Symmetry breaking in ideal magnetohydrodynamics: the role of the velocity
aa r X i v : . [ m a t h . A P ] F e b Symmetry breaking in ideal magnetohydrodynamics:the role of the velocity
Dimitri Cobb and Francesco Fanelli , Université de Lyon, Université Claude Bernard Lyon 1
Institut Camille Jordan – UMR 5208
43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, FRANCE [email protected] , [email protected] March 1, 2021
Abstract
The ideal magnetohydrodynamic equations are, roughly speaking, a quasi-linear symmetric hyperbolic system ofPDEs, but not all the unknowns play the same role in this system. Indeed, in the regime of small magnetic fields,the equations are close to the incompressible Euler equations. In the present paper, we adopt this point of viewto study questions linked with the lifespan of strong solutions to the ideal magnetohydrodynamic equations. Firstof all, we prove a continuation criterion in terms of the velocity field only. Secondly, we refine the explicit lowerbound for the lifespan of -D flows found in [11], by relaxing the regularity assumptions on the initial magneticfield. Keywords: ideal MHD; velocity field; Elsässer variables; blow-up criterion; lifespan.
In this paper, we are concerned with ideal incompressible magnetohydrodynamics (MHD forshort), which are governed by the following system of equations:(1) ∂ t u + ( u · ∇ ) u + ∇ (cid:18) Π + 12 | b | (cid:19) = ( b · ∇ ) b∂ t b + ( u · ∇ ) b − ( b · ∇ ) u = 0div( u ) = 0 . These equations describe the motion of an ideal incompressible magnetofluid, that is an inviscid,perfectly conducting and incompressible fluid which is subject to a self-generated magnetic field.We set these equations on the whole d -dimensional space R d . The vector fields u, b : R × R d −→ R d are, respectively, the velocity and magnetic fields of the fluid, while the scalar quantity Π : R × R d −→ R is the hydrodynamic pressure.Since the 1930s, this set of equations has been the subject of intense studies by physicistsand, for the past thirty years, by mathematicians. Amongst the many problems that have beenexplored (wave propagation, magnetofluid topology, stationary solutions and their stability, etc .),the well-posedness theory has proven to be one of the most challenging, as it is still unknown1hether system (1) possesses global solutions, even in the case of two dimensions of space. Thiscontrasts very much with the case of the incompressible Euler equations, with no magnetic field,where the existence and uniqueness of a global regular solution has been proved for planar solutions d = 2 . The presence of the magnetic field makes all the methods that work for the Euler systeminoperable for ideal MHD.In this article, we extend some results we previously obtained in [11] regarding the lifespan ofBesov-Lipschitz solutions. In doing so, we will highlight the fact that the velocity field u playsa special role in (1), in that we will require different levels of regularity on u and b to prove acontinuation criterion and a lower bound on the lifespan of solutions. We point out that this typeof results owes to the particular nature of ideal MHD, as it reaches further than the standardtheory of quasi-linear symmetric hyperbolic systems, where all unknowns must recieve a similartreatment due to the symmetric nature of the system.To begin with, let us present, in the next subsection, some generalities on the ideal MHDsystem (1); there, we will also recall some well-posedness results obtained in [11]. Those resultsconstitute the starting point of the present analysis. Roughly speaking, equations (1) can be viewed as a first-order quasi-linear symmetric hyperbolicsystem (of course, this is not completely correct, due to the presence of the pressure term). Thus,it is natural to study well-posedness questions in a functional framework based on finite energyconditions.However, the structure of the equations is much richer than that, as may be highlighted by achange of unknowns. Specifically, by introducing the so-called
Elsässer variables (2) α = u + b and β = u − b , the ideal MHD system (1) can be recasted into the following system of transport equations:(3) ∂ t α + ( β · ∇ ) α + ∇ π = 0 ∂ t β + ( α · ∇ ) β + ∇ π = 0div( α ) = div( β ) = 0 . In the above, π and π are two possibly distinct scalar functions, which enforce the two indepen-dent divergence-free conditions div( α ) = 0 and div( β ) = 0 . While it is clear that all solutions ofthe ideal MHD system also solve (3), with in addition ∇ π = ∇ π = ∇ (Π+ | b | / , the converse isnot, in general, true without imposing some kind of condition the solutions must satisfy at infinity | x | −→ + ∞ . For instance, if the solution ( α, β ) of (3) lies in some L p space, where ≤ p < + ∞ ,then it can be shown that ( u, b ) , obtained inverting transformation (2), solves (1). We refer tothe discussion in Section 4 of [11] for more on this issue; see also [9] for the statement of a sharpequivalence result.To the best of our knowledge, the Elsässer formulation (3) of the ideal MHD equations wasinvolved, in a way or another, in all well-posedness results for system (1) obtained so far, startingfrom the the very first works of Schmidt [16] and Secchi [17]. We refer e.g. to [5], [8], [3], [11]and references therein for more recent studies. It should be noted that equations (3) are basicallya system of transport equations. This makes it possible to propagate integrability assumptionsother than L , and to solve the ideal MHD system (provided the equivalence between equations(1) and (3) holds) in spaces based on L p conditions, for any p ∈ ]1 , + ∞ ] .The previous observation was used in [11] to solve the ideal MHD system in endpoint Besovspaces B s ∞ ,r included in the space of globally Lipschitz functions. The result is given in the nextstatement (see Theorems 2.1 and 2.4 in [11]). We remark that the finite energy condition on the2olutions is there, mainly to guarantee the equivalence between the original system (1) and itsElsässer formulation (3). We refer to Theorem 4.3 in [11] for more details about that issue. Theorem 1.1.
Let d ≥ . Let ( s, r ) ∈ R × [1 , + ∞ ] satisfy either s > , or s = r = 1 . Then theideal MHD system (1) is well-posed, locally in time, in the space X sr := (cid:26) ( u, b ) ∈ (cid:16) B s ∞ ,r ( R d ) (cid:17) (cid:12)(cid:12)(cid:12) div ( u ) = div ( b ) = 0 and u , b ∈ L ( R d ) (cid:27) , and there exists a T > such that the flow map t (cid:0) u ( t ) , b ( t ) (cid:1) belongs to C (cid:0) [0 , T ]; X sr (cid:1) if ≤ r < + ∞ , to C w (cid:0) [0 , T ]; X s ∞ (cid:1) if r = + ∞ . In addition, if T < + ∞ and (4) Z T (cid:16)(cid:13)(cid:13) ∇ u ( t ) (cid:13)(cid:13) L ∞ + (cid:13)(cid:13) ∇ b ( t ) (cid:13)(cid:13) L ∞ (cid:17) d t < + ∞ , then ( u, b ) can be continued beyond T into a solution of (1) with the same regularity. Solving in critical spaces becomes particularly important in the case of space dimension d = 2 ,because, in that setting, one can show an improved lower bound on the lifespan of the solutions.We point out that this bound does not rely on classical quasi-linear hyperbolic theory, but is reallytied to the special structure of the equations. The precise estimate is contained in the followingstatement (this corresponds to Theorem 2.6 of [11]) Theorem 1.2.
Consider an initial datum (cid:0) u , b (cid:1) such that u , b ∈ L ( R ) ∩ B ∞ , ( R ) , with div ( u ) = div ( b ) = 0 . Then, the lifespan T > of the corresponding solution ( u, b ) of the -Dideal MHD problem (1) , given by Theorem 1.1, enjoys the following lower bound: T ≥ C (cid:13)(cid:13) ( u , b ) (cid:13)(cid:13) L ∩ B ∞ , log C log C log C (cid:13)(cid:13) ( u , b ) (cid:13)(cid:13) L ∩ B ∞ , k b k B ∞ , , where C > is a “universal” constant, independent of the initial datum. The interest for the previous statement comes from the fact that it implies an “asymptoticallyglobal” well-posedness result, in the following sense: if, for some ε > , one has k b k B ∞ , ∼ ε ,then the lifespan T ε > of the corresponding solution verifies the property T ε −→ + ∞ for ε → + . This is consistent with the fact that, in the regime ε → + , the ideal MHD system (1)reduces to the incompressible Euler equations, which are globally well-posed in -D.A phenomenom of this type has already been observed in [12] for the non-homogeneous incom-pressible Euler system, where, in the regime of near constant densities, the lifespan of the uniqueBesov-Lipschitz solution can be shown to tend to infinity, with an explicit lower bound for thelifespan. See also [13] for a similar result for a quasi-incompressible Euler system. However, wehave to remark that the lower bound of Theorem 1.2 does not hold in a critical setting (namely,at the level of B ∞ , regularity), and requires higher smoothness assumptions for both initial data u and b .We will comment a little bit more on the contents of Theorems 1.1 and 1.2 in the nextsubsection, when presenting an overview of our main results. The previous Theorems 1.1 and 1.2 deal with the velocity field and the magnetic field in a quitesymmetric way (apart, of course, in the explicit lower bound on the lifespan in the second state-ment, where b plays a special role). However, looking at the equations hints that the velocity u and the magnetic field b do not play exactly the same role in system (1), despite the symmetricstructure of the system: in fact, the magnetic field equation is bilinear in ( u, b ) .3ur main purpose here is to push forward this observation as far as we can, especially in twodirections. First of all, we aim at finding a continuation criterion in terms only of u . Secondly,we want to establish a lower bound on the lifespan of the solutions in dimension d = 2 , whichrequires additional regularity B ∞ , only on the initial velocity field u .We will explain better the improvements in both directions here below. Before doing this, wewant to clarify that, in our analysis, we will need to resort again to the Elsässer system (3). Thus,in order to guarantee the equivalence between (1) and (3), we place ourselves in the same settingadopted in [11], namely we will always work in the framework of finite energy solutions of the idealMHD equations. Besides, this framework will enable us to use quite freely the Leray projectionoperator P , and perform B s ∞ ,r estimates on the projected system. This differs from the approachemployed in [11], which was based on recasting the equations in the vorticity formulation. Theorem 1.1, and especially the continuation condition (4), points at a classical phenomenon inthe context of quasilinear symmetric hyperbolic systems: the lifespan T ∗ of solutions may becharacterized by the finiteness of their L T ∗ ( W , ∞ ) norms. In fact, as for the Beale-Kato-Majdacontinuation criterion [2] for the Euler equations, it is possible to show (see [5]) that a time T > is prior to explosion, namely T < T ∗ , if and only if Z T (cid:16) k ω k L ∞ + k j k L ∞ (cid:17) d t < + ∞ , where ω = curl ( u ) and j = curl ( b ) are the vorticity and electrical current matrices of the fluid.A number of refinements to that criterion exist for ideal MHD, in different spaces. We refer e.g. to [6] or [8] for results in that spirit.However, we have to remark that, contrary to the momentum equation, the magnetic fieldequation is linear with respect to b . So, we may wish for a continuation criterion based on thevelocity alone. In this respect, we will prove that T < T ∗ as long as(5) Z T (cid:13)(cid:13) ∇ u ( t ) (cid:13)(cid:13) L ∞ d t < + ∞ . The fact that one needs second order derivatives of u in the previous criterion is reasonable. Thisloss of derivatives has to be ascribed to the hyperbolic nature of the system: bounding ∇ b in L ∞ requires a control on ∇ u , since estimates cannot be closed in a L ∞ setting.Similarly, we notice that the Elsässer variables α = u + b and β = u − b solve linear equations(3) too. This makes it possible to establish a continuation criterion based on either α or β . Moreprecisely, we prove that T < T ∗ as long as(6) Z T (cid:13)(cid:13) ω ± j (cid:13)(cid:13) B ∞ , d t < + ∞ . In fact, we will prove that (cid:13)(cid:13) ω + j (cid:13)(cid:13) L T ( B ∞ , ) < + ∞ if and only if (cid:13)(cid:13) ω − j (cid:13)(cid:13) L T ( B ∞ , ) < + ∞ , so thatthe ± sign in (6) is not ambiguous. Of course, this is not really surprising, as the magnetic fieldis a pseudovector: the equations remain unchanged when substituting − b to b .To conclude this part, we remark that the continuation cirterion (5), although it requires acontrol of the second derivative ∇ u , can be formulated with the L ∞ norm, unlike (6) whichuses the B ∞ , one. This is a consequence of the fact that we work at critical regularity B ∞ , ,combined with a fundamental property of the magnetic field equation: it naturally preserves thedivergence-free property div( b ) = 0 in time. This means that no addition of a gradient term isneeded to keep the magnetic field solenoidal, resulting in a simpler evolution equation.4 .2.2 Improved lower bound for the lifespan of solutions As already mentioned, there is no global well-posedness theory for the ideal MHD system, evenin the case of two dimensions of space. However, as highlighted by Theorem 1.2 above, we expectbetter behaviour from the solutions in the regime of small magnetic fields, as, in this case, thesystem is close to the -D Euler equations.We remark that, for Theorem 1.2 to hold, the initial data must possess at least B ∞ , regularity.Indeed, the method of the proof (which is contained [11]) required to find lower order estimates(namely, in B ∞ , ) for the magnetic field, and use it as a measure of how close the solution ( u, b ) is to the Euler system: if b is small, then ( u, b ) almost solves the Euler equations. This was aproblem, as the magnetic field equation involves first order derivatives of the velocity, so that, ina non-Hilbertian functional framework, such estimates can only be based on higher order ones for u , whence the regularity assumption on the initial data.In this paper, we use a different method to relax this regularity requirement: we will only needthe initial velocity field u to be B ∞ , , while b ∈ B ∞ , will suffice. To achieve this improvement,we will instead compare directly the ideal MHD system to the Euler equations, which we knowhave a global solution v at the level of regularity of the initial datum u . In doing this, instead ofusing the magnetic field to measure the proximity with the Euler system, we introduce Elsässer-type variables δα = u + b − v and δβ = u − b − v, which have the nice property of solving a set of transport equations whose forcing terms onlyinvolve derivatives with respect to the Euler solution v .Before moving on, let us comment a bit further on the physical nature of -D ideal MHD.While -D Euler equations can easily be understood as simply describing a planar fluid, thatis not quite so with MHD, which has an inherent three-dimensional nature: the magnetic fieldcirculates around the electrical current, so they cannot be simultaneously coplanar.In -D MHD, the fluid evolves in a plane, but the electrical current is always normal to theplane of the fluid, so that the magnetic field will indeed be planar. In particular, the electricalcurrent may always be represented as a scalar function j = ∂ b − ∂ b . This is of course analogousto the fact that planar fluids have a -D vorticity that is, at all times, normal to the plane ofmotion. Structure of the paper
Before concluding this introduction, we give a short overview of the paper.In the next section, we introduce some tools from Fourier analysis and Littlewood-Paley theory,which we will need in our analysis. Section 3 is devoted to the statement and proof of some newcontinuation criteria, as described in Paragraph 1.2.1 above. Finally, the improved lower boundon the lifespan of the solutions in two space dimensions, requiring higher regularity on the initialvelocity only, will be the topic of Section 4.
Acknowledgements
The authors wish to express their deep gratitude to Raphaël Danchin, whose interesting remarks about apreliminary version of their previous work [11] motivated the present study.The work of the second author has been partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissement d’Avenir” (ANR-11-IDEX-0007),and by the projects BORDS (ANR-16-CE40-0027-01), SingFlows (ANR-18-CE40-0027), all operated bythe French National Research Agency (ANR). Both authors have been partially supported by the projectCRISIS (ANR-20-CE40-0020-01), operated by the French National Research Agency (ANR). Fourier analysis toolbox
In this section, we give a summary of the harmonic analysis tools we will use throughout thisarticle. We start by giving the main ideas of Littlewood-Paley analysis and paradifferential cal-culus, and then we present their use in the theory of transport equations. We conclude with ashort section on the Leray projector, where we present some useful inequalities involving it.If not otherwise specified, we refer to Chapter 2 of [1] for full details on this part.
Here we recall the basic principles of Littlewood-Paley theory. We focus only on the R d case, eventhough a similar analysis can be performed also in the case of the torus T d .First of all, let us introduce a non-homogeneous dyadic partition of unity with respect to theFourier variable. We fix a smooth radial function χ supported in the ball B (0 , , equal to in aneighborhood of B (0 , and such that r χ ( r e ) is nonincreasing over R + for all unitary vectors e ∈ R d . Set ϕ ( ξ ) = χ ( ξ ) − χ (2 ξ ) and ϕ j ( ξ ) := ϕ (2 − j ξ ) for all j ≥ . The dyadic blocks (∆ j ) j ∈ Z are defined by ∆ j := 0 if j ≤ − , ∆ − := χ ( D ) and ∆ j := ϕ (2 − j D ) if j ≥ . We also introduce the following low frequency cut-off operator:(7) S j u := χ (2 − j D ) = X k ≤ j − ∆ k for j ≥ . Note that S j is a convolution operator. More precisely, if we denote F ( f ) = b f the Fouriertransform of a function f and F − the inverse Fourier transform, after defining K := F − χ and K j ( x ) := F − (cid:2) χ (2 − j · ) (cid:3) ( x ) = 2 jd K (2 j x ) , we have, for all j ∈ N and all tempered distributions u ∈ S ′ , that S j u = K j ∗ u . Thus the L normof K j is independent of j ≥ , hence S j maps continuously L p into itself, for any ≤ p ≤ + ∞ .With this preparation, the following Littlewood-Paley decomposition of tempered distributionsholds true:(8) ∀ u ∈ S ′ , u = X j ≥− ∆ j u in S ′ . Next, let us recall the so-called
Bernstein inequalities , which explain the way derivatives acton spectrally localised functions.
Lemma 2.1.
Let < r < R . A constant C exists so that, for any non-negative integer k , anycouple ( p, q ) in [1 , + ∞ ] , with p ≤ q , and any function u ∈ L p , we have, for all λ > , Supp b u ⊂ B (0 , λR ) = ⇒ k∇ k u k L q ≤ C k +1 λ k + d (cid:16) p − q (cid:17) k u k L p ;Supp b u ⊂ { ξ ∈ R d | rλ ≤ | ξ | ≤ Rλ } = ⇒ C − k − λ k k u k L p ≤ k∇ k u k L p ≤ C k +1 λ k k u k L p . The second Bernstein inequality may be extended to Fourier multipliers whose symbol are ho-mogeneous functions. This is particularly useful when dealing with the Leray projection operator.
Lemma 2.2.
Let σ : R d \{ } −→ C be a smooth homogeneous function of degree m ∈ Z . Then,for all j ≥ and p ∈ [1 , + ∞ ] , we have ∀ u ∈ S ′ , k σ ( D )∆ j f k L p ≤ C jm k ∆ j u k L p . Throughout we agree that f ( D ) stands for the pseudo-differential operator u
7→ F − [ f ( ξ ) b u ( ξ )] . Definition 2.3.
Let s ∈ R and ≤ p, r ≤ + ∞ . The non-homogeneous Besov space B sp,r = B sp,r ( R d ) is defined as the subset of tempered distributions u for which k u k B sp,r := (cid:13)(cid:13)(cid:13)(cid:0) js k ∆ j u k L p (cid:1) j ≥− (cid:13)(cid:13)(cid:13) ℓ r < + ∞ . In this article, we mainly work with the B m ∞ , spaces ( m = 0 , , ), which are embedded inusual spaces of bounded functions: we have B m ∞ , ֒ → W m, ∞ . In particular, the space B ∞ , is contained in the space of globally Lipschitz functions W , ∞ . In this subsection, we recall some useful results from paradifferential calculus. We mainly focus onthe Bony paraproduct decomposition (after J.-M. Bony, see [4]) and on some basic commutatorestimates.We start by introducing the paraproduct operator. Formally, the product of two tempereddistributions u and v may be decomposed into u v = T u ( v ) + T v ( u ) + R ( u, v ) , where we have defined T u ( v ) := X j S j − u ∆ j v and R ( u, v ) := X j X | k − j |≤ ∆ j u ∆ k v . The above operator T is called “paraproduct” whereas R is called “remainder”. The paraproductand remainder operators have many nice continuity properties. The following ones will be ofconstant use in this paper. Proposition 2.4.
For any ( s, p, r ) ∈ R × [1 , + ∞ ] and t > , the paraproduct operator T mapscontinuously L ∞ × B sp,r in B sp,r and B − t ∞ , ∞ × B sp,r in B s − tp,r . Moreover, the following estimates hold: kT u ( v ) k B sp,r ≤ C k u k L ∞ k∇ v k B s − p,r and kT u ( v ) k B s − tp,r ≤ C k u k B − t ∞ , ∞ k∇ v k B s − p,r . For any ( s , p , r ) and ( s , p , r ) in R × [1 , + ∞ ] such that s + s > , /p := 1 /p + 1 /p ≤ and /r := 1 /r + 1 /r ≤ , the remainder operator R maps continuously B s p ,r × B s p ,r into B s + s p,r . In the case s + s = 0 , provided r = 1 , the operator R is continuous from B s p ,r × B s p ,r with values in B p, ∞ . The consequence of this proposition is that the spaces B s ∞ ,r are Banach algebras as long as s > . Instead, notice that the space B ∞ ,r is not an algebra.Now, we switch to considering some commutator estimates. The first one is contained in thenext statement (see Lemma 2.100 and Remark 2.101 in [1]). Lemma 2.5.
Assume that v ∈ B s ∞ ,r with ( s, r ) ∈ R × [1 , + ∞ ] satisfying s > , or s = r = 1 .Denote by (cid:2) v · ∇ , ∆ j (cid:3) f = ( v · ∇ )∆ j − ∆ j ( v · ∇ ) the commutator between the transport operator v · ∇ and the frequency localisation operator ∆ j . Then we have ∀ f ∈ B s ∞ ,r , js (cid:13)(cid:13)(cid:2) v · ∇ , ∆ j (cid:3) f (cid:13)(cid:13) L ∞ . c j (cid:16) k∇ v k L ∞ k f k B s ∞ ,r + k∇ v k B s − ∞ ,r k∇ f k L ∞ (cid:17) , where (cid:0) c j (cid:1) j ≥− is a sequence belonging to the unit ball of ℓ r . ξ = 0 . Lemma 2.6.
Let κ be a smooth function on R d \ { } , which is homogeneous of degree m awayfrom a neighborhood of . Then, for a vector field v such that ∇ v ∈ L ∞ , one has: ∀ f ∈ B s ∞ ,r , (cid:13)(cid:13)(cid:2) T v , κ ( D ) (cid:3) f (cid:13)(cid:13) B s − m +1 ∞ ,r . k∇ v k L ∞ k f k B s ∞ ,r . Additional commutator estimates, involving the Leray projection operators, are postponed toSubsection 2.4.
In this section, we focus on transport equations in non-homogeneous Besov spaces. We refer toChapter 3 of [1] for a complete presentation of the subject. We study the initial value problem(9) ( ∂ t f + v · ∇ f = gf | t =0 = f . We will always assume the velocity field v = v ( t, x ) to be a Lipschitz divergence-free function, i.e. div( v ) = 0 . It is therefore practical to formulate the following definition: the triplet ( s, p, r ) ∈ R × [1 , + ∞ ] is said to satisfy the Lipschitz condition if the inequality(10) s > dp or s = dp and r = 1 . holds. As we have explained above, this implies the embedding B sp,r ֒ → W , ∞ .The main well-posedness result concerning problem (9) in Besov spaces is contained in thefollowing statement, stated in the case p = + ∞ (the only relevant one for our analysis). We recallhere that, when X is Banach, the notation C w (cid:0) [0 , T ]; X (cid:1) refers to the space of functions whichare continuous in time with values in X endowed with its weak topology. Theorem 2.7.
Let ( s, r ) ∈ R × [1 , + ∞ ] satisfy the Lipschitz condition (10) with p = + ∞ . Givensome T > , let g ∈ L T ( B s ∞ ,r ) . Assume that v ∈ L T ( B s ∞ ,r ) and that there exist real numbers q > and M > for which v ∈ L qT ( B − M ∞ , ∞ ) . Finally, let f ∈ B s ∞ ,r be an initial datum.Then, the transport equation (9) has a unique solution f in: • the space C (cid:0) [0 , T ]; B s ∞ ,r (cid:1) , if r < + ∞ ; • the space (cid:16)T s ′ . As discovered by Vishik [18] and, with a different proof, by Hmidi and Keraani [14], theprevious statement can be improved when the Besov regularity index is s = 0 , provided div( v ) = 0 .Precisely, under these conditions, the estimate in Theorem 2.7 can be replaced by an inequalitywhich is linear with respect to k∇ v k L T ( L ∞ ) . Theorem 2.8.
Assume that ∇ v ∈ L T ( L ∞ ) and that v is divergence-free. Let r ∈ [1 , + ∞ ] . Thenthere exists a constant C = C ( d ) such that, for any solution f to problem (9) in C (cid:0) [0 , T ]; B ∞ ,r (cid:1) ,with the usual modification of C into C w if r = + ∞ , we have k f k L ∞ T ( B ∞ ,r ) ≤ C (cid:26) k f k B ∞ ,r + k g k L T ( B ∞ ,r ) (cid:27) (cid:18) Z T k∇ v ( τ ) k L ∞ d τ (cid:19) . .4 Leray projection This paragraph is concerned with the Leray projection operator P , which is defined as P := Id + ∇ ( − ∆) − div in the sense of Fourier multipliers, namely ∀ f ∈ S , \ ( P f ) j ( ξ ) = b f j ( ξ ) − X k ξ j ξ k | ξ | b f k ( ξ ) . The operator P can also be seen as a singular integral operator. Thus, Calderón-Zygmundtheory may be applied to prove that P defines a bounded operator in the L p −→ L p topology, forany < p < + ∞ .In the endpoint space L ∞ , the Leray projector is no longer defined as a Fourier multiplier,because of the singularity of its symbol at ξ = 0 . However, for j ∈ { , ..., d } , one may give senseof the operator ∆ − P ∂ j in L ∞ , by using the integrability properties of the fundamental solutionof the Laplacian. In fact, we have the following result, which corresponds to Proposition 8 of [15]. Proposition 2.9.
Let j ∈ { , ..., d } . We have a bounded operator ∆ − P ∂ j : L ∞ −→ L ∞ . With this proposition at hand, we can prove the following statement. It is a commutatorestimate between a transport operator and the Leray projector in the critical Besov space B ∞ , .It corresponds to Lemma 2.5 in [10], although that result did not deal with the endpoint exponent p = + ∞ . Lemma 2.10.
Let f, g ∈ L ∩ B ∞ , be two vector fields such that div ( f ) = div ( g ) = 0 . Thefollowing inequality holds true: (cid:13)(cid:13)(cid:2) f · ∇ , P (cid:3) g (cid:13)(cid:13) B ∞ , . k f k B ∞ , k g k B ∞ , . Proof.
We start by noticing that, thanks to the regularity assumption on both f and g , thecommutator is well-defined, since P always acts on L vector fields. This also guarantees us that P g = g . Using this fact, we can write the Bony decomposition for the products involved in thecommutator: we get (cid:2) f · ∇ , P (cid:3) g = d X k =1 (cid:16)(cid:2) T f k , P (cid:3) ∂ k g + T P ∂ k g ( f k ) − P T ∂ k g ( f k ) + R ( P ∂ k g, f k ) − P R ( ∂ k g, f k ) (cid:17) = d X k =1 (cid:16)(cid:2) T f k , P (cid:3) ∂ k g + T ∂ k g ( f k ) − P ∂ k T g ( f k ) + R ( ∂ k g, f k ) − P ∂ k R ( g, f k ) (cid:17) . The first, second and fourth summand are easy to bound, by using Lemma 2.6 and Proposition 2.4.The only new difficulty comes from the third and fifth terms, which involve the Leray projection.However, since the operator P is always written in composition with a derivative ∂ k in those terms,we may use Proposition 2.9 to write k P ∂ k T g ( f k ) k B ∞ , ≤ k ∆ − P ∂ k T g ( f k ) k L ∞ + X m ≥ m k ∆ m P T ∂ k g ( f k ) k L ∞ . kT g ( f ) k L ∞ + X m ≥ m k ∆ m P T ∂ k g ( f k ) k L ∞ . m ≥ , the operator ∆ m P is bounded on L ∞ . Thus, we mayuse Proposition 2.4 to obtain m k ∆ m P T ∂ k g ( f k ) k L ∞ . k f k B ∞ , k g k B ∞ , c m , for a suitable sequence (cid:0) c m (cid:1) m ∈ N ∈ ℓ of unitary norm. In the end, we get the estimate k P ∂ k T g ( f k ) k B ∞ , . k f k B ∞ , k g k B ∞ , . For the fifth term P ∂ k R ( g, f k ) , we can proceed in a similar way. This completes the proof of thesought bounds for the commutator.From the previous lemma, we immediately deduce the next result. Corollary 2.11.
Let f ∈ L ∩ B ∞ , and v ∈ L ∩ B ∞ , be two divergence-free vector fields. Thefollowing inequality holds true: k P ( f · ∇ ) v k B ∞ , . k f k B ∞ , k v k B ∞ , . Proof.
For proving the previous statement, it is enough to write P ( f · ∇ ) v = ( f · ∇ ) v − (cid:2) f · ∇ , P (cid:3) v , where we have used also the fact that div ( v ) = 0 . The first term in the right-hand side can beestimates directly, whereas we use the bounds of Lemma 2.10 for the second one. In this section, we state and prove our main results concerning continuation criteria for solutionsof the ideal MHD equations (1). In Subsection 3.1, we focus on a criterion based only on thevelocity field, while in Subsection 3.2 we will present a continuation criterion in terms of theElsässer variables.
The main result of this section is the following statement.
Theorem 3.1.
Let ( u , b ) ∈ L ( R d ) ∩ B ∞ , ( R d ) be a set of divergence-free initial data. Consider T > such that the ideal MHD system, supplemented with those initial data, has a unique solution ( u, b ) in the space C (cid:0) [0 , T [ ; L ( R d ) ∩ B ∞ , ( R d ) (cid:1) .Then this solution may be continued beyong the time T provided that (11) Z T (cid:13)(cid:13) ∇ u ( t ) (cid:13)(cid:13) L ∞ d t < + ∞ . Proof.
As far as continuation results go (keep in mind Theorem 1.1), we already know that thesolution may be prolonged beyond time T if we have(12) Z T (cid:16) k∇ u k L ∞ + k∇ b k L ∞ (cid:17) d t < + ∞ . Therefore, we only have to show that the integral in (12) is finite under condition (11).We start by recalling that, by a simple energy method, we have(13) sup t ∈ [0 ,T [ (cid:16) k u ( t ) k L + k b ( t ) k L (cid:17) . k u k L + k b k L .
10y using this bound together with the Bernstein inequalities, we can estimate k∇ u k L ∞ ≤ k ∆ − ∇ u k L ∞ + X m ≥ k ∆ m ∇ u k L ∞ . k u k L + X m ≥ − m k ∆ m ∇ u k L ∞ (14) . (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L + k∇ u k L ∞ . Thus, under condition (11) we deduce that k∇ u k L T ( L ∞ ) . T (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L + k∇ u k L T ( L ∞ ) < + ∞ .It remains us to show that also k∇ b k L T ( L ∞ ) is finite. This will be a consequence of the factthat b solves a linear transport equation, which we may differentiate to obtain estimates on thefirst derivative ∇ b . Precisely, for j = 1 , , we have ∂ t ∂ j b + ( u · ∇ ) ∂ j b = − ( ∂ j u · ∇ ) b + ( ∂ j b · ∇ ) u + ( b · ∇ ) ∂ j u. A basic L ∞ -estimate immediately gives, for all ≤ t < T , the bound(15) k∇ b ( t ) k L ∞ ≤ k∇ b k L ∞ + Z t n k∇ u k L ∞ k∇ b k L ∞ + k b k L ∞ k∇ u k L ∞ o d τ The term k∇ u k L ∞ has already been estimated in (14). Arguing similarly, we can find an upperbound for k b k L ∞ which involve only the quantities we have at our disposal: by separating lowand high frequencies, we have k b k L ∞ ≤ k ∆ − b k L ∞ + X m ≥ k ∆ m b k L ∞ . k b k L + X m ≥ − m k ∆ m ∇ b k L ∞ . (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L + k∇ b k L ∞ . Plugging (14) and the previous bound into (15), we obtain an integral inequality which is linearwith respect to k∇ b k L ∞ : for all ≤ t < T , we have sup τ ∈ [0 ,t ] k∇ b ( τ ) k L ∞ . k∇ b k L ∞ + (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L Z t k∇ u k L ∞ d τ + Z t k∇ b k L ∞ (cid:16) (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L + k∇ u k L ∞ (cid:17) d τ. By using Grönwall’s lemma, we deduce that k∇ b k L ∞ must be bounded as long as condition (11)is fulfilled. So the integral (12) must therefore also be finite while (11) holds.We conclude this part with a remark concerning a continuation criterion in terms of themagnetic field only. Remark 3.2.
The evolution of the velocity field is dictated by the momentum equation, whichis quadratic with respect to u . This implies that it is not possible, in general, to find the kindof linear estimates that would yield a continuation criterion dispensing of any condition on thevelocity field. However, in the special case of space dimension d = 2 , the vorticity equation islinear in u : we have ∂ t ω + u · ∇ ω = b · ∇ j . Therefore, in that case it is possible to bound u , or ω , given good enough bounds on the magneticfield. This leads to a continuation criterion based only on b : we have T < T ∗ as long as Z T k j k B ∞ , d t < + ∞ . .2 A continuation criterion based on the Elsässer variables As we have explained in the introduction, the magnetic field is not the only quantity whichsolves a linear equation: the Elsässer variables also do so. This means that we are able to find acontinuation criterion based only on either α = u + b or β = u − b . Theorem 3.3.
Let ( u , b ) ∈ L ( R d ) ∩ B ∞ , ( R d ) be a set of divergence-free initial data. Consider T > such that the ideal MHD system, supplemented with those initial data, has a unique solution ( u, b ) in the space C (cid:0) [0 , T [ ; L ( R d ) ∩ B ∞ , ( R d ) (cid:1) .Then, if we denote ω = curl ( u ) and j = curl ( b ) , one has (16) Z T (cid:13)(cid:13) ω + j (cid:13)(cid:13) B ∞ , d t < + ∞ ⇐⇒ Z T (cid:13)(cid:13) ω − j (cid:13)(cid:13) B ∞ , d t < + ∞ . In addition, in the case those integrals are finite, the solution ( u, b ) may be continued beyond T into a solution belonging to the same regularity class.Proof. We already know, from Proposition 5.7 in [11], that the solution may be continued beyondthe time T if and only if(17) Z T n k∇ ( u + b ) k L ∞ + k∇ ( u − b ) k L ∞ o d t < + ∞ . Throughout this proof, we assume that(18) Z T (cid:13)(cid:13) ω + j (cid:13)(cid:13) B ∞ , d t < + ∞ . We are going to show that this condition is enough to ensure that also(19) Z T (cid:13)(cid:13) ω − j (cid:13)(cid:13) B ∞ , d t < + ∞ , and that the finiteness of both integrals implies (17).The exact same argument will apply also when considering the quantity ω − j , whence theclaimed equivalence.We start the proof by remarking that, splitting into low and high frequencies as done in (14)and using the Biot-Savart law f k = ( − ∆) − d X j =1 ∂ j (cid:2) curl ( f ) (cid:3) jk , it is easy to show that, for any divergence-free vector field f , one has(20) k∇ f k L ∞ ≤ k f k B ∞ , . k f k L + k curl ( f ) k B ∞ , . In the above equations, we have denoted curl ( f ) the matrix such that (cid:2) curl ( f ) (cid:3) jk = ∂ j f k − ∂ k f j ,with the usual identification curl ( f ) = ∇ × f in dimension d = 3 , and curl ( f ) = ∂ f − ∂ f indimension d = 2 . Thus, after noticing that k α k L ≤ (cid:13)(cid:13)(cid:0) u, b (cid:1)(cid:13)(cid:13) L ≤ (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L in view of the energy inequality (13), we get(21) k∇ ( u + b ) k L ∞ ≤ k u + b k B ∞ , . (cid:13)(cid:13)(cid:0) u , b (cid:1)(cid:13)(cid:13) L + k ω + j k B ∞ , . β = u − b solves the linear equation(22) ∂ t β + ( α · ∇ ) β = (cid:2) α · ∇ , P (cid:3) β , where P is the Leray projector onto the space of divergence-free vector fields, as introduced inSubsection 2.4.By applying the dyadic block ∆ j , for j ≥ − , to equation (22), we obtain(23) (cid:16) ∂ t + α · ∇ (cid:17) ∆ j β = ∆ j (cid:2) α · ∇ , P (cid:3) β + (cid:2) α · ∇ , ∆ j (cid:3) β. The second commutator in the right-hand side of this equation can be estimated with the help ofLemma 2.5, while we can resort to Lemma 2.10 for bounding the first one. Then, we produce L ∞ estimates for the dyadic blocks ∆ j β appearing in the transport equation (23): we have j k ∆ j β k L ∞ . j k ∆ j β k L ∞ + Z T c j ( t ) k α k B ∞ , k β k B ∞ , d t , for suitable sequences (cid:0) c j ( t ) (cid:1) j ≥− belonging to the unit sphere of ℓ . By summing the previousinequality over all j ≥ , we get k β k L ∞ T ( B ∞ , ) . k β k B ∞ , + Z T k α k B ∞ , k β k B ∞ , d t. At this point, we estimate the B ∞ , norm of α = u + b by using (21), and we finally infer anintegral inequality which is linear with respect to k β k B ∞ , : k β k L ∞ T ( B ∞ , ) . k β k B ∞ , + Z T (cid:16) k α k L + k ω + j k B ∞ , (cid:17) k β k B ∞ , d t . Thus, we may use Grönwall’s lemma to end the proof.
Remark 3.4.
Because of the presence of the pressure terms in the equations and of the factthat we work in a critical regularity framework, we are unable to obtain a continuation criteriondepending only on the L ∞ norm of ∇ ( u ± b ) .However, in a subcritical regularity framework B s ∞ ,r with s > , we believe that the samemethod of [2] applies to give a continuation criterion in terms of the finiteness of the norm k curl ( u ± b ) k L T ( L ∞ ) . We do not treat the extension of our result in this direction here. In this section, we present a refinement of Theorem 1.2 of the introduction. As in that result, weexhibit a lower bound for the lifespan of the solutions in space dimension d = 2 , which impliesthat the lifespan tends to + ∞ when the size of the magnetic field tends to . The point is thatwe require higher regularity on the initial velocity field only. We present here the precise statement concerning the improved lower bound for the lifespan ofthe solutions in two space dimensions.
Theorem 4.1.
Let ( u , b ) ∈ L ( R ) be a set of divergence-free initial data such that u ∈ B ∞ , ( R ) and b ∈ B ∞ , ( R ) . hen, the lifespan T > of the corresponding solution ( u, b ) ∈ C (cid:0) [0 , T [ ; L ( R ) ∩ B ∞ , ( R ) (cid:1) of the -D ideal MHD system (1) , given by Theorem 1.1, enjoys the following lower bound: (24) T ≥ C k u k L ∩ B ∞ , log ( C log " C log C k u k L ∩ B ∞ , k b k B ∞ , ! , where C > is a constant independent of the initial data. Before proving the previous statement, a couple of remarks are in order.
Remark 4.2.
Our result is stated in two dimensions of space. This is crucial, as the proof relieson the existence of global solutions to the Euler system. However, this is the only point in ourargument that is specific to d = 2 , and our proof may be adapted to all dimensions d ≥ to showthe following fact: if we denote by T E > the lifespan of the unique B ∞ , solution v of the Eulerproblem with initial datum v | t =0 = u , and if we set k b k B ∞ , = ε , then the lifespan T ε of thesolution ( u, b ) to the ideal MHD system (1) satisfies T ε −→ T E as ε → + . Remark 4.3.
For proving Theorem 4.1, we use the fact that ( u, b ) is a finite energy solution onlyto recast the ideal MHD system into the Elsässer variables. However, at the quantitative level,the magnetic energy k b k L plays absolutely no role in the computations.Thus, our approach, and the result, can be adapted to other situations where the solutionshave infinite energy, but one can use the equivalence of the original ideal MHD system with itsprojected counterpart ( i.e. the system obtained after application of the Leray projector P to theequations) and with the Elsässer formulation. We refer to [9] for more details on that topic. This subsection is devoted to the proof of Theorem 4.1. The main idea of our method is tocompare the ideal MHD system to the classical homogeneous Euler system, which is known to beglobally well-posed in R in our functional framework.We divide the proof into three steps. Step 1: solving the Euler equations.
To begin with, we solve the incompressible Eulerequations with initial datum u . More precisely, let v : R × R −→ R be the unique globalsolution of the initial value problem(25) ∂ t v + ( v · ∇ ) v + ∇ p = 0div( v ) = 0 v | t =0 = u which lies in the class C (cid:0) R + ; L ( R ) ∩ B ∞ , ( R ) (cid:1) . We refer e.g. to Chapter 7 of [1] for details.In the rest of the proof, we need the following lemma. Even though the estimate containedtherein is well-known, we were not able to find a precise reference for it. Therefore, we also providea full proof. Lemma 4.4.
Set V = k u k L ∩ B ∞ , . Then, for all T > , we have the following inequality: sup t ∈ [0 ,T ] k v ( t ) k L ∩ B ∞ , ≤ C V exp (cid:0) C T V e C T V (cid:1) , for some numerical constant C > , independent of u . roof. Since the following energy conservation holds true, namely(26) ∀ t ≥ , k v ( t ) k L ≤ k v k L , we only have to bound the Besov norm of v . For this, we resort to the vorticity form of the Eulerequations. Define the vorticity Ω = ∂ v − ∂ v of the flow v , which can be recovered from Ω bythe -D Biot-Savart law v = −∇ ⊥ ( − ∆) − Ω . Then, Ω solves the pure transport equation(27) ∂ t Ω + v · ∇ Ω = 0 , with Ω | t =0 = Ω := ∂ v , − ∂ v , . First of all, we find B ∞ , bounds for Ω , by using the linear transport estimates of Theorem2.8: we get, for any T > , the bound(28) k Ω k L ∞ T ( B ∞ , ) . k Ω k B ∞ , (cid:18) Z T k∇ v k L ∞ d t (cid:19) . To control the norm of the gradient ∇ v appearing in this estimate, we resort to the inequalityexhibited in (20): by combining the latter with (26), and then using Grönwall’s lemma, from (28)we obtain k Ω k B ∞ , . k Ω k B ∞ , (cid:16) T k u k L (cid:17) exp (cid:16) cT k Ω k B ∞ , (cid:17) . So, by adding the energy k v k L to both sides of this inequality, we obtain a first order estimateof the solution v , namely k v k L ∞ T ( B ∞ , ) . k u k L + k Ω k B ∞ , (cid:16) T k u k L (cid:17) exp (cid:16) cT k Ω k B ∞ , (cid:17) . V (1 + T V ) e cT V . V e cT V . Next, we differentiate the vorticity equation (27) to obtain a second order bound on v . For k = 1 , , we get ∂ t ∂ k Ω + u · ∇ ∂ k Ω = − ∂ k v · ∇ Ω . Applying the linear estimate of Theorem 2.8 one more time, we find k∇ Ω k L ∞ T ( B ∞ , ) . k∇ Ω k B ∞ , + X k =1 , Z T k ∂ k v · ∇ Ω k B ∞ , d t (cid:18) Z T k∇ v k L ∞ d t (cid:19) . The only difficulty in using the previous inequality is that the space B ∞ , is not an algebra.Therefore we must be careful with the first integral, which involves the second order derivative ∇ Ω . However, using the Bony decomposition and Proposition 2.4, we find that the functionproduct is a continuous map in the B ∞ , × B ∞ , −→ B ∞ , topology. Therefore, by adding theenergy k v k L to both sides of the previous inequality, we find an integral estimate for the quantity φ ( t ) := k v k L ∞ t ( L ∩ B ∞ , ) , namely φ ( T ) . (cid:26) V + Z T φ ( t ) V e ctV d t (cid:27) (cid:18) Z T V e ctV d t (cid:19) . V e cT V (cid:26) Z T φ ( t ) d t (cid:27) . An application of Grönwall’s lemma to this last inequality ends the proof.
Step 2: using the Elsässer formulation.
This having been established, we come back to theideal MHD system (1). Thanks to the finite energy condition on the solution ( u, b ) , we can recastthe equations in Elsässer variables, as done in Subsection 1.1, and compare the Elsässer system(3) to the homogeneous Euler equations (25). 15n order to do so, we start by defining the difference functions δα = α − v = ( u − v ) + b and δβ = β − v = ( u − v ) − b . By using equations (3) and (25), we find that the couple ( δα, δβ ) satisfies the system(29) ∂ t ( δα ) + ( β · ∇ ) δα + ( δβ · ∇ ) v + ∇ ( δπ ) = 0 ∂ t ( δβ ) + ( α · ∇ ) δβ + ( δα · ∇ ) v + ∇ ( δπ ) = 0div( δα ) = div( δβ ) = 0 , where, for k ∈ { , } , we have set δπ k = π k − p as being the differences of the pressure functions.The difference functions δα and δβ are very well suited for our purposes, as their initial valuesdepend only on the initial magnetic field: δα (0) = b and δβ (0) = − b . In addition, estimating δα and δβ will immediately provide control for the solution ( α, β ) of theElsässer system (3), thanks to explicit bounds for the regular solution v . With that in mind, weseek to estimate the B ∞ , norms of ( δα, δβ ) . To do this, we see from (29) that we will need B ∞ , bounds on v , which are given in Lemma 4.4 above.Let j ≥ − . By applying the Leray projection operator P , followed by the dyadic block ∆ j ,to the first two equations in (29), we obtain(30) ∂ t ∆ j ( δα ) + ( β · ∇ )∆ j ( δα ) + ∆ j P ( δβ · ∇ ) v = ∆ j (cid:2) β · ∇ , P (cid:3) δα + (cid:2) β · ∇ , ∆ j (cid:3) δα∂ t ∆ j ( δβ ) + ( α · ∇ )∆ j ( δβ ) + ∆ j P ( δα · ∇ ) v = ∆ j (cid:2) α · ∇ , P (cid:3) δβ + (cid:2) α · ∇ , ∆ j (cid:3) δβ . We now perform L ∞ estimates on that system. By using the commutator estimates of Lemmas2.5 and 2.10 and the bounds of Corollary 2.11, we get j k ∆ j ( δα, δβ ) k L ∞ . j k ∆ j b k L ∞ + Z t c j ( τ ) k ( δα, δβ ) k B ∞ , (cid:16) k v k B ∞ , + k ( α, β ) k B ∞ , (cid:17) d τ, where (cid:0) c j ( τ ) (cid:1) j ≥− are sequences all belonging to the unit sphere of ℓ . By applying the Minkowskiinequality to this last inequality, we obtain(31) k ( δα, δβ ) k B ∞ , . k b k B ∞ , + Z t k ( δα, δβ ) k B ∞ , (cid:16) k v k B ∞ , + k ( α, β ) k B ∞ , (cid:17) d τ . Step 3: final estimates.
With the previous estimate (31) at hand, we can conclude the proofof Theorem 4.1. In order to simplify the next computations, we define, for all
T > , the quantities E ( T ) = sup t ∈ [0 ,T ] (cid:13)(cid:13) ( δα ( t ) , δβ ( t )) (cid:13)(cid:13) B ∞ , and φ ( T ) = sup t ∈ [0 ,T ] k v ( t ) k B ∞ , . For completing the proof, it remains us to estimate the B ∞ , norm of the solution ( α, β ) andfind an inequality for E ( t ) , since φ ( t ) is finite at every time t > . Now, we observe that α = δα + v and β = δβ + v , which implies (cid:13)(cid:13)(cid:0) α ( t ) , β ( t ) (cid:1)(cid:13)(cid:13) B ∞ , ≤ E ( t ) + φ ( t ) . Using this estimate in (31), we infer the integral inequality E ( t ) . E + Z t E ( τ ) (cid:16) E ( τ ) + φ ( τ ) (cid:17) d τ = E + Z t E ( τ )d τ + Z t E ( τ ) φ ( τ )d τ.
16o get rid of the linear part in this inequality, namely the last summand in the right-hand side,we start by using Grönwall’s lemma. Thus we get, for all
T > , the bound E ( T ) . (cid:18) E + Z T E ( t )d t (cid:19) e cT φ ( T ) , where c > is some irrelevant numerical constant. Next, in order to bound E ( T ) on some timeinterval, we define the time T ∗ > as T ∗ := sup (cid:26) T > (cid:12)(cid:12)(cid:12) Z T E ( t )d t ≤ E (cid:27) . So, for all times ≤ T ≤ T ∗ , we must have the bound E ( T ) . E e cT φ ( T ) . Therefore, by definitionof T ∗ , we must have the inequality e cT ∗ φ ( T ∗ ) T ∗ ≥ E . This inequality alone proves that the time T ∗ on which the solution is known to satisfy uniform B ∞ , estimates is arbitrarily large if E is made as small as necessary. However, to find morequantitative inequalities, we need to use the upper bound for φ ( T ) provided by Lemma 4.4: wefind that, for all T ≤ T ∗ , one has Z T E ( t ) d t . E V T V exp n cT V exp (cid:0) cT V e cT V (cid:1) o . Thus, by definition of T ∗ , for T = T ∗ we must have E V T ∗ V exp n cT ∗ V exp (cid:16) cT ∗ V e cT ∗ V (cid:17) o ≥ E . By using the inequality x ≤ e x − in the previous estimate and applying the logarithm functionthree times, we prove the theorem. References [1] H. Bahouri, J.-Y. Chemin, R. Danchin: “Fourier analysis and nonlinear partial differential equa-tions” . Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of MathematicalScinences), Springer, Heidelberg, 2011.[2] J. Beale, T. Kato, A. Majda:
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