GLOBAL REGULARITY OF LOGARITHMICALLY SUPERCRITICAL 3-D LAMHD-ALPHA SYSTEM WITH ZERO DIFFUSION

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1 GLOBAL REGULARITY OF LOGARITHMICALLY SUPERCRITICAL 3-D LAMHD-ALPHA SYSTEM WITH ZERO DIFFUSION KAZUO YAMAZAKI Abstract. We study the three-dimensional Lagrangian-averaged magnetohydrodynamicsalpha system with zero diffusion. Despite the dissipation strength at the logarithmically supercritical level, using dyadic decomposition techniques we show that given initial data sufficiently smooth, the solution pair remains smooth for all time. This settles the global regularity case suggested by the authors in [7]. Keywords: Magnetohydrodynamics system, Navier-Stokes equations, global regularity, Besov spaces, criticality. Introduction We study the following generalized Lagrangian-averaged magnetohydrodynamicsα (LAMHD-α) system: v t + (u )v + v j u j + (π + ) b + νl v = (b b), (a) j= b + (u )b (b )u η b =, t (b) subjected to the following filtration, divergence-free and initial conditions respectively: v = ( α )u, u = b =, (v, b)(x, ) = (v, b )(x) () where L is a Fourier operator with its symbol Lf(ξ) = m(ξ) ˆf(ξ), m(ξ) = ξ r g( ξ ), r R+ (3) with g : R + R + a radially symmetric, non-decreasing function such that g. We denote by v, u, b : R 3 R + R 3 the velocity, filtered velocity and magnetic vector fields respectively and π : R 3 R + R the pressure scalar field. The parameters ν, η represent viscosity and diffusion constants while α > the MSC : 35B65, 35Q35, 35Q86 Washington State University, Department of Mathematics, Pullman, WA , U.S.A., Phone: , kyamazaki@math.wsu.edu

2 KAZUO YAMAZAKI length-scale parameter displaying the width of the filters. We denote the components of v, u as v = (v, v, v 3 ), u = (u, u, u 3 ). Hereafter let us denote t, i for t, x i, i =,, 3 respectively and Λ = ( ) a fractional Laplacian. The classical Lagrangian-averaged Navier-Stokes-α (LANS-α) model which is (a)-(b) at b, m(ξ) = ξ was introduced by the authors in [6] as an excellent closure model of turbulence in infinite channels and pipes; we also refer to [7] for the closely related Leray-α model (cf. also [9] for comparison with Bardina model). The LAMHD-α system (a)-(b) at r =, g was introduced by the authors in [6] in belief that the filtration in the magnetic field is unnecessary. The LANS-α MHD system originates from the classical MHD system which describes the motion of electrically conducting fluids and has found much applications in various applied sciences such as astrophysics, plasma physics and geophysics. In particular, since the fundamental work of [], the mathematical analysis of the MHD system has found much attraction. When ν, η >, r =, g, the authors in [6] showed that the system (a)-(b) admits a unique weak solution pair. In fact, the authors in [37] extended this global well-posedness result to the fourth dimension (cf. also [35]). In [5], the author studied the classical MHD system with fractional Laplacians on the dissipative and diffusive terms and the authors in [8] followed suit for the LANS-α model. (cf. also [3, 33, 34] for such fractional Laplacian cases). The advantage of the fractional Laplacians is that it allows us to distinguish the cases among the subcritical, critical and supercritical with respect to rescaling of solutions. In particular, the author in [3] showed that the N-dimensional Navier-Stokes equations (NSE) for N 3, when the dissipative term is replaced by a fractional Laplacian with the power of + N 4, admits a global regularity result despite logarithmic worsening similar to that of (3) and (4) below. The investigation of the global regularity issue in such a logarithmically supercritical case has attracted much attention since then; we refer to e.g. [3,, 4, 5, 9,,, 4, 6, 8, 9]. We now motivate our study. The authors in [7] showed that when ν >, η =, g, r = 3, there exists a unique global strong solution pair for the system (a)- (b). This is the endpoint case with the critical dissipation strength which requires substantially non-trivial techniques such as Brezis-Wainger type inequality from [] due to the complete lack of diffusion (cf. also [8, 36] for such partially inviscid cases); we note that in case r > 3, the proof is significantly easier due to the subcriticality strength of dissipation. The authors suggested difficulty in obtaining the logarithmic improvement of their result (see Remark pg. 586 [7]). The purpose of this manuscript is to give an affirmative solution to this issue. and g be a radially symmetric non- Theorem.. Let ν >, η =, r = 3 decreasing function such that g(τ) c ln(τ) ln(ln(e + τ)) satisfying e dτ g(τ) ln(τ)τ =. (4) Then given v, b H 5 (R 3 ), there exists a unique classical solution pair v, b L ([, ); H 5 (R 3 )) to (a)-(b) that satisfies () and (3). Remark.. () The proof of Theorem. was inspired largely by [4, 5] which in turn was motivated by the work such as [5]. However, we remark that the proof in [4, 5] took advantage of the reformulation of (a)-(b)

3 LAMHD-ALPHA SYSTEM 3 after taking a curl in two-dimensional case. Unfortunately the same strategy does not work because there is no clear advantage in taking curls under the three-dimensional setting as in the current manuscript. Due to this issue, some new estimates were needed. () We wish to make a curious remark on the following mathematical problem. In [3], the author showed that the following N-dimensional (N 3) generalized Boussinesq system with zero diffusion admits a global regularity result: { t u + (u )u + π + νλ α u = θe N, t θ + (u )θ =, with ν >, α + N 4. We refer interested readers to [3] for details; we remark that after this result in [3] was published, the author was informed of the similar result in [3] obtained independently. This result may be seen as an extension of the similar results on the two-dimensional case in [, ]. Interestingly, the method in the current manuscript does not seem to go through in the aim to improve these results in [3, 3] logarithmically, as shown in Theorem.. We hope that this problem becomes settled in future works. In the Preliminaries section, we set up notations and state key lemmas; thereafter, we prove our theorem.. Preliminaries We use the notation A a,b B to imply that there exists a positive constant c that depends on a, b such that A cb, similarly with A a,b B. For brevity, we also write f to imply fdx. R 3 Let us recall the notion of Besov spaces (cf. [4]). We denote by S(R 3 ) the space of Schwartz functions and S (R 3 ) its dual. We define S = {φ S, φ(x)x γ dx =, γ =,,,...}. R 3 Its dual S is given by S = S/S = S /P where P is the space of polynomials. For k Z we define A k = {ξ R 3 : k < ξ < k+ }. It is well-known that there exists a sequence {Φ k } in S(R 3 ) such that supp ˆΦ k A k, ˆΦk (ξ) = ˆΦ ( k ξ) or Φ k (x) = k3 Φ ( k x) and k= ˆΦ k (ξ) = { if ξ R 3 \ {}, if ξ =. To define the inhomogeneous Besov space, we let Ψ C (R 3 ) be such that = ˆΨ(ξ) + ˆΦ k (ξ), f = Ψ f + Φ k f (5) k= k=

4 4 KAZUO YAMAZAKI for any f S. With that, we set k f = { Ψ f if k =, Φ k f if k =,,,..., and define for any s R, p, q [, ], the inhomogeneous Besov space where f B s p,q = In particular B s, = H s. The following lemma will be useful: B s p,q = {f S : f B s p,q < } { ( k= (ks k f L p) q ) q if q <, sup k< ks k f L p if q =. Lemma.. (cf. [4]) Bernstein s Inequality: Let f L p (R 3 ) with p q and < r < R. Then for all s Z + {}, and λ >, there exists a constant C s > such that { sup γ =s γ f L q C s λ s+3( p q ) f L p if supp ˆf {ξ : ξ λr}, Cs λ s f L p sup γ =s γ f L p C s λ s f L p if supp ˆf {ξ : λr ξ λr}, and if we replace derivative γ by the fractional derivative, the inequalities remain valid only with trivial modifications. Finally, we recall the product estimate on the homogeneous Sobolev space and the commutator estimate: Lemma.. (cf. []) Let f W s,p L q, g W s,p L q, s, < p k <, < q k, p k + q k = p, k =,. Then fg Ẇ s,p ( f Ẇ s,p g L q + f L q g Ẇ s,p ). Lemma.3. (cf. [3]) Let f, g be smooth such that f L p, Λ s g L p, Λ s f L p3, g L p4, p (, ), p = p + p = p 3 + p 4, p, p 3 (, ), s >. Then Λ s (fg) fλ s g L p ( f L p Λ s g L p + Λ s f L p 3 g L p 4 ). 3. Proof of Theorem.: A Priori Estimates Without loss of generality, we assume ν = η = α =. Proposition 3.. Under the hypothesis of Theorem., the solution to the system (a)-(b) in [, T ] satisfies the following bound: sup t [,T ] ( v L + b L )(t) + T Lv Ldτ.

5 LAMHD-ALPHA SYSTEM 5 Proof. We take L -inner products of (a), (b) with (u, b) and use () which e.g. leads to (u )v u + v j u j u =, j= and integrate in time to obtain sup t [,T ] ( u L + u L + b L )(t) + T Lu L + L u Ldτ. (6) Now we take L -inner products of (a)-(b) with (v, b) and sum to work on t( v L + b L ) + Lv L (7) = v j u j v + (b )b u (b )b u + (b )u b + (u )b b = + j= v j u j v + (b )u b j= i,j,k= k b i i b j k u j + k b i i u j k b j u ( v + b + b ) u L ( v L + b L + b L ) u L ( + v L + b L ) k u i i b j k b j due to integration by parts, (), Hölder s inequalities and (6). Now we estimate for M R + to be determined subsequently, u L k u L k k M k 3 g( k ) k u L g( k ) + k >M k( ) k( 5 ) g( k k u L g( k ) ) by (5), and Bernstein s inequalities. We continue this estimate by

6 6 KAZUO YAMAZAKI u L g(m) k L u L + k g ( k ) k M g(m) k M k >M L u L k >M + k g 4 ( k ) k >M k g(m) ln(m) L u L + M L u L L u L (8) L u L where we used the hypothesis that g is nondecreasing, Plancherel theorem, (3), Hölder s inequalities and (4). Therefore, applying (8) in (7) gives t( v L + b L ) + Lv L (9) ( g(m) ) ln(m) L u L + M L u L ( + v L + b L ). Choosing M c(e + v L + b L ) for c > sufficiently large, using Young s inequality and absorbing the dissipative term lead to t M + Lv L g(m) ln(m)(e + L u L)M. () Therefore, integrating over time [, t] for t [, T ] gives M(t) M() dτ g(τ) ln(τ)τ t T e + L u L dτ e + L u Ldτ by (6). Taking sup over t [, T ] on the left hand side and considering the condition (4), we see that Integrating (), using () we obtain sup ( v L + b L)(t). () t [,T ] T Lv Ldτ. This completes the proof of Proposition 3.. Proposition 3.. Under the hypothesis of Theorem., the solution to the system (a)-(b) in [, T ] satisfies the following bound:

7 LAMHD-ALPHA SYSTEM 7 sup t [,T ] ( v L + b L )(t) + T L v Ldτ. Proof. We take L -inner products of (a)-(b) with ( v, b) to obtain in sum We estimate t( v L + b L ) + L v L () = (u )v v + v j u j v (b )b v (u )v v = j= ((u )b) b + ((b )u) b. i,j,k= k u i i v j k v j u L v L (3) by integration by parts, () and Hölder s inequalities where we further compute u L u L + k k u L (4) + k= k( ) k( 5 ) g( k ) k u L g( k ) + L u L by (5), Lemma. and (6). Hence, applying (4) in (3) gives (u )v v u L v L (5) Next, ( + L u L ) v L. v j u j v = j= j,k= k v j u j k v + v j k u j k v v u + v u v where the first term may be estimated identically as in (3). For the second term, v u v v L 6 u L 3 v L v L u L 3 by Hölder s inequalities and homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ). We see that we can estimate u L 3 + k k( 3 ) k u L

8 8 KAZUO YAMAZAKI due to (5) and Lemma.; hence, the previous estimates in (4) can be repeated to reach Therefore, we have shown u L 3 + L u L. (6) Next, by () j= v j u j v ( + L u L ) v L. (7) (b )b v = (b )b u (b )b u (8) where (b )b u b L 6 b L u L 3 + L u L (9) by Hölder s inequalities, homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ), Proposition 3. and (6). The higher order non-linear term is more difficult but can be handled as follows: (b )b u () = i,j,k= k b i i b j k u j + b i ikb j k u j b L b L 6 u L 3 + b L 6 b L u L 3 b L u L 3 due to Hölder s inequalities, homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ) and Proposition 3.. We can estimate u L 3 k u L 3 () k k k( ) k( 3 ) g( k ) g(k ) k u L L v L by (5), Lemma. and Hölder s inequality. Hence, with () applied to () and Young s inequality, we obtain (b )b u b L L v L 4 L v L + c b L. () Next, relying on a commutator estimate we compute

9 LAMHD-ALPHA SYSTEM 9 ((u )b) b = [ ((u )b) u b] b (3) ( u L b L + u L 3 b L 6) b L ( + L u L ) b L by (), Hölder s inequality, Lemma.3, homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ), (4) and (6). Finally for the last term in (), we compute ((b )u) b ( b L u L + b L 6 u L 3) b L (4) ( + L u L ) b L + L v L b L 4 L v L + c( + L u L ) b L by Hölder s inequality, Lemma., homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ), (4), Proposition 3., () and Young s inequality. In sum of (5), (7), (9), (), (3), (4) applied to (), we obtain after absorbing the dissipative term t ( v L + b L ) + L v L (5) ( + L u L ) ( + v L + b L ). Hence Gronwall s inequality with Proposition 3. completes the proof of Proposition 3.. Proposition 3.3. Under the hypothesis of Theorem., the solution to the system (a)-(b) in [, T ] satisfies the following bound: sup t [,T ] ( Λ 5 v L + Λ6 b L )(t) + T LΛ 5 v Ldτ. Proof. We apply Λ 5, Λ 6 on (a)-(b), take L -inner products with Λ 5 v, Λ 6 b respectively to obtain in sum, t( Λ 5 v L + Λ6 b L ) + LΛ5 v L (6) = Λ 5 ((u )v) Λ 5 v + Λ 6 ((u )b) Λ 6 b j= We first estimate Λ 5 (v j u j ) Λ 5 v + Λ 5 ((b )b) Λ 5 v + Λ 6 ((b )u) Λ 6 b.

10 KAZUO YAMAZAKI Λ 5 ((u )v) Λ 5 v + Λ 6 ((u )b) Λ 6 b (7) = (Λ 5 ((u )v) u Λ 5 v) Λ 5 v + (Λ 6 ((u )b) u Λ 6 b) Λ 6 b ( u L Λ 5 v L + Λ 5 u L 6 v L 3) Λ 5 v L + ( u L Λ 6 b L + Λ 6 u L 3 b L 6) Λ 6 b L u L ( Λ 5 v L + Λ6 b L ) + Λ5 u L 6 v L 3 Λ 5 v L + Λ 6 u L 3 b L Λ 6 b L by (), Hölder s inequalities, Lemma.3 and homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ). We further estimate u L u 6 L Λ 3 u 5 6 L + v L, (8) Λ 5 u L 6 v L 3 v 4 L Λ 5 v 3 4 L v 7 8 L Λ 5 v 8 L + Λ 5 v L, (9) Λ 6 u L 3 u 4 L Λ 7 u 3 4 L + Λ 5 v L, (3) due to Gagliardo-Nirenberg inequalities, (6) and Proposition 3.. Thus, applying (8), (9), (3) in (7) and using Proposition 3., we obtain Λ 5 ((u )v) Λ 5 v + Λ 6 ((u )b) Λ 6 b + Λ 5 v L + Λ6 b L. (3) Next, Λ 5 (v j u j ) Λ 5 v ( Λ 5 v L u L + v L 6 Λ 5 u L 3) Λ 5 v L (3) j= Λ 5 v L + v L ( + Λ5 v L ) Λ 4 v L + Λ 5 v L by Hölder s inequality, Lemma., (8), homogeneous Sobolev embedding of Ḣ (R 3 ) L 6 (R 3 ), (3), Proposition 3. and Young s inequality. Next, Λ 5 ((b )b) Λ 5 v Λ 5 div(b b) L Λ 5 v L (33) Λ 6 b L b L Λ 5 v L Λ 6 b L b 4 L b 3 4 L Λ 5 v L Λ 5 v L + Λ6 b L where we used Hölder s inequality, Lemma., Gagliardo-Nirenberg inequality, (6), Proposition 3. and Young s inequality. Finally,

11 LAMHD-ALPHA SYSTEM Λ 6 ((b )u) Λ 6 b ( Λ 6 b L u L + b L Λ 6 u L ) Λ 6 b L (34) Λ 6 b L u L + b 4 L b 3 4 L Λ 7 u L Λ 6 b L Λ 6 b L + Λ5 v L Λ 6 b L Λ 5 v L + Λ6 b L by Hölder s inequality, Lemma., Gagliardo-Nirenberg inequality, (8), (6), Proposition 3. and Young s inequality. Applying (3), (3), (33), (34) in (6) gives t( Λ 5 v L + Λ6 b L ) + LΛ5 v L + Λ5 v L + Λ6 b L. After absorbing, integrating in time completes the proof of Proposition Proof of Theorem.: Local Theory With a priori estimates achieved in Propositions 3., 3. and 3.3, it is a standard procedure to complete the proof of Theorem.. For completeness we sketch the proof of the local theory. We recall the mollification of J ɛ f of f L p (R 3 ), p by ( ) x y (J ɛ f)(x) = ɛ R 3 ρ f(y)dy, ɛ > ɛ 3 where ρ( x ) C, ρ, R 3 ρdx =. We regularize the system (a)-(b) by t v ɛ + J ɛ ((J ɛ u ɛ ) (J ɛ v ɛ )) + 3 j= J ɛ((j ɛ vj ɛ (J ɛu ɛ j ))) + (π ɛ + bɛ ) + J ɛ (L J ɛ v ɛ ) = J ɛ ((J ɛ b ɛ ) (J ɛ b ɛ )), t b ɛ + J ɛ ((J ɛ u ɛ ) (J ɛ b ɛ )) J ɛ ((J ɛ b ɛ ) (J ɛ u ɛ )) =, u ɛ = b ɛ =, v ɛ = ( )u ɛ and define for convenience ( ) y ɛ v ɛ, θ ɛ b ɛ uɛ For any ɛ >, using properties of mollifiers, one can show via the Picard Theorem, the global existence of the regularized solution y ɛ with its regularity of y ɛ C ([, ); H 5 (R 3 ) H 5 (R 3 )). Thereafter, through the process of obtaining a uniform bound locally in time, and showing that {θ ɛ } is Cauchy in C([, T ]; L (R 3 )), and then using Alaoglu s theorem, one arrives at the existence of the local solution pair v, b L ([, T ]; H 5 (R 3 )). We omit further details referring to [7] where the local theory for the NSE and the Euler equations using mollifiers is described in detail. u ɛ b ɛ

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