REGULARITY RESULTS ON THE LERAY-ALPHA MAGNETOHYDRODYNAMICS SYSTEMS

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1 REGULARITY RESULTS ON THE LERAY-ALPHA MAGNETOHYDRODYNAMICS SYSTEMS DURGA KC AND KAZUO YAMAZAKI Abstract. We study certain generalized Leray-alha magnetohydrodynamics systems. We show that the solution airs of velocity and magnetic fields to this system in two-dimension reserve their initial regularity in two cases: dissiation logarithmically weaker than a full Lalacian and zero diffusion, zero dissiation and diffusion logarithmically weaker than a full Lalacian. These results extend revious results in [4]. Moreover, we show that for a certain three-dimensional Leray-alha magnetohydrodynamics system, sufficient condition of regularity may be reduced to a horizontal gradient or a artial derivative in just one direction of the magnetic field, reducing comonents from the results in []. Keywords: Magnetohydrodynamics system, Navier-Stokes equations, regularity, Besov saces. Introduction We study the following two and three-dimensional Leray-alha magnetohydrodynamics MHD) systems resectively v t + u )v + π + νl v = b )b, b t + u )b b )u + ηl b =, v + u )v + π ν v = b )b, t a) b + u )b b )v η b =. t b) a) b) To both these systems, we subject the following relation between u and v, the incomressibility conditions and their initial data v = α )u, u = b =, v, b)x, ) = v, b )x). ) We denote by v the N-dimensional velocity vector field, N = or, u the filtered velocity, b the magnetic vector field and π the ressure scalar field. We also denote Both authors exress gratitude to the referee for valuable comments that imroved the manuscrit greatly, and Professor Jiahong Wu for his teaching. The second author additionally exresses his gratitude to Professor David Ullrich for his teaching. MSC: 5B65; 5Q5; 5Q86.

2 DURGA KC AND KAZUO YAMAZAKI ν, η the viscosity and diffusivity constants resectively and L defined by Lfξ) = mξ) ˆfξ), mξ) ξ g ξ ). 4) Finally, α > is the length-scale arameter reresenting the width of the filters. Hereafter for simlicity let us write t = t, i = x i, i =,, and denote the Lebesgue saces L, homogeneous and inhomogeneous Sobolev saces Ẇ m,, W m, equied with their norms L, Ẇ m,, W m, resectively. We briefly discuss the rich history concerning these systems. Firstly, the authors in [8] introduced LANS α model, which is well-known for its remarkable erformance as a closure model of turbulence in infinite channels and ies as their solutions give excellent agreement with emirical data for a wide range of large Reynolds numbers. Insired by the work in [8], the authors in [9] roosed the Leray-α model, the system a)-b) at b, g. Subsequently, the authors in [] roosed the systems a)-b) at g, a)-b) at g along with the three-dimensional LANS-α MHD system. As the LANS α and Leray α models introduced in [8] and [9] resectively model the Navier-Stokes equations NSE), the systems a)-b), a)-b) model the MHD system. The MHD system describes the motion of electrically conducting fluids and has broad alications in alied sciences such as astrohysics, geohysics and lasma hysics. It has been studied mathematically by many mathematicians for long time e.g. [6]); in articular, very recently the global regularity issue of the two-dimensional case has attracted much attention e.g. [4] and references found therein). In a three-dimensional eriodic domain, the authors in [] roved the existence of the unique weak solution air to the three-dimensional LANS-α MHD system and also obtained some convergence result as α +. In fact, it was shown in [4] that when ν, η >, g, this global well-osedness result of the LANS-α MHD system remains valid even in four-dimensional case cf. also [4]). In [], the authors obtained the global regularity results of the two-dimensional LANS-α MHD system in case of zero dissiation but with full Lalacian in the diffusion or zero diffusion but with full Lalacian in the dissiation. Subsequently in [4], the global regularity results in a similar case of ν >, η =, g and ν =, η >, g for the solution to the system a)-b) were also obtained. We also refer readers to [, 5, 7, 8, 4] for closely related results. On the other hand, for the system a)- b), desite the filtration, the global regularity result even in the three-dimensional case remains oen cf. []). For this urose, the authors in [] obtained Serrintye regularity criteria cf. [7]) in terms of the magnetic field b, in articular b. The urose of this manuscrit is two-fold. We first extend the global regularity results of the systems a)-b) by [4] logarithmically and reduce the regularity criteria of [] comonent-wise. We denote a horizontal gradient in R by h =,, ) and resent our results. Theorem.. Suose ν >, η = and g : R + R + is a radially symmetric, non-decreasing function such that gτ) c lne + lnτ)) lnτ). Then for any v, b H 4 R ) and any T >, there exists a unique solution air v, b) to the

3 LERAY-ALPHA MHD SYSTEMS system a)-b), ) such that su v H 4 R ) + b H 4 R ))t) + t [,T ] T Lv H 4 R ) dτ <. Theorem.. Suose ν =, η > and g : R + R + is a radially symmetric, nondecreasing function such that gτ) c lne + lnτ)). Then for any v, b H 4 R ) and any T >, there exists a unique solution air v, b) to the system a)- b), ) such that su v H 4 R ) + b H 4 R ))t) + t [,T ] T Lb H 4 R ) dτ <. We immediately remark that the hyotheses of Theorems. and. imly resectively dτ gτ) =, 5) lnτ)τ e e dτ g τ) lnτ)τ =. 6) Theorem.. Suose ν, η >, v, b) C[, T ); H 4 R )) is a solution to the system a)-b), ) for a given v, b H 4 R ) and T h b r L dτ <, + r +,. 7) 4 Then v, b) remains in the same class of regularity on [, T ] for some T > T. Theorem.4. Suose ν, η >, v, b) C[, T ); H 4 R )) is a solution to the system a)-b), ) for a given v, b H 4 R ) and { T b r L dτ <, + r + 4, if < 6, 8), if 6 <. Then v, b) remains in the same class of regularity on [, T ] for some T > T. Remark.. ) Global regularity results in the logarithmically suercritical regime was initiated by the author in [8] on the defocusing nonlinear wave equation and [9] on the NSE and insired many others to extend these results to different models: e.g. [5] on the Euler equations, [5] on the Boussinesq system, [] on the MHD system. We note that e.g. in the case of Theorem., we may take based on 4), and hence mξ) = L = ξ lne + ln ξ )) lne + ln ) )). ) Comonent reduction results of regularity criteria has also caught much attention recently and there is abundance of work in the literature by now. We refer readers to only its subset and references found therein: [,, 4] for the NSE, [, ] for the MHD system, [4] for active scalars.

4 4 DURGA KC AND KAZUO YAMAZAKI ) The numerical analysis results in [, 5] suggested that the velocity field lays more dominant role than the magnetic field in reserving the regularity of the solution air for the classical MHD system. In harmony with these suggestions, there are various regularity criteria for the three-dimensional MHD system in terms of only the velocity field e.g. [4, 9]) but not on the magnetic vector field alone. However, the regularity criteria in Theorems. and.4 rely only on a few comonents of b and the system a)- b) aroaches the three-dimensional classical MHD system as α +. We remark that in comarison, the authors in [] in articular obtained a regularity condition for the classical three-dimensional MHD system in terms of h v and h b or h v and b. As already mentioned, in [] the authors already obtained a Serrin-tye regularity criteria; in articular, T b r L dτ <, + =, <, r which is analogous to the scaling-invariant criterion for the three-dimensional NSE. Obtaining a comonent reduction tye result such as 7) and 8) and imroving the uer bounds of + 4 in 7) and + 4, in 8) back u to is extremely imortant, and very recently for the three-dimensional NSE, the authors in [7] obtained such a result via very sohisticated techniques of anisotroic Littlewood-Paley theory see also [6]). It should be a roblem of great interest if analogous results for the system a)-b) may be obtained in terms of only a few comonents of b. 4) The roof of Theorems. and. is insired by the work in [, ]; we emhasize however that the method in [] will not go through as we are considering the endoint-case which was omitted in the work of []. Our roof required significant extension of the method in []. This is due to the fact that while one may searate the velocity field in the non-linear terms of the system a)-b), it seems difficult to searate the magnetic field. Our roof may be alicable in other circumstances as well e.g. [9]). The roofs of Theorems. and.4 were insired by the work in [, 6]. In the Preliminaries section, let us set u notations, state some key facts and useful lemmas. Thereafter, we rove our claims.. Preliminaries Without loss of generality, let us assume α = ν = η = and use the notation A a,b B, A a,b B to imly that there exists a non-negative constant c that deends on a, b such that A cb, A = cb resectively. We write the vorticity as w v and fractional Lalacian Λ ). We now state some lemmas, secifying dimension when it is imortant. Lemma.. cf. [6]) Let f be divergence-free vector field such that f L,, ). Then f L c f L.

5 LERAY-ALPHA MHD SYSTEMS 5 Lemma.. cf. [7]) Let f W δ, L q, g W δ, L q, δ, < k <, < q k, k + q k =, k =,. Then fg Ẇ δ, f Ẇ δ, g L q + f L q g Ẇ δ, ). Lemma.. cf. [8]) Let f, g be smooth such that f L, Λ s g L, Λ s f L, g L 4,, ), = + = + 4,,, ), s >. Then Λ s fg) fλ s g L f L Λ s g L + Λ s f L g L 4 ). Lemma.4. cf. []) Let f L R ) W s, R ) where s R such that [, ), < s. Then f L R ) s, f L R ) + f H R ) log + f W s, R )) + ). Lemma.5. cf. []) Suose φ C R ) and µ, λ <, µ + λ >, + γ = µ + λ. Then φ LγR) φ L λ R ) φ L λ R ) φ L µ R ). Moreover, for r 6, φ L r R ) cr) φ 6 r r L R ) φ r r L R ) φ r r L R ) φ r r L R ). Lemma.6. cf. []) Let f, g, h Cc R ). Then γ fgh c f γ L q R ) if γ γ L s R ) g γ L R ) j, k, )g γ L R ) h L R ) R where < γ, q, s,, and. γ q + s = and i, j and k are any combinations of Lemma.7. Let < and a, b. Then a + b) a + b ). Let us recall the notion of Besov saces in R cf. [6]). We denote by SR ) the Schwartz class functions and S, its dual. We define { } S = φ SR ), φx)x r dx =, r =,,,.... R Its dual S is given by S = S/S For k Z we define = S /P where P is the sace of olynomials. A k = {ξ R : k < ξ < k+ }. It is well-known that there exists a sequence {Φ k } SR ) such that { su ˆΦ k A k, ˆΦk ξ) = ˆΦ k if ξ R ξ) and ˆΦ \ {}, k ξ) = if ξ =. Consequently, for any f S, k= k= Φ k f = f. To define the inhomogeneous Besov saces, we let Ψ C R ) be such that = ˆΨξ) + ˆΦ k ξ), Ψ f + Φ k f = f, 9) k= k=

6 6 DURGA KC AND KAZUO YAMAZAKI for any f S. With that, we set if k, k f = Ψ f if k =, Φ k f if k =,,,..., and define for any s R,, q [, ], the inhomogeneous Besov saces B,qR s ) = {f S : f B s,q R ) < }, { k= where f B s,q R ) = ks k f L R )) q ) q if q <, su k< ks k f L R ) if q =. The following lemma will be useful in obtaining estimates: Lemma.8. cf. [6]) Bernstein s inequality: Let f L R ) with q and < r < R. Then for all k Z + {}, and λ >, there exists a constant C k such that { su γ =k D γ f Lq R ) C k λ k+ q ) f L R ), λ k C k f L R ) su γ =k D γ f L R ) C k λ k f L R ), if su ˆf { ξ λr}, su ˆf {λr ξ λr} resectively. Moreover, if we relace the derivative D γ by the fractional Lalacian Λ γ, the inequalities remain valid only with trivial modifications.. A riori estimates for Theorem. We first obtain the basic energy conservation: taking L -inner roducts on a)- b) at ν =, η = with u, b) and integrating in time we obtain su t [,T ] T u L + u L + b L )t) + Lu L + L u L dτ u,b. ).. v L + b L -estimate for Theorem.. Proosition.. Suose ν =, η = and g : R + R + is a radially symmetric, non-decreasing function such that gτ) c lne + lnτ)) lnτ) and hence satisfies 5). Then the solution air to the system a)-b), ) in [, T ] satisfies su t [,T ] v L + b L )t) + T Lv Ldτ. Proof. Let us denote by At) e+ v L + b L )t). We take L -inner roducts on a)-b) with v, b) so that in sum we can estimate t v L + b L ) + Lv L = b )b v b )u b + u )b b = b )u b + b b u + b u b u b b b L u L + b L u L u L At) )

7 LERAY-ALPHA MHD SYSTEMS 7 by Hölder s inequalities and ). Using Littlewood-Paley decomosition 9), Bernstein s inequalities, Plancherel theorem, 4) and the fact that g is non-decreasing, we obtain for some M > to be determined subsequently u L k u L k k M k g k ) k u L g k ) + k M k >M k >M k k g k ) k u L g k ) gm) k L u L + k g k ) k Lv L ; we also used the fact that u L v L. We bound this by Hölder s inequalities: u L gm) k L u L ) k M + k >M k k M k g 4 k ) k Lv L k >M gm) lnm) L u L + M Lv L. k >M Thus, taking ) into ) we obtain, choosing M = cat) for c sufficiently large, t v L + b L ) + Lv L c gcat)) ) lncat)) L u L + cat)) Lv L At) Lv L + cgcat)) lncat)) + L u L )At) by Young s inequalities. Subtracting Lv L from both sides, we obtain t v L + b L ) + Lv L gcat)) lncat)) + L u L)At) ) which imlies t [, T ], cat) ca) dτ gτ) lnτ)τ t by ). This imlies according to 5) + L u L )dτ T + L u L)dτ su At). 4) t [,T ] Now we go back to ) and integrate in time over [, T ] to obtain T Lv Ldτ A) + su gcaτ)) lncaτ))aτ) τ [,T ] ) T due to ) and 4). This comletes the roof of Proosition.. + L u L)dτ

8 8 DURGA KC AND KAZUO YAMAZAKI.. w L + b L -estimate for Theorem.. Proosition.. Suose ν =, η = and g : R + R + is a radially symmetric, non-decreasing function such that gτ) c lne + lnτ)) lnτ) and hence satisfies 5). Then the solution air to the system a)-b), ) in [, T ] satisfies su t [,T ] w L + b L )t) + T Proof. Taking a curl on a) leads to Lw Ldτ. t w + u )w + L w = b )j + u ) u u ) u ). 5) Taking L -inner roducts of 5) with w, alying Λ on b) and taking L -inner roducts with Λ b, we obtain t w L + Lw L = b )jw + u ) u u ) u )w I i, 6a) Firstly, i= t b L = II i. i= by Hölder s inequality. We estimate w L 8 Λ u )b) Λ b + I b L 8 j L w L 8 k Λ b )u) Λ b 6b) k 4 g k ) kw L g k ) Lw L 7) by 9), Bernstein s inequalities, 4) and Hölder s inequalities. By 7) we deduce I b H b L Lw L Lw L 4 + c b L 8) due to Sobolev embedding of H R ) L 8 R ), ) and Proosition.. Similarly, by Hölder s inequalities, 7), Sobolev embedding of H R ) L 8 R ), ) and Proosition., we obtain I u L 8 v L w L 8 u H w L Lw L Lw L 4 + c w L. 9) Next, we first rewrite using divergence-free roerty of u and then estimate II = [Λ u )b) u Λ b] Λ b u L Λ b L + Λ u L 4 b L 4) Λ b L u L b L + Λ u L w L b L b L + u L ) w L + b L )

9 LERAY-ALPHA MHD SYSTEMS 9 by Hölder s inequality, Lemma., Gagliardo-Nirenberg inequality, Proosition. and Young s inequality. Now we estimate u L u L + k k u L ) u L + k k k k g k ) k u L g k ) + k k g k ) k L u L + Lv L by 9), Bernstein s inequality, Plancherel theorem and Hölder s inequality. Therefore, II + Lv L ) w L + b L). ) Finally, II Λ b L u L + b L Λ u L ) Λ b L ) + Lv L ) Λ b L + b L w L b L + Lv L ) Λ b L + b L + b H log + b H ) + ) w L b L w L + b L ) + Lv L ) log + w L + b L ) by Hölder s inequality, Lemma., ), Lemma.4, ), Young s inequality and Proosition.. In sum of 6a), 6b), 8), 9), ), ), we obtain after subtracting Lw L from both sides, t w L + b L ) + Lw L ) w L + b L ) + Lv L ) log + w L + b L ). By Proosition. this leads to su w L + b L)t) t [,T ] from which integrating in time on ) comletes the roof of Proosition.... Higher regularity for Theorems.. We aly Λ 4 on a), Λ 5 on b), take L -inner roducts with Λ 4 v, Λ 5 b) resectively to estimate t Λ 4 v L + Λ5 b L ) + LΛ4 v L 4) = [Λ 4 u )v) u Λ 4 v] Λ 4 v [Λ 5 u )b) u Λ 5 b] Λ 5 b + Λ 4 b )b) Λ 4 v + due to the incomressibility of u. Firstly, Λ 5 b )u) Λ 5 b 4 III i III u L Λ 4 v L + Λ 4 u L 4 v L 4) Λ 4 v L 5) [ u L + w L ) Λ 4 v L + Λ u L Λ 6 u L v 5 6 L Λ 4 v 6 L ] Λ 4 v L + Λ 4 v L by Hölder s inequality, Lemma., Sobolev embedding of H R ) L R ), the Gagliardo-Nirenberg inequalities, ), Proosition. and Young s inequalities. i=

10 DURGA KC AND KAZUO YAMAZAKI Reeating these alications on III to III leads to III u L Λ 5 b L + Λ 5 u L 4 b L 4) Λ 5 b L 6) ) u L + w L ) Λ 5 b L + Λ u 6 L Λ 6 u 5 6 L b L b L Λ 5 b L + Λ 4 v L + Λ5 b L. On the other hand, III Λ 4 b L 4 b L 4 + b L Λ 5 b L ) Λ 4 v L 7) ) Λ b 6 L Λ 5 b 5 6 L b L b L + b L + b L ) Λ 5 b L Λ 4 v L + Λ 4 v L + Λ5 b L by Hölder s inequality, Lemma., the Gagliardo-Nirenberg inequalities, Sobolev embedding of H R ) L R ), ), Proositions. and. and Young s inequalities. Finally, III 4 Λ 5 b L u L + b L Λ 6 u L ) Λ 5 b L 8) Λ 5 b L u L + w L ) + b L + b L ) Λ 4 v L ) Λ 5 b L Λ 4 v L + Λ5 b L by Hölder s inequality, Lemma., Sobolev embedding of H R ) L R ), Proositions.,. and Young s inequalities. Alying 5), 6), 7) and 8) in 4), Gronwall s inequality imlies the H 4 -bound of v, b). 4. A riori estimates for Theorem. 4.. v L + b L -estimate for Theorem.. We have analogously to ), by taking L -inner roducts of a)-b) at ν =, η = with u, b), su t [,T ] T u L + u L + b L )t) + Lb L dτ u,b. 9) We rove the following roosition: Proosition 4.. Suose ν =, η = and g : R + R + is a radially symmetric, non-decreasing function such that gτ) c lne + lnτ)) and hence satisfies 6). Then the solution air to the system a)-b), ) in [, T ] satisfies su t [,T ] v L + b L )t) + T L b Ldτ.

11 LERAY-ALPHA MHD SYSTEMS Proof. We take L -inner roducts of a) with v and b) with b, sum and estimate t v L + b L ) + L b L ) = b )b v + u )b b b )u b = b )b v u b b + b u b + b ) u b b L b L + v L ) + u L b L by Hölder s and Young s inequalities. We emhasize that in contrast to the case of Theorem. where we were able to searate u in every non-linear term and estimate u L in Besov sace as in ), we are unable to searate b in every non-linear term because we will not be able to handle b. Thus, we must estimate u L b L differently. We first estimate similarly to ), b L k M k g k ) kb L g k ) + k M k >M k >M k k g k ) kb L g k ) ) gm) k Lb L + k g k ) k L b L gm) lnm) Lb L + M L b L by 9), Bernstein s inequality, Plancherel theorem, 4) and Hölder s inequalities. Next, by Gagliardo-Nirenberg inequality and 9), Now b L u L b L u L u L b L v L b L. ) k M k g k ) k b L g k ) + k >M k >M k >M k k g k ) k b L g k ) ) gm)m lnm) Lb L + k 7k g 4 k ) L b L gm)m lnm) Lb L + M L b L by 9), Bernstein s inequality, 4) and Hölder s inequalities. Hence, ) alied to ) and Lemma.7 give v L b L v L g M)M lnm) Lb L + M 6 L b L ). 4) Considering ), ), ) and 4), we see that there exists a constant c > so that t At) + L b L c gm) ) lnm) Lb L + M L b L At) 5) ) + c A 6 t) g M)M lnm) Lb L + M 6 L b L.

12 DURGA KC AND KAZUO YAMAZAKI Now we take M = cat) for c > large so that c M L b L At) L b L by Young s inequality and + cat), Thus, c A 6 t)m 6 L b L L b L. t At) + L b L cgcat)) lncat))at) Lb L + L b L 6) Subtracting L b L from both sides, we obtain + cat) + At)g cat)) lncat)) Lb L. t At) + L b L g cat)) lncat))at)e + Lb L). 7) Thus, we obtain cat) dτ g τ) lnτ)τ ca) due to 9). By 6) this imlies t e + Lb L dτ T e + Lb Ldτ 8) su At). 9) t [,T ] Integrating in time on 7) and using 9) and 9) comletes the roof of Proosition w L + b L -estimate for Theorem.. Proosition 4.. Suose ν =, η = and g : R + R + is a radially symmetric, non-decreasing function such that gτ) c lne + lnτ)) and hence satisfies 6). Then the solution air to the system a)-b), ) in [, T ] satisfies su t [,T ] w L + b L )t) + T Proof. We take a curl on a) to obtain as in 5) L b Ldτ. t w + u )w = b )j + u ) u u ) u ). 4) We take L -inner roducts on 4) with w, aly Λ on b), take L -inner roducts of b) with Λ b to obtain t w L + b L ) + L b L 4) = b )jw + [ u ) u u ) u ]w Firstly, where we used that as in 7), Λ u )b) Λ b + Λ b )u) Λ b 4 IV i. IV b L 8 j L 8 w L L j L w L 4) j L 8 L j L, b L 8 b H 4) i=

13 LERAY-ALPHA MHD SYSTEMS by Sobolev embedding of H R ) L 8 R ), 9) and Proosition 4.. Next, IV u L u L w L 44) u L + u L log + u H )) w L log + w L + b L ) w L + b L ) by Hölder s inequality, Lemma.4, 9) and Proosition 4.. Next, IV = [Λ u )b) u Λ b] Λ b 45) u L Λ b L + Λ u L 4 b L 4) Λ b L u L + u L log + u H )) Λ b L + v L w L b L Λ b L log + w L + b L ) w L + Λ b L ) due to the incomressibility of u, Hölder s inequality, Lemma., Gagliardo-Nirenberg inequalities, Lemma.4, 9), Proosition 4. and Young s inequality. Finally, IV 4 Λ b L u L + b L Λ u L ) Λ b L 46) log + w L + Λ b L ) w L + Λ b L ) by Hölder s inequality, Lemmas. and.4, 9) and Proosition 4.. In sum of 4), 4), 44), 45) and 46), using Young s inequality we obtain t w L + b L ) + L b L 47) L j L w L + log + w L + b L ) w L + b L ) L b L + c log + w L + b L ) w L + b L ). Subtracting L b L from both sides, we have Integrating in time we obtain t log + log + w L + b L)). 48) su w L + b L)t) 49) t [,T ] and hence integrating on 47), using 49) comletes the roof of Proosition Higher regularity for Theorem.. In the Higher Regularity estimate for Theorem., we only used the uniform bound of su t [,T ] v H + b H )t) from ), Proositions. and., and not its dissiation. Thus, the same bound from 9), Proositions 4. and 4. allow us to obtain the higher regularity estimate needed for Theorem..

14 4 DURGA KC AND KAZUO YAMAZAKI 5. Proofs of Theorems. and. In this section, we rove the local existence and uniqueness of the solution air v, b) to a)-b) if v, b ) H 4 R ). We define the mollification J ɛ f of f L R ), by ) x y J ɛ f)x) = ɛ R ρ fy)dy, ɛ >, ρ x ) C, ρ, ρdx =. ɛ R We consider for ν, η, t v ɛ + J ɛ J ɛ u ɛ ) J ɛ v ɛ )) + π ɛ + νj ɛ L 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 )) + ηj ɛ L b ɛ = J ɛ J ɛ b ɛ ) J ɛ u ɛ )), with u ɛ = b ɛ =. We further rewrite ) y ɛ v ɛ u ɛ, y ɛ H 4 vɛ H + 4 bɛ H 4, θɛ u ɛ, b ɛ t y ɛ = F ɛ y ɛ ), v ɛ = )u ɛ, y ɛ x, ) = yx) ɛ = v, b )x), ) F ɛ y ɛ P Jɛ [J ) ɛ u ɛ ) J ɛ v ɛ )] + P J ɛ [J ɛ b ɛ ) J ɛ b ɛ ] Jɛ L v ɛ J ɛ [J ɛ u ɛ ) J ɛ b ɛ )] + J ɛ [J ɛ b ɛ ) J ɛ u ɛ )] Jɛ L b ɛ. b ɛ 5a) 5b) Using Picard s theorem, we can show the global existence of the unique solution y ɛ. Thereafter, we can show that y ɛ is uniformly bounded in L [, T ]; H 4 R ) H 4 R )) for some T >, {θ ɛ } is Cauchy in C[, T ]; L ), rely on Banach Alaoglu s therorem to obtain the desired result. We omit details and refer readers to []. 6. Proof of Theorems. and.4 In this section, we set v = v, v, v ), b = b, b, b ), and for brevity W t) h v L + hb L )t), Y t) h v L + hb L )t), Xt) v L + b L )t), Zt) v L + b L )t). 6.. H -estimate for the roof of Theorem.. We resent the roof in case, ); the cases =, also go through via a straight-forward modification. Taking L -inner roducts of a)-b) with v, b) resectively and integrating in time lead to su t [,T ] T v L + b L )t) + v L + b L dτ v,b. 5) We denote by h i= i and take L -inner roducts on the equations a)-b) with h v, h b) resectively to obtain t h v L + hb L ) + hv L + hb L 5) = u )v h v u )b h b + b )b h v + b )v h b 4 I i. i=

15 LERAY-ALPHA MHD SYSTEMS 5 We estimate I + I u L v L h v L + b L h b L ) Y t) 4 + cx 5) by Hölder s inequalities, Sobolev embedding of H R ) L R ), 5) and Young s inequalities. We first integrate by arts on I, I 4 : I + I 4 = h b b h v + h b v h b 54) due to ). Thus, using Hölder s and Gagliardo-Nirenberg and Young s inequalities we estimate I + I 4 h b L h v b L L + h b v L L ) 55) h b L h v h v L L b L + h b Y 4 + c hb L v L + b L ). L h b L v L ) Taking 5) and 55) into 5) and subtracting Y t) from both sides lead to Integrating in time leads to W t) + t t W + Y X + h b Y τ)dτ W ) + + t L X. 56) h b L Xτ)dτ 57) due to 5). Next, we take L -inner roducts of a)-b) with v, b) to estimate We estimate similarly to 5) t v L + b L ) + v L + b L 58) = u )v v u )b b + b )b v + b )v b 4 II i. i= II + II u L v L v L + b L b L ) Z 4 + cx. 59) Next, we first decomose as follows: II + II 4 = b )b h v + b )v h b + b )b v + b )v b 6) where for the first two integrals, we have the estimate used in 56). Thus, we focus on the third and fourth integrals on which we first decomose and estimate: writing

16 6 DURGA KC AND KAZUO YAMAZAKI b h = b, b, ) and integrating by arts, using incomressibility b )b v + b )v b 6) = b h h )b v + b h h )v b + b b v) = b h h )b v + b h h )v b b b v) = b h h b v + b h h v b + b h h b v) + h b h ) b v) = b h h b v + b h h v b + h b h ) b v) + h b h ) b v) h b L b L 4 v L 4 + h v L b L 4 h b L + h v L ) b L b L 6 + v L v L 6) h b L + h v L ) b L h b L b L + v L h v L v L ) by Hölder s inequalities, interolation inequalities and Lemma.5 with γ = 6, λ = µ =. Thus, with this, 56) alied to 6) along with 59) in 58) we obtain t X + Z + h b L )X + W X 4 Y Z 4. 6) We integrate 6) in time and estimate Xt) + + t + su τ [,t] + t t t Zτ)dτ 6) + h b L W τ) t h b Zτ)dτ + c )Xτ)dτ ) 4 t ) t ) 4 Xτ)dτ Y τ)dτ Zτ)dτ t ) t ) L 4 Xτ)dτ + + h b L Xτ)dτ Zτ)dτ + t h b L t ) 4 Xτ)dτ + h b L Xτ)dτ by Hölder s inequality, 5), 57), Young s inequality and Lemma.7. Now t ) 4 t 8 h b L Xτ)dτ ) h b Xτ)dτ by Young s inequality and 5). After subtracting Zτ)dτ from both sides, Gronwall s inequality comletes the roof of Theorem The H -estimate for the roof of Theorem.4. L t )

17 LERAY-ALPHA MHD SYSTEMS Case < 6. We work on 5): tw + Y = 4 I i 64) for which we use the estimates on I + I as in 5). For I and I 4, we use Lemma.6 to obtain I + I 4 b L b L h b L i= h v L + v L h v L h b L ) Y 4 + c b L X 65) by 5) and Young s inequalities. Thus, from 5) and 65) alied to 64), after subtracting Y t) from both sides and integrating in time we obtain W t) + t Y τ)dτ W ) + + by 5). Next, as in 58) we have t t b tx + Z = Xτ)dτ + L Xτ)dτ t b L Xτ)dτ 66) 4 II i 67) where for II + II we use the estimate from 59) while as in 6) we decomose II + II 4 = b )b h v + b )v h b + b )b v + b )v b. Again, for the first two integrals, we can follow the same estimate we obtained from 65) whereas from 6) we have b )b v + b )v b h b L + h v L ) b L h b L b L + v L h v L v L ). Thus, for any ɛ > arbitrary small we can obtain II + II 4 ɛy + c b L X + ) W X 4 Y Z 4. 68) Hence, for ɛ > sufficiently small so that by continuity of Riesz transform in L R ), ɛy Z 4, we obtain from 67) after subtracting Z from both sides, i= t X + Z X + b L X + W X 4 Y Z 4. 69)

18 8 DURGA KC AND KAZUO YAMAZAKI Integrating in time we obtain Xt) + + t + su τ [,t] + t t t b Zτ)dτ 7) L Xτ)dτ ) 4 t ) t ) 4 Xτ)dτ Y τ)dτ Zτ)dτ t ) t ) L 4 Xτ)dτ + + b L Xτ)dτ Zτ)dτ W τ) t b Zτ)dτ + c + t b L t ) 4 Xτ)dτ + b L Xτ)dτ by Hölder s inequalities, 66), Young s inequality and Lemma.7 where we can estimate as before due to 5) t ) 4 b L Xτ)dτ t b 8 ) L Xτ)dτ. Gronwall s inequality comletes the roof of Theorem.4 in case < Case > 6. The roof of the case = is simler; we assume < below. We take L -inner roducts on a)-b) with v, b) to estimate t v L + b L ) + v L + b L 7) = u )v v u )b b + b )b v + b )v b. Firstly, u )v v + u )b b 7) u L v L v L + b L b L ) v L + b L 8 ) + cx by Hölder s inequality, Sobolev embedding of H R ) L R ), 5) and Young s inequalities. Next, after integrating by arts twice we estimate b )b v + b )v b 7) = b v b + b b v b L v L b + b L L v ) L b L v L b b L L + b L v L v L ) v L + b L 8 + c b L b 6 L + v 6 L )

19 LERAY-ALPHA MHD SYSTEMS 9 where we used ), Hölder s, Gagliardo-Nirenberg and Young s inequalities. Thus, alying 7) and 7) in 7) and integrating in time we obtain v L + b L )t) + T T + c b L dτ ) v L + b Ldτ 74) T b L + v L dτ ) by Hölder s inequality and 5). Next, we take L -inner roducts of a)-b) with v, b) and estimate as in 58) and 59) to obtain tx + Z Z 4 + cx + II + II 4. Next, by Hölder s inequality, Lemma.5 with r =, we estimate 4 II i b L v L 75) i= ) b L b 6 L b 6 L b 6 L ) v L v 6 L v 6 L b 6 L Z + c b 6 5 L b 5 L b 5 L v 5 L Z + cx b L + v L ) where we also used the Young s inequalities. Thus, Gronwall s inequality using 5) and 74) comletes the roof of Theorem.4 for the case <. 6.. Higher regularity for Theorems. and.4. Alying Λ 4 to both a), b) and taking L -inner roducts with Λ 4 v, Λ 4 b), we obtain t Λ 4 v L + Λ4 b L ) + Λ5 v L + Λ5 b L = [Λ 4 u )v) u )Λ 4 v] Λ 4 v [Λ 4 u )b) u )Λ 4 b] Λ 4 b + [Λ 4 b )b) b )Λ 4 b] Λ 4 v + [Λ 4 b )v) b )Λ 4 v] Λ 4 b u L Λ 4 v L + Λ 4 u L v L ) Λ 4 v L 6 + u L Λ 4 b L + Λ 4 u L b L ) Λ 4 b L 6 + b L Λ 4 b L Λ 4 v L 6 + b L Λ 4 v L + Λ 4 b L v L ) Λ 4 b L 6 u L u L Λ 4 v L + Λ 4 u L v L v L ) Λ 5 v L + b L b L Λ 4 b L Λ 5 v L + b L b L Λ 4 v L + Λ 4 b L v L v L ) Λ 5 b L Λ5 v L + Λ 5 b L + c v L + b L ) Λ 4 v L + Λ4 b L )

20 DURGA KC AND KAZUO YAMAZAKI by Hölder s inequalities, Lemma., the Gagliardo-Nirenberg inequalities, Sobolev embedding of Ḣ R ) L 6 R ). Subtracting Λ5 v L + Λ 5 b L and alying Gronwall s inequality imlies in articular the H 4 -bound of v, b) Proof of Theorems. and.4. Very similarly to the roofs of Theorems. and. in Section 5, given v, b H 4 R ), it can be shown that there exists a unique solution air v, b) C[, T ); H 4 R )) for some T >. Under the hyothesis of 7) or 8), we have shown the bound on su t [,T ] v H4 R ) + b H4 R ))t) and hence by a standard argument we can restart the local theory at time T to extend the solution to T > T. We omit further details. References [] H. Brezis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 98), [] C. Cao, E.S. Titi, Global regularity criterion for the D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., ), [] C. Cao, J. Wu, Two regularity criteria for the D MHD equations, J. Differential Equations, 48 ), [4] C. Cao, J. Wu, B. Yuan, The D incomressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 4), [5] D. Chae, P. Constantin, J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrohic equations, Arch. Ration. Mech. Anal., ), 5-6. [6] J.-Y. Chemin, Perfect Incomressible Fluids, Oxford lecture series in mathematics and its alications, Oxford, 4, 998. [7] J.-Y. Chemin, P. Zhang, On the critical one comonent regularity for -D Navier-Stokes system, Ann. Sci. Éc. Norm. Suér, to aear. [8] S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi, S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel flow, Phys. Rev. Lett., 8 998), [9] A. Cheskidov, D. D. Holm, E. Olson, E. S. Titi, On a Leray alha model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 46 5), [] B.-Q. Dong, Z. Zhang, The BKM criterion for the D Navier-Stokes equation via two velocity comonents, Nonlinear Anal. Real World Al., ), [] J. Fan, T. Ozawa, Regularity criteria for the magnetohydrodynamic equations with artial viscous terms and the Leray-α-MHD model, Kinet. Relat. Models, 9), 9-5. [] J. Fan, T. Ozawa, Global Cauchy roblem for the -D magnetohydrodynamic-α models with artial viscous terms, J. Math. Fluid Mech., ), 6-9. [] A. Hasegawa, Self-organization rocesses in continuous media, Adv. Phys., 4 985), -4. [4] C. He, Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 5), [5] T. Hmidi, On a maximum rincile and its alication to logarithmically critical Boussinesq system, Anal. PDE, 4 ), [6] X. Jia, Y. Zhou, Regularity criteria for the D MHD equations via artial derivatives, Kinet. Relat. Models, 5 ), [7] T. Kato, Liaunov functions and monotonicity in the Navier-Stokes equation, Functionalanalytic methods for artial differential equations, lecture notes in mathematics, 45 99), 5-6. [8] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Al. Math., 4 988), [9] D. KC, K. Yamazaki, Logarithmically extended global regularity result of Lans-alha MHD system in two-dimension, J. Math. Anal. Al., 45 5), [] J. S. Linshiz, E. S. Titi, Analytical study of certain magnetohydrodynamic-α models, J. Math, Phys., 48, ). [] A. J. Majda, A. L. Bertozzi, Vorticity and Incomressible Flow, Cambridge University Press, Cambridge,. [] L. Ni, Z. Guo, Y. Zhou, Some new regularity criteria for the D MHD equations, J. Math. Anal. Al., 96 ), 8-8.

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