h ) − ∂ kq h q h p ! ∂ p ( ∂ m − kq h ) , and ϕ im , a i ∈ C α (Ω) are defined as a := − h q ) / h q σh p − gh − Q )(1 + h q ) / h q σ , a := (1 + h q ) / σh p ,ϕ m := − σ (1 − ∂ q ) − tr m − X k =1 (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h p ! ,ϕ m := − gσ (1 − ∂ q ) − tr m − X k =0 (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h ! ,ϕ m := 1 σ (1 − ∂ q ) − tr (1 + h q ) / m − X k =0 (cid:18) m − k (cid:19) ( ∂ k +1 q h p ) ∂ m − k − q h p ! ,ϕ m := − σ (1 − ∂ q ) − tr h p m − X k =0 (cid:18) m − k (cid:19) ( ∂ k +1 q h q ) ∂ m − k − q ( h q (1 + h q ) / ) ! ,ϕ m := − σ (1 − ∂ q ) − tr (2 gh − Q ) m − X k =0 (cid:18) m − k (cid:19) ( ∂ k +1 q h q ) ∂ m − k − q ( h q (1 + h q ) / ) ! . We denote in this section by C i , i ∈ N , universal constants which are independent of m and the function h considered in Proposition 5.4. Moreover, we use K i , i ∈ N , to denote alsoconstants that are independent of m , but may depend on k ∂ lq h k α with ≤ l ≤ . Proof of Proposition 5.4.
The proof follows by using the induction principle. We will only showthat ∂ q h belongs to C α (Ω) and that it solves the system (5.2) when m = 1 . The general ∂ mq h solves the first equation of (5.2) in the weak sense and the two boundary conditions pointwise. APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 15 induction step follows by using a similar argument as in this first induction step (see e.g. theproof of [15, Proposition 2.1]).To begin, we observe that for all ε ∈ (0 , , the horizontal translation h ε ∈ C α (Ω) definedby h ε ( q, p ) := h ( q + ε, p ) for ( q, p ) ∈ Ω is also a weak solution of (3.5) and (2.6). Subtractingthe relations satisfied by h ε from those satisfied by h , we find that the function u ε := ( h ε − h ) /ε belongs to C α (Ω) and it is a weak solution of the equation ( a ε ∂ q u ε ) q + ( a ε ∂ p u ε ) q + ( a ε ∂ q u ε ) p + ( a ε ∂ p u ε ) p = 0 in Ω , (5.3)whereby a ε = 1 h ε,p , a ε = − h q h p h ε,p , a ε = − h q + h ε,q h ε,p , a ε = ( h p + h ε,p )(1 + h q )2 h p h ε,p . Because of (2.6), the equation (5.3) is uniformly elliptic when ε ∈ (0 , is sufficiently small.Furthermore, u ε also satisfies the boundary conditions (cid:26) u ε = (1 − ∂ q ) − tr ( b u ε + b u ε,q + b u ε,p ) on p = 0 ,u ε = 0 on p = p , (5.4)with b i given by b :=1 − g (1 + h q ) / σ , b := ( h p + h ε,p )(1 + h q ) / σh p H p ,b := − ( h q + h ε,q ) P i =0 (1 + h q ) i (1 + h ε,q ) − i σh ε,p ((1 + h q ) / + (1 + h ε,q ) / ) − (2 gh ε − Q )( h q + h ε,q ) P i =0 (1 + h q ) i (1 + h ε,q ) − i σ ((1 + h q ) / + (1 + h ε,q ) / ) . In view of (5.3) and (5.4), we may use Schauder estimates for weak solutions of Dirichlet prob-lems, cf. [17, Theorem 8.33] to conclude that there exists a positive constant K , which isindependent of ε, such that k u ε k α ≤ K (cid:0) k u ε k + k (1 − ∂ q ) − tr ( b u ε + b u ε,q + b u ε,p ) k α (cid:1) (5.5)for sufficiently small ε. We now prove that k u ε k α may be bounded from above by a constantwhich is independent of ε. Indeed, the mean value theorem implies that k u ε k ≤ k h q k . (5.6)On the other hand, taking into account that (1 − ∂ q ) − ∈ L ( C α/ ( S ) , C α/ ( S )) and using thealgebra property of C α/ ( S ) , we get k (1 − ∂ q ) − tr ( b u ε + b u ε,q + b u ε,p ) k α ≤ C k (1 − ∂ q ) − tr ( b u ε + b u ε,q + b u ε,p ) k α/ ≤ C k tr ( b u ε + b u ε,q + b u ε,p ) k α/ ≤ K k tr u ε k α/ , (5.7)with C and K independent of ε. The well-known interpolation property of the Hölder spaces ( C ( S ) , C α ( S )) θ, ∞ = C α/ ( S ) if θ = 2 + α α ) , cf. e.g. [29], implies, via Young’s inequality, that k tr u k α/ ≤ C k tr u k − θ k tr u k θ θ ≤ δ k u k α + C ( δ ) k u k (5.8)for all δ > and all u ∈ C α (Ω) , the constant C ( δ ) being positive. Particularly, if we choose δ := (2 K K ) − , the relations (5.5)-(5.8) yield that k u ε k α ≤ K k h q k (1 + K C ( δ )) (5.9)for all sufficiently small ε , the right-hand side of (5.9) being independent of ε. Since u ε convergespointwise to h q , we find, by using (5.9), a subsequence of ( u ε k ) k which converges to h q in C (Ω) . The uniform bound (5.9) implies that in fact h ∈ C α (Ω) . Finally, passing to the limit k → ∞ in (5.3) and (5.4) we recover, in view of h q ∈ C α (Ω) , the relations (5.2) with m = 1 . (cid:3) The following lemma will be one of the main tools when estimating the norm of the solution ∂ mq h of (5.2). Lemma 5.5.
Let n , N ∈ N satisfy ≤ n ≤ N , and assume that ∂ nq u i ∈ C α (Ω) for all ≤ n ≤ N and ≤ i ≤ . If there exists a constant L ≥ and a real number r ∈ [0 , n ] suchthat k ∂ nq u i k α ≤ L n − r ( n − n )! for all n ≤ n ≤ N , then we find a constant C = C ( n ) > with the property that k ∂ nq ( u u ) k α ≤ C X i =1 n − X l =0 k ∂ lq u i k α ! L n − r ( n − n )! , (5.10) k ∂ nq ( u u u ) k α ≤ C X i =1 n − X l =0 k ∂ lq u i k α ! L n − r ( n − n )! , (5.11) k ∂ nq ( u u u u u ) k α ≤ C X i =1 n − X l =0 k ∂ lq u i k α ! L n − r ( n − n )! (5.12) for all n ≤ n ≤ N . Remark . We will use the assertions of the Lemma 5.5 several times in this paper with n ∈ { , , } . Since the constant C depends only on n , it is useful to define C := max { C (2) , C (3) , C (5) } . (5.13)It is important to stress that the quantities on the right-hand side of (5.10)-(5.12) contain onlyderivatives of u i which have order less than n . Proof of Lemma 5.5.
Leibniz’s rule implies that for all n ≤ n ≤ N we have ∂ nq ( u u ) = n X k =0 (cid:18) nk (cid:19) ( ∂ kq u ) ∂ n − kq u . (5.14)Taking into account that k u u k α ≤ k u k α k u k α , we find for all n ≤ n ≤ min { n − , N } that (cid:13)(cid:13) ∂ nq ( u u ) (cid:13)(cid:13) α ≤ n ! n − n X k =0 + n − X k = n − n +1 + n X k = n k ∂ kq u k α k ∂ n − kq u k α ≤ n ! n − n X k =0 k ∂ kq u k α L n − k − r ( n − k − n )! ! + n ! n − X k =0 k ∂ kq u k α ! APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 17 + n ! n X k = n L k − r ( k − n )! k ∂ n − kq u k α ≤ n ! n − X k =0 k ∂ kq u k α ! L n − r ( n − n )! . (5.15)On the other hand, if N ≥ n and n ≤ n ≤ N , then we split the sum (5.14) as follows ∂ nq ( u u ) = n − X k =0 + n − n X k = n + n X k = n − n +1 (cid:18) nk (cid:19) ( ∂ kq u ) ∂ n − kq u , (5.16)and obtain from the hypothesis that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X k =0 (cid:18) nk (cid:19) ( ∂ kq u ) ∂ n − kq u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ n − X k =0 (cid:18) nk (cid:19) L n − k − r ( n − k − n )! k ∂ kq u k α ≤ L n − r ( n − n )! n − X k =0 k ∂ kq u k α ! n n ( n − n + 2) n ≤ C n − X k =0 k ∂ kq u k α ! L n − r ( n − n )! . (5.17)The third sum of (5.16) can be estimated by the same expression (5.17). Finally, the secondterm of (5.16) is estimated as follows (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − n X k = n (cid:18) nk (cid:19) ( ∂ kq u ) ∂ n − kq u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ n − n X k = n (cid:18) nk (cid:19) L k − r ( k − n )! L n − k − r ( n − k − n )! ≤ L n − r ( n − n )! n − n X k = n n n ( n − k − n + 1) n ( k − n + 1) n ≤ L n − r ( n − n )! n − n +1 X k =1 n n ( n − n + 2 − k ) n k n ≤ C L n − r ( n − n )! , (5.18)since, using the inequality n ( n − n + 2) ≥ n for n ≥ n , we have n − n +1 X k =1 n n ( n − n + 2 − k ) n k n ≤ n n n − n +1 X k =1 ( n − n + 2) n ( n − n + 2 − k ) n k n ≤ n ) n ∞ X k =1 k n , (5.19)the last series being finite as n ≥ . Gathering (5.15), (5.17), and (5.18), we have established(5.10). We next apply the estimate (5.10) to the functions v := u u C (cid:16) P i =1 P n − l =0 k ∂ lq u i k α (cid:17) and v := u and obtain (5.11). The last claim (5.12) follows by applying (5.10) to the functions w := u u u C (cid:16) P i =1 P n − l =0 k ∂ lq u i k α (cid:17) and w := u u C (cid:16) P i =4 P n − l =0 k ∂ lq u i k α (cid:17) . (cid:3) Because the functions f m , g m , and ϕ im contain derivatives of /h p , when estimating theirnorms we make use of the following result. Lemma 5.7.
Assume that ∂ nq u ∈ C α (Ω) for all ≤ n ≤ N , with N ≥ , and let C be theconstant defined by (5.13) . If there exists a constant L with L ≥ k ∂ q (1 /u ) k α + k ∂ q (1 /u ) k / α + C X l =0 (2 k ∂ lq (1 /u ) k α + k ∂ lq u k α ) ! (5.20) and k ∂ nq u k α ≤ L n − ( n − for all ≤ n ≤ N , and if inf Ω u > , then we have k ∂ nq (1 /u ) k α ≤ L n − / ( n − for all ≤ n ≤ N. (5.21) Proof.
In view of (5.20), it is clear that (5.21) is satisfied when n = 3 . So, let us assume that N ≥ and that (5.21) is satisfied for all ≤ n ≤ m − , whereby m ≤ N is arbitrarily chosen.We only need to prove that (5.20) holds for m too. Therefore, we write ∂ mq (1 /u ) = ∂ m − q ( u u u ) , whereby u := − ∂ q u and u = u := 1 /u. Our hypothesis, the induction assumption, the relation(5.20), and the fact that
L > , yield that k ∂ nq u k α = k ∂ n +1 q u k α ≤ L n − ( n − , k ∂ nq u k α = k ∂ nq (1 /u ) k α ≤ L n − / ( n − ≤ L n − ( n − for all ≤ n ≤ m − . Therefore, we may use the estimate (5.11) of Lemma 3.2 (with r = 1 , n = 2 , and N = m − ) to obtain, in view of (5.20), that k ∂ mq (1 /u ) k α = k ∂ m − q ( u u u ) k α ≤ C X l =0 (2 k ∂ lq (1 /u ) k α + k ∂ lq u k α ) ! L m − ( m − ≤ L m − / ( m − , which is the desired estimate. (cid:3) In the proof of Theorem 5.1, we also need to estimate, when considering the boundary terms ϕ im , expressions containing derivatives of (1 + h q ) / . To this end, we need, additionally to theLemma 5.5 and 5.7, the following result. APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 19
Lemma 5.8.
Assume that ∂ nq u ∈ C α (Ω) for all ≤ n ≤ N , with N ≥ , and let C be theconstant given by (5.13) . If there exists a constant L satisfying L ≥k ∂ q ((1 + u ) − / ) k / α + k ∂ q ((1 + u ) − / ) k α + k ∂ q u k α + C X l =0 k ∂ lq u k α ! + C X l =0 k ∂ lq ((1 + u ) − / ) k α + X l =0 k ∂ lq u k α + X l =0 k ∂ l +1 q u k α ! + C X l =0 k ∂ lq (1 + u ) k α + X l =0 k ∂ lq ((1 + u ) − / ) k α ! , (5.22) such that k ∂ nq u k α ≤ L n − ( n − for all ≤ n ≤ N , then we have k ∂ nq ((1 + u ) / ) k α ≤ L n − / ( n − for all ≤ n ≤ N. (5.23) Proof.
We first prove that k ∂ nq ((1 + u ) − / ) k α ≤ L n − / ( n − for all ≤ n ≤ N. (5.24)With our choice of L , it is clear that (5.24) is satisfied when n = 3 . Let us now presuppose that N ≥ and that (5.24) is true for all ≤ n ≤ m − , whereby m satisfies ≤ m ≤ N. It sufficesto prove that (5.24) holds true for n = m. To this end, we observe that ∂ mq ((1 + u ) − / ) = − ∂ m − q (cid:16) ((1 + u ) − / ) uu q (cid:17) , and that (5.22) together with the induction assumption imply k ∂ nq ((1 + u ) − / ) k α ≤ L n − / ( n − ≤ L n − ( n − , k ∂ nq u k α ≤ L n − ( n − , k ∂ nq u q k α ≤ k ∂ n +1 q u k α ≤ L n − ( n − for all ≤ n ≤ m − . These inequalities allow us to use the estimate (5.12) of Lemma 5.5 (with r = 1 , n = 2 , and N = m − ) in order to obtain that k ∂ mq ((1 + u ) − / ) k α = k ∂ m − q (((1 + u ) − / ) uu q ) k α ≤ C X l =0 k ∂ lq ((1 + u ) − / ) k α + X l =0 k ∂ lq u k α + X l =0 k ∂ lq u q k α ! × L m − ( m − ≤ L m − / ( m − , when L satisfies (5.22). This proves (5.24).In order to prove (5.23), we observe that our hypothesis together with (5.22) and the estimate(5.10) of the Lemma 5.5 (with r = 2 , n = 3 , and N = N ) imply that k ∂ nq (1 + u ) k α = k ∂ nq u k α ≤ C X l =0 k ∂ lq u k α ! L n − ( n − ≤ L n − / ( n − for all ≤ n ≤ N . Whence, invoking (5.24) and the estimate (5.10) of Lemma 5.5 (with r = 7 / , n = 3 , and N = N ), we deduce that k ∂ nq ((1 + u ) / ) k α = k ∂ nq ((1 + u )(1 + u ) − / ) k α ≤ C X l =0 k ∂ lq (1 + u ) k α + X l =0 k ∂ lq (1 + u ) − / k α ! L n − / ( n − ≤ L n − / ( n − for all ≤ n ≤ N, the last inequality being a consequence of our choice for L . This is the desiredclaim. (cid:3) Finally, we come to the proof of our second main result stated in Theorem 5.1.
Proof of Theorem 5.1.
The proof uses the induction principle. To this end, let C be the constantdefined by (5.13). We first pick a positive constant L which satisfies L ≥k ∂ q (1 /h p ) k α + k ∂ q (1 /h p ) k / α + C X l =0 (2 k ∂ lq (1 /h p ) k α + k ∂ lq h p k α ) ! + k ∂ q ((1 + h q ) − / ) k / α + k ∂ q ((1 + h q ) − / ) k α + k ∂ q h q k α + C X l =0 k ∂ lq h q k α ! + C X l =0 k ∂ lq ((1 + h q ) − / ) k α + X l =0 k ∂ lq h q k α + X l =0 k ∂ l +1 q h q k α ! + C X l =0 k ∂ lq (1 + h q ) k α + X l =0 k ∂ lq ((1 + h q ) − / ) k α ! + k ∂ q h q k α + X l =0 k ∂ lq h k α , (5.25)and observe that L ≥ . Moreover, this choice ensures that the estimate (5.1) is satisfied, forthis fixed L , when ≤ m ≤ . We next assume that (5.1) is true for all ≤ n ≤ m − whereby m ≥ , and are left to prove, possibly under some additional constraints on L (see (5.42)),that (5.1) holds also for m. Recalling the Proposition 5.4, we know that ∂ mq h is the solution ofthe elliptic boundary value problem (5.2). Proceeding as in the proof of Proposition 5.4, theSchauder estimate [17, Theorem 8.33] together with the inequality (5.8) show that the solution ∂ mq h of (5.2) can be estimated as follows k ∂ mq h k α ≤ K k ∂ mq h k + k f m k α + k g m k α + X i =1 k ϕ im k α ! . (5.26)Whence, we are left to prove that the right-hand side of (5.26) can be bounded from aboveby L m − ( m − . To establish this property, we notice that the induction assumption impliesthat max {k ∂ nq h q k α , k ∂ nq h p k α } ≤ k ∂ nq h k α ≤ L n − ( n − for ≤ n ≤ m − . (5.27)It is now immediate to observe, due to (5.25), that L ≥ max {k ∂ q h q k α , k ∂ q h p k α } , so that we alsohave max {k ∂ nq h q k α , k ∂ nq h p k α } ≤ L n − / ( n − for ≤ n ≤ m − . (5.28) APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 21
On the other hand, the relation satisfied by L , the estimate (5.27), and the assumption (2.6)on h , guarantee that the function u := h p satisfies the assumptions of the Lemma 5.7 (with N = m − ). Therefore, we conclude that k ∂ nq (1 /h p ) k α ≤ L n − / ( n − for all ≤ n ≤ m − , (5.29)which implies, in view of L ≥ k ∂ q (1 /h p ) k α that k ∂ nq (1 /h p ) k α ≤ L n − / ( n − for all ≤ n ≤ m − . (5.30)With these preparations, we start and estimate the right-hand side of (5.26). We begin byobserving that k ∂ mq h k ≤ k ∂ m − q h k α ≤ L m − ( m − . (5.31)Next we consider the expressions k f m k α and k g m k α . The arguments used for bounding thesequantities are quite similar, and we will present them in detail only when estimating a repre-sentative term of f m . Indeed, recalling (5.28), (5.30), and the estimates (5.10)-(5.12) of Lemma5.5 (with r = 3 / , n = 2 , and N = m − ), we conclude that there exists a constant K > such that max ((cid:13)(cid:13)(cid:13)(cid:13) ∂ nq (cid:18) h p (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) α , (cid:13)(cid:13)(cid:13)(cid:13) ∂ nq (cid:18) h q h p (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) α , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ nq h q h p !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ) ≤ K L n − / ( n − (5.32)for all ≤ n ≤ m − . With this observation at hand, it is not difficult to see that all the termsdefining f m and g m can be estimated by using the same arguments as when dealing with thefollowing representative term of f mm − X k =1 (cid:18) m − k (cid:19) ∂ kq (cid:18) h p (cid:19) ∂ q ( ∂ m − kq h ) = X k =1 + m − X k =2 + m − X k = m − ! (cid:18) m − k (cid:19) ∂ kq (cid:18) h p (cid:19) ∂ q ( ∂ m − kq h ) . In view of (5.32) and of the induction assumption, we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k =1 + m − X k = m − ! (cid:18) m − k (cid:19) ∂ kq (cid:18) h p (cid:19) ∂ q ( ∂ m − kq h ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ K L m − / ( m − . On the other hand, using additionally the relations (5.19) and (5.27), we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =2 (cid:18) m − k (cid:19) ∂ kq (cid:18) h p (cid:19) ∂ q ( ∂ m − kq h ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ m − X k =2 (cid:18) m − k (cid:19) (cid:13)(cid:13)(cid:13)(cid:13) ∂ kq h p (cid:13)(cid:13)(cid:13)(cid:13) α (cid:13)(cid:13)(cid:13) ∂ m − kq h (cid:13)(cid:13)(cid:13) α ≤ K L m − / m − X k =2 (cid:18) m − k (cid:19) ( k − m − k − ≤ K L m − / ( m − m − X k =2 ( m − ( m − k − ( k − ≤ K L m − / ( m − . Proceeding in the same way with the remaining terms of f m and g m , we end up with k f m k α + k g m k α ≤ K L m − / ( m − . (5.33) We are left to estimate the terms k ϕ im k α , ≤ i ≤ . To this end, we observe that our choiceof the constant L and the induction assumption k ∂ nq h k α ≤ L n − ( n − for ≤ n ≤ m − yield, via Lemma 5.8, that k ∂ nq ((1 + h q ) / ) k α ≤ L n − / ( n − for all ≤ n ≤ m − . (5.34)Since (5.27) and the induction assumption imply max {k ∂ nq h q k α , k ∂ nq h p k α } ≤ L n − / ( n − for ≤ n ≤ m − , (5.35)we find together with (5.34) and the relations (5.10)-(5.12) of Lemma 5.5 (with r = 3 / , n = 3 ,and N = m − ) that max n(cid:13)(cid:13)(cid:13) ∂ nq (cid:0) h q (cid:1) / (cid:13)(cid:13)(cid:13) α , (cid:13)(cid:13)(cid:13) ∂ nq (cid:0) h q (cid:1) / (cid:13)(cid:13)(cid:13) α o ≤ K L n − / ( n − , max n(cid:13)(cid:13)(cid:13) ∂ nq (cid:16) h q (cid:0) h q (cid:1) / (cid:17)(cid:13)(cid:13)(cid:13) α , (cid:13)(cid:13)(cid:13) ∂ nq (cid:16) h q (cid:0) h q (cid:1) / (cid:17)(cid:13)(cid:13)(cid:13) α o ≤ K L n − / ( n − (5.36)for all ≤ n ≤ m − . On the other hand, the relation (5.29) and the estimates (5.10)-(5.11) ofLemma 5.5 (with r = 3 / , n = 3 , and N = m − ) yield max n(cid:13)(cid:13) ∂ nq (cid:0) /h p (cid:1)(cid:13)(cid:13) α , (cid:13)(cid:13) ∂ nq (cid:0) /h p (cid:1)(cid:13)(cid:13) α o ≤ K L n − / ( n − (5.37)for all ≤ n ≤ m − . In view of (5.35)-(5.37) and of k ∂ nq h k α ≤ K L n − / ( n − for ≤ n ≤ m, (5.38)the C α − norm of the functions ϕ m and ϕ m can be estimated by the same quantity. Moreprecisely, we have k ϕ m k α ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =1 (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α , and we split the sum in the latter sum as follows m − X k =1 (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h p = X k =1 + m − X k =3 + m − X k = m − ! (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h p . Recalling (5.36) and (5.37), we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k =1 + m − X k = m − ! (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ K L m − / ( m − . When estimating the middle sum we take advantage of (5.19), (5.36), and (5.37) to find (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =3 (cid:18) mk (cid:19) ( ∂ kq (1 + h q ) / ) ∂ m − kq h p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ m − X k =3 (cid:18) mk (cid:19) (cid:13)(cid:13)(cid:13) ∂ kq (1 + h q ) / (cid:13)(cid:13)(cid:13) α (cid:13)(cid:13)(cid:13)(cid:13) ∂ m − kq h p (cid:13)(cid:13)(cid:13)(cid:13) α ≤ K m − X k =3 (cid:18) mk (cid:19) L k − / ( k − L m − k − / ( m − k − ≤ K L m − ( m − m − X k =3 m ( k − ( m − k − ≤ K L m − ( m − . APILLARY-GRAVITY WATER WAVES WITH BOUNDED VORTICITY 23
The arguments being also true when estimating ϕ m , we conclude that X i =1 k ϕ im k α ≤ K L m − / ( m − . (5.39)Finally, recalling (5.35)-(5.37), one can easily see that the norms k ϕ im k α , i ∈ { , , } , maybe bounded by the same quantity. Indeed, we have that k ϕ m k α ≤ K (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =0 (cid:18) m − k (cid:19) ( ∂ k +1 q h p ) ∂ m − k − q h p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α , and we split the sum on the right-hand side of the latter inequality as follows m − X k =0 (cid:18) m − k (cid:19) ( ∂ k +1 q h p ) ∂ m − k − q h p = X k =0 + m − X k =2 + m − X k = m − ! (cid:18) m − k (cid:19) ( ∂ k +1 q h p ) ∂ m − k − q h p . The relations (5.35) and (5.37) imply that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k =0 + m − X k = m − ! (cid:18) m − k (cid:19) ( ∂ k +1 q h p ) ∂ m − k − q h p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ K L m − / ( m − and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =2 (cid:18) m − k (cid:19) ( ∂ k +1 q h p ) ∂ m − k − q h p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α ≤ m − X k =2 (cid:18) m − k (cid:19) k ∂ k +1 q h p k α (cid:13)(cid:13)(cid:13)(cid:13) ∂ m − k − q h p (cid:13)(cid:13)(cid:13)(cid:13) α ≤ K L m − m − X k =2 (cid:18) m − k (cid:19) ( k − m − k − ≤ K L m − ( m − m − X k =2 m ( k − ( m − k − ≤ K L m − ( m − , meaning that X i =3 k ϕ im k α ≤ K L m − / ( m − . (5.40)Gathering (5.26), (5.31), (5.33), (5.39), and (5.40), we conclude that k ∂ mq h k α ≤ K (1 + K + K + K ) L m − / ( m − , (5.41)the constants K i being independent of m and L . Therefore, we may require, additionally to(5.25), that the constant L should also satisfy L ≥ K (1 + K + K + K ) . (5.42)This additional restriction and (5.41) lead to the desired conclusion. (cid:3) We conclude the paper with the proof of our main existence result.
Proof of Theorem 2.1.
The proof of Theorem 2.1 follows by combining the assertions of theLemmas 3.2 and 4.7 and that of the Theorems 4.6 and 5.1. (cid:3)
Acknowledgement
A.-V. Matioc was supported by the ERC Advanced Grant “Nonlinearstudies of water flows with vorticity” (NWFV).
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Institut für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria
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