ON THE GLOBAL REGULARITY ISSUE OF THE TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS SYSTEM WITH MAGNETIC DIFFUSION WEAKER THAN A LAPLACIAN

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1 ON THE GOBA REGUARITY ISSUE OF THE TWO-DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM WITH MAGNETIC DIFFUSION WEAKER THAN A APACIAN KAZUO YAMAZAKI Abstract. In this manuscript, we discuss the recent developments in the research direction concerning whether the solution to the two-dimensional magnetohydrodynamics system with certain velocity dissipation and magnetic diffusion strengths preserves the high regularity of a given initial data for all time or exhibits a blowup in finite time. In particular, we address an open problem in case the magnetic diffusion is weaker than a full aplacian. In short, we consider this system with both velocity dissipation and magnetic diffusion strength measured in terms of fractional aplacians with certain powers, and point out the following gap in the results among the current literature. In case the power of the fractional aplacian representing the magnetic diffusion is one, the global well-posedness follows as long as the velocity dissipation is present regardless of how weak its strength may be, and hence the sum of the two powers need to only be more than one ([6, 3]). On the other hand, once the power of the fractional aplacian representing the magnetic diffusion drops below one, in order to ensure the system s global well-posedness, the sum of the powers from the dissipation and diffusion must be equal to or more than two, improved only logarithmically ([, 4, 7]). We discuss this issue in more detail, explain its reasoning, and provide some basic regularity criteria to gain better insight to this difficult direction of research. Keywords: BMO space; Fourier transform; fractional aplacian; magnetohydrodynamics system; regularity.. Introduction One of the most well-known outstanding open problems in mathematical analysis questions whether the solution to the following N-dimensional Navier-Stokes equations (NSE), in case N = 3, admits a unique solution with finite kinetic energy for all time or if there exists an initial data with finite kinetic energy such that its corresponding solution experiences a blowup in finite time: u + (u )u + π ν u = 0, t (a) u = 0, u(x, 0) u 0 (x), (b) where we denoted by u : R N R + R N, π : R N R + R, the velocity, the pressure fields respectively, and ν > 0 represents viscosity. This system at ν = 0 recovers the Euler equations. 00MSC : 35B65; 35Q35; 35Q6

2 KAZUO YAMAZAKI Hereafter let us write t = t, x i = i for i N, x = (x,..., x N ), as well as f = f(x)dx. Furthermore, for brevity we use the notations A R N a,b B, A a,b B to imply that there exists a constant c(a, b) that depends on a, b such that A cb, A = cb respectively. Finally, we denote the fractional aplacian, Λ r ( ) r, r R, defined through a Fourier symbol of ξ r so that Λ r f(ξ) = ξ r ˆf(ξ). et us now couple (a) with Maxwell s equation from electromagnetism to write down the generalized magnetohydrodynamics (MHD) system as t u + (u )u + π + νλ α u = (b )b, t b + (u )b + ηλ β b = (b )u, (a) (b) u = b = 0, (u, b)(x, 0) (u 0, b 0 )(x), (c) where we denoted by b : R N R + R N, the magnetic field and η 0 the magnetic diffusivity. et us call the generalized MHD system at ν, β > 0, α = β =, the classical MHD system. As we will see, it is important to observe here that (a) at α = is precisely (a) with the forcing term of (b )b. It is also worth noting that if one achieves in proving that given an arbitrary initial data (u 0, b 0 ) sufficiently smooth, its corresponding solution (u, b) preserves its regularity for all time, then one may take b 0 0, and deduce by uniqueness the smooth solution u for the NSE (a)-(b). In this way, it may be argued that proving the global well-posedness of the MHD system (a)-(c) is more difficult than that of the NSE (a)- (b). Now similarly to many other systems of equations in physics, the solutions to both the NSE (a)- (b) and the MHD system (a)- (c) admit a rescaling property, specifically that e.g. in the latter case, if (u, b)(x, t) (and π(x, t)) solves the generalized MHD system (a)-(c) for γ = α = β, then so does (u λ, b λ )(x, t) λ γ (u, b)(λx, λ γ t), λ R +, (3) (and π λ (x, t) λ 4γ π(λx, λ γ t)). Moreover, taking (R N )-inner products of (a)-(b) with (u, b), making use of the divergence-free property of both u and b from (c), the following identity that describes that conservation of energy and cumulative energy dissipation and diffusion may be shown: ( u + b ) t (t) + ν Λ α u + η Λβ b dτ = u 0 + b 0 (4) 0 for all t in the time interval [0, T ] over which the solution exists. It is well-known that while other bounded quantities of the solutions exist (see [5]), those in (4) seem to be the most useful upon a priori estimates in the effort to prove higher regularity of the solution. Studying the rescaled solution, e.g. u λ (x, t), in this most useful bounded quantity deduces u λ (t) = λ4γ N u(λ γ t) (5) by (3), where 4γ N = 0 if and only if γ = + N 4, indicating that + N 4 is a kind of a threshold for the powers α, β such that relying only on these bounds from (4) in the a priori estimates, one can hope to prove higher regularity only if both α and β are equal to or more than + N 4. Indeed, this is precisely what was proven in [3]; the result therein at N = in particular recovers the pioneering work of [4, 6] that the NSE (a)-(b), and the classical MHD system respectively are both

3 D MHD SYSTEM 3 globally well-posed. Additionally, let us mention that the works of [9,, 4, 7] proved logarithmic improvement and in particular the results from [, 4, 7] showed that in the case of the generalized MHD system (a)-(c), while α + N 4 remains necessary, the condition on β + N 4 can be relaxed to only β 0 and α + β + N. That is, the lower bound on β may be shifted toward α although not vice versa, rooting from the fact that upon the H (R N )-estimate on the system (a)-(c), every non-linear term that must be estimated involves u, but not necessarily b and hence the support from dissipation plays a more important role than the diffusion in most cases, although not all as we will see. et us formally state the result from [4] for completeness: Theorem.. ([4, Theorem.], see also [, 7] for the case η = 0) Consider the following N-dimensional generalized MHD system: t u + (u )u + π + ν u = (b )b, t b + (u )b + η b = (b )u, u = b = 0, (u, b)(x, 0) (u 0, b 0 )(x), where the Fourier operators i, i {, }, are defined to have the Fourier symbols of i f(ξ) = m i (ξ) ˆf(ξ) with m i (ξ) conditioned to obey the lower bounds of m (ξ) ξ α g (ξ), m (ξ) ξ β g (ξ) for all sufficiently large ξ and g i : R + [, ) are radially symmetric, nondecreasing functions. If ν > 0 and η > 0, α + N 4, β > 0, α + β + N, and dτ τ(g (τ) + = +, g (τ)) then given any initial data (u 0, b 0 ) H s (R N ), s > + N, this system has a unique global classical solution (u, b). From our previous discussion, we understand that breaking the threshold of α + N 4, β + N 4 more than logarithmically is extremely difficult and may require some bounded quantity better than those in (4). For this reason, it is understood among experts that N-dimensional generalized MHD system (a)-(c) is energy-critical if α + β = + N, energy-subcritical if α + β > + N and energysupercritical if α + β < + N, which in particular implies that the two-dimensional (d) NSE as well as the d classical MHD system are energy-critical. A remarkable feature of the NSE (a)-(b) in the d case is that such a bounded quantity better than those in (4) in fact exists. Indeed, even in the case of the Euler equations, denoting by w u and applying a curl operator on (a) with ν = 0 shows that the evolution of w is governed by a transport equation of t w + (u )w = 0. (6) Thus, multiplying (6) by w p w, p, integrating over R and using that (u )w w p w = (u ) w p = 0 (7) p

4 4 KAZUO YAMAZAKI due to (b) lead to p t w p p = 0. (8) Using that p t w p = p w p t w p p, dividing (8) by w p p, taking p + shows that the sup t [0,T ] w(t) is bounded by w(0). With this bound attained, applying Λ s to (a) with ν = 0, s >, taking (R )-inner products with Λ s u, employing (36) from emma 3. and emma 3. lead immediately to t Λ s u = [Λ s ((u )u) (u )Λ s u] Λ s u u Λ s u ( u + w log ( + Λ s u ) + ) Λ s u ; hence, the higher regularity is attained via Gronwall s inequality applied to (9). This recovers the classical results from [35] (see also [0]). A natural question arises whether the favorable formulation of the vorticity equation may exploited in order to improve the results on the MHD system from [6, 3] that required α, β in the d case. In this endeavor, applying a curl operator on (a)-(b), one arrives at t w + (u )w + νλ α w = (b )j, t j + (u )j + ηλ β j (9) (0a) = (b )w + [ b ( u + u ) u ( b + b )] (0b) where j b represents current density. A quick comparison of (0a) with (6) shows that (0a) is still a transport equation, but now forced by (b )j while dissipated fractionally by νλ α w. Thus, upon an p (R )-estimate of w, in contrast to (8), we are faced with p t w p + ν Λ α w w p w = (b )j w p w. () p Therefore, we see heuristically that if η > 0 and β is sufficiently large so that (b )j is sufficiently smooth due to (4) and thus bounded, then the higher regularity should be attained via an analogous computations in (7)-(9). Moreover, even if β is not sufficiently large, if ν > 0 and α > 0 is adequately large, then w p w within (b )j w p w may be appropriately handled using (4) and thus higher regularity may follow. Following this intuitive argument, an explosive amount of work appeared very recently improving results one after another. On one hand, for the case ν = 0 and hence α = 0, Tran, Yu and Zhai in [] showed that β > suffices in order to prove the global well-posedness. Jiu and Zhao in [8] and the author in [6] independently improved this result to β > 3. Thereafter, Jiu and Zhao in [9] and Cao, Wu and Yuan in [] independently showed that β > in fact suffices; the approach of the former was taking advantage of the property of a heat kernel, while the latter the Besov space techniques (see also [5]). On the other hand, for the case η > 0, β =, Tran, Yu and Zhai in [] showed that α suffices. Subsequently, Yuan and Bai in [34], as well as the author in [8], independently improved this result to α > 3. Thereafter, Ye and Xu in [33] improved from α > 3 to α 4. Finally, the authors in [6] proved that α > 0 suffices (also [3]). At the time of writing this current manuscript, the most difficult

5 D MHD SYSTEM 5 remaining open problem is the case of ν = 0 and hence α = 0 while η > 0, β =, which would represent a complete extension of the classical results from [0, 35] to the MHD system. We also refer to [0] for numerical analysis, and [7, 3] for regularity criteria results. A common ingredient in the hypothesis of every one of these recent new results on the d generalized MHD system is that η > 0, β. This is because applying a curl operator on (a) not only gives a favorable vorticity formulation in the form of a forced dissipative transport equation in (0a), but furthermore if one combines an (R )-estimate of w with that of j, even the forcing term (b )j vanishes due to a remarkable cancellation. Indeed, upon taking (R )-estimates of (0a)-(0b) with (w, j) respectively, summing them up leads to t( w + j ) + ν Λα w + η Λβ j = [ b ( u + u ) u ( b + b )]j () where we made use of the facts that not only (u )ww = (u ) w = 0, (u )jj = (u ) j = 0, (3) but also (b )jw + (b )wj = (b )(wj) = 0 (4) due to (c). The right-hand side of () may be estimated in the following simple and in fact an optimal fashion: t( w + j ) + ν Λα w + η Λβ j = [ b ( u + u ) u ( b + b )]j b 4 u j 4 j j w by Hölder s inequality, Gagliardo-Nirenberg s inequality and emma 3.3. This implies that if η > 0, β = so that from (4) we would have T 0 b dτ, then Young s inequality applied in (5) separating η j would allow us to close this ( w + j )(t)-estimate over [0, T ]. et us point out, very importantly for our subsequent discussion, that from the first inequality of (5) we chose u to play the role of being estimated because u w, and we chose b 4 and j 4 to share the role of to be absorbed to the diffusive term η Λ β j and to be square-integrable in time. This immediate H (R )-bound of (u, b) in the d case is precisely the reason why the door was opened for all the recent results in [, 6, 7, 8, 9,, 6, 8, 3, 3, 33, 34]. Now let us consider the same estimate in (5) with β < and α + β < so that a global well-posedness in such a case will be an improvement of the work in [0, 35]. One may follow the analogous idea in (5) and use Hölder s inequality to bound the right-hand side of () by [ b ( u + u ) u ( b + b )]j b 4 u 4 j. (6) (5)

6 6 KAZUO YAMAZAKI However, we immediately realize that b 4 from (6) cannot possibly be interpolated between b being estimated, and Λ β b which is to be squareintegrable in time, because β <. Therefore, this term must be interpolated as β β b 4 b Λ β b β, (7) i.e. partially being estimated and partially to be absorbed to the diffusive term η Λ β j. However, this application of Gagliardo-Nirenberg s inequality in (7) requires β. On the other hand, since the interpolation of (7) did not make use of terms that are square-integrable in time from (4), it seems most efficient to interpolate u 4 from (6) between the being estimated and to be squareintegrable in time as follows: u 4 u α 3 α Λ α u α. (8) However, this application of Gagliardo-Nirenberg s inequality requires α 3 and thus together with the requirement from (7), we already have the restriction of α+β. Various other options of Hölder s inequality in (6) are possible; however, upon many attempts, every one of them seems to require α + β ; in particular, the extreme case of [ b ( u + u ) u ( b + b )]j b u j immediately requires α > so that u u + Λ α u becomes integrable over [0, T ]. Heuristically the problem is that the right-hand side of () is more or less b u ; if this were u b, then H (R )-estimate of (u, b) may be immediately attained with just ν > 0, α = and η = 0. Unfortunately upon many attempts to make some cancellations to reverse such roles of u and b, we found this task very difficult. Indeed, although in the literature there have been various identities due to unique cancellations via appropriate additions and subtractions, taking advantage of divergence-free conditions (e.g. [3, emma.3], [30, Proposition.]), vorticity formulation is really remarkable in that upon applying a curl operator on the non-linear term, ((u )u) = ((u )u ) ((u )u ) = u u + u u + (u ) u u u u u (u ) u = (u )w, cancellations occur within itself. One does not even need to add or subtract with another equation to make this happen! One noteworthy idea, that has proven to be useful in the literature, is to consider a symmetric form of S w + j, S w j so that the roles of u and b are indeed somehow reversible. That is, we assume γ = α = β, ν = η = λ, estimate t S + ((u b) )S + λλ γ S = u b + u b + b u b u, (9a) t S + ((u + b) )S + λλ γ S = u b u b b u + b u, (9b)

7 D MHD SYSTEM 7 where we used that we may rewrite [ b ( u + u ) u ( b + b )] = u b + u b + b u b u, (0) and hope that we can complete the estimate of S + S with γ < so that by triangle inequality we may conclude the bound of w + j. Unfortunately this strategy also seems to require γ ; e.g. assuming γ <, taking (R )-inner products with S in (9a), we may estimate t S + λ Λγ S ( u b +γ + u b +γ + b u +γ + b u ) +γ Λγ S λ 4 Λγ S + c u b γ λ 4 Λγ S + c Λ γ u b by Hölder s inequality, Hardy-ittlewood-Sobolev theorem (e.g. [8, pg. 9]), Young s inequality and Sobolev embedding of Ḣ γ (R ) γ (R ). For this estimate to be closed, in comparison with (4), we will need γ α which is if and only if α and because we assumed γ = α = β, this requires α + β. Therefore, we conclude that there has been this large gap, for a technical reason, that if η > 0, β, the best results in the literature (e.g. [, 5, 6, 9, 3]) requires only α + β >. But once we reduce our hypothesis to η > 0, β <, then the requirement suddenly jumps to α + β from the classical results ([6, 3]), improvable only logarithmically in [, 4, 7]. In the rest of this paper we provide a basic Serrin-type regularity criterion for the solution to be smooth ([7]) to gain better insight to this difficult case when η > 0, β < and α + β < : Theorem.. et N =, ν > 0, η > 0, β [, ), α [, β) and (u, b) C([0, T ); H s (R )) ([0, T ); H s+α (R )) C([0, T ); H s (R )) ([0, T ); H s+β (R )) () be the solution pair to the generalized MHD system (a)-(c) for a given (u 0, b 0 ) H s (R ), s >. Suppose the solution (u, b) over the time interval of [0, T ] satisfies T 0 u r pdτ < where p + r + p ( ), β β p, () where the case p = + amounts to the condition of r. Then (u, b) remains in the same regularity class as () on [0, T ] for some T > T. Theorem.3. et N =, ν > 0, η > 0, β [, ), α [, β) and (u, b) in the regularity class of () be the solution pair to the generalized MHD system (a)-(c) for a given (u 0, b 0 ) H s (R ), s >. Suppose the solution (u, b) over the

8 8 KAZUO YAMAZAKI time interval of [0, T ] satisfies T 0 b r pdτ < { where p + + r p + p or T 0 j BMO dτ <. ( ) β, for β ( p <, α ), for max{, α } < p <, Then (u, b) remains in the same regularity class as () on [0, T ] for some T > T. Remark.. Improvements on the space of initial data from H s (R ) for s > is certainly possible. We choose not to pursue this direction of research here. We also emphasize that lim β + p ( β ) =, indicating again that heuristically the well-posedness of the generalized MHD system (a)-(c) under the hypotheses of Theorem. and Theorem.3 may not be too far out of reach. Remark.. A natural question is whether we may obtain a regularity criteria in terms of u or b instead of their gradients as in () and (3) respectively. We choose to leave this in the future direction of research, only commenting that naive attempts actually seem to fail immediately. This is because a standard procedure of obtaining such a regularity criteria e.g. in terms of u instead of u for the MHD system starts with first integrating by parts and taking away the derivative on u. However, if we work on the H (R )-estimate of (u, b), then although by taking (R )-inner products on (a)-(b) with ( u, b) respectively, one may compute t( u + b ) + ν Λα u + η Λβ b = (u )u u + (u )b b (b )b u + (b )u b = i,j,k= u i k ( i b j k b j ) u j k ( k b i i b j ) + u j i ( k b i k b j ), it will be difficult to close this estimate due to the second-order derivatives on b and the hypothesis that β <. Similarly in an attempt to estimate (R )-norm of w and j, one immediately realizes that integration by parts in the right side of () leads to second-order derivatives on b which we will not be able to estimate because β < : [ b ( u + u ) u ( b + b )]j = u (( b )j) + u (( b )j) + u (( b + b )j). (3). Proofs of Theorem. and Theorem.3 ocal theory may be proven using mollifiers in a standard way (e.g. [5]); thus, in this manuscript, we choose to focus on a priori estimates.

9 D MHD SYSTEM 9.. H (R )-bound. Proposition.. Under the hypothesis of Theorem., the solution (u, b) over the time interval [0, T ] satisfies sup t [0,T ] ( w + j )(t) + T 0 Λ α w + Λβ j dτ <. (4) Proof. Firstly, let us assume p ( β, ) in () and continue from (5), and estimate [ b ( u + u ) u ( b + b )]j u p b b p p βp βp u p b b Λ β b η Λβ j + c u βp βp p b βp (5) by Hölder s inequality, Gagliardo-Nirenberg s inequality and Young s inequality. Applying (5) in (5), subtracting η Λβ j from both sides leads to t ( w + j ) + ν Λα w + η Λβ j u βp βp p b (6) from which Gronwall s inequality completes the proof of Proposition. in case p ( β, ). Next, the case p = β is immediate as well because [ b ( u + u ) u ( b + b )]j u β b Λ β b η Λβ b + c u β b by Hölder s inequality, Sobolev embedding Ḣβ (R ) β (R ) and Young s inequality. Finally, in the case p = + we immediately obtain [ b ( u + u ) u ( b + b )]j u b by Hölder s inequality. After applying these estimates in (5), Gronwall s inequality deduces (4) and completes the proof of Proposition.. Proposition.. Under the hypothesis of Theorem.3, the solution (u, b) over the time interval [0, T ] satisfies sup t [0,T ] ( w + j )(t) + T 0 Λ α w + Λβ j dτ <. (7)

10 0 KAZUO YAMAZAKI ( ) Proof. Firstly, let us consider the case p + r + p β, β continue again from (5) and estimate [ b ( u + u ) u ( b + b )]j < p <. We u b p b p p βp βp u b p b Λ β b βp η Λβ j + c b βp βp p ( u + b ) by Hölder s inequality, Gagliardo-Nirenberg s inequality and Young s inequality. After applying (8) in (5), Gronwall s inequality leads to (7) in this case. The case in which p = β is immediately done similarly: [ b ( u + u ) u ( b + b )]j (8) u b β b β η Λβ b + c u b β by Hölder s inequality, Sobolev embedding Ḣβ (R ) β (R ) and Young s inequality. Secondly, let us consider the case p + r + p We estimate from (5): [ b ( u + u ) u ( b + b )]j u p b p b p αp αp u αp Λ α u b b p ν Λα u + c( u + b ) b αp αp p ( ) α, max{, α } < p <. by Hölder s inequality, Gagliardo-Nirenberg s inequality and Young s inequality. After applying (9) in (5), Gronwall s inequality deduces (7). astly, for the case T 0 j BMOdτ <, we estimate [ b ( u + u ) u ( b + b )]j = [ u b + u b + b u b u ]j (30) u b j BMO j BMO ( u + b ) by (0), duality of BMO space and the Hardy space H, [4, Theorem II.] and Young s inequality. After applying (30) to (5), Gronwall s inequality deduces (7) and completes the proof of Proposition.... Higher regularity from H (R )-bound. In this subsection, we raise the regularity from H (R )-bound to H s (R ). As we emphasized in the Section, this is the relatively easier part of the proof; the difficulty in case β < is the H (R )- estimate which we already accomplished in Proposition. and Proposition.. (9)

11 D MHD SYSTEM Proposition.3. Under the hypothesis of Theorem. or Theorem.3, the solution (u, b) over the time interval [0, T ] satisfies sup t [0,T ] ( w + j )(t) + T 0 Λ α w + Λβ j dτ <. (3) Proof. et us first assume the more difficult case α <. We take (R )-inner products on (0a)-(0b) with ( w, j) to deduce t( w + j ) + ν Λα w + η Λβ j = u w w u j j + b j w (3) + Λ β [ b ( u + u ) u ( b + b )]Λ +β j after integration by parts. For clarify, let us estimate the first three integrals and the last integral separately. Firstly, u w w u j j + b j w u α w w α + u β j j β + b α j w α Λ α u w Λ α w + Λ β u j Λ β j (33) + Λ α b j Λ α w ν 8 Λα w + η 8 Λβ j + c( w + j )( Λ α w + Λ β w + Λ α j ) by Hölder s inequalities, Sobolev embedding of Ḣ γ (R ) γ (R ), Ḣ γ (R ) γ (R ), emma 3.3 and Young s inequality. Secondly, Λ β [ b ( u + u ) u ( b + b )]Λ +β j b u Ḣ β Λ β j ( Λ β b β u β + b β Λ β u β ) Λβ j η 8 Λβ j + c( w + j )( Λ β w + Λ β j ) (34) by Hölder s inequality, emma 3.4, emma 3.3, Sobolev embeddings of Ḣ β (R ) β (R ) and Ḣ β (R ) β (R )) and Young s inequality. Applying (33) and (34) in (3), using that α, β by hypothesis, Gronwall s inequality type argument using Proposition. and Proposition. deduces (3). The case in which α [, β) is easier and it suffices for us to modify within the estimate of (33) as follows: u w w + b j w u 4 w w 4 + b β j w β ν 8 Λα w + η 8 Λβ j + c( w + j )( + Λα w + Λβ j )

12 KAZUO YAMAZAKI by Hölder s inequalities, the Sobolev embeddings of H α (R ) 4 (R ), Ḣβ (R ) β (R ), H α (R ) β (R ), and Young s inequality. Applying this estimate, along with the estimate of u j j η 8 Λβ j + c Λ β w j which may be seen from (33), and (34) to (3) deduces (3) via Gronwall s inequality. The proof of Proposition.3 is now complete. Proofs of Theorem. and Theorem.3. The proofs of Theorem. and Theorem.3 follow immediately from Proposition.3, emma 3. and (37) from emma 3.. Indeed, taking (R )-inner products of (a)-(b) with (Λ s u, Λ s b), results in t( Λ s u + Λs b ) + ν Λs+α u + η Λs+β b = Λ s ((u )u) Λ s u + Λ s ((b )b) Λ s u Λ s ((u )b) Λ s b + Λ s ((b )u) Λ s b Λ s ((u )u) (u )Λ s u Λ s u + Λ s ((b )b) (b )Λ s b Λ s u (35) + Λ s ((u )b) (u )Λ s b Λ s b + Λ s ((b )u) (b )Λ s u Λ s b u Λ s u + b Λs b Λ s u + u Λ s b ( u + b ) ln(e + Λ s u + Λs b )( Λs u + Λs b ), where we used that (u )Λ s u Λ s u = 0, (u )Λ s b Λ s b = 0, (b )Λ s b Λ s u + (b )Λ s u Λ s b = 0, due to (c). Gronwall s inequality applied to (35), along with Proposition.3 completes the proofs of Theorem. and Theorem Appendix For completeness, we state lemmas that have been used: emma 3.. ([]; see also [6, emma.3 and Appendix], and [9, emma.7 and Appendix]) et f H s (R ), s >, satisfy f = 0, f (R ). Then f ( f + f log ( + f H s) + ). (36) Similarly let f (R ) W s,p (R ) where s R such that p [, ), p < s. Then f ( f + f H log ( + f W s,p) + ). (37) emma 3.. ([]) et f, g be smooth such that f p (R ), Λ s g p (R ), Λ s f p3 (R ), g p4 (R ), where p (, ), p = p + p = p 3 + p 4, p, p 3 (, ), and s > 0. Then Λ s (fg) fλ s g p ( f p Λ s g p + Λ s f p 3 g p 4 ).

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