SECOND PROOF OF THE GLOBAL REGULARITY OF THE TWO-DIMENSIONAL MHD SYSTEM WITH FULL DIFFUSION AND ARBITRARY WEAK DISSIPATION

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1 SECOND PROOF OF THE GOBA REGUARITY OF THE TWO-DIMENSIONA MHD SYSTEM WITH FU DIFFUSION AND ARBITRARY WEAK DISSIPATION KAZUO YAMAZAKI Abstract. In regards to the mathematical issue of whether a system of equations admits a unique solution for all time or not, given an arbitrary initial data sufficiently smooth, the case of the magnetohydrodynamics system may be arguably more difficult than that of the Navier-Stokes equations. In the last several years, an explosive amount of work by many mathematicians was devoted to make progress toward the global well-posedness of the two-dimensional magnetohydrodynamics system with diffusion in terms of a full aplacian but with zero dissipation; nevertheless, this problem remains open. The purpose of this manuscript is to provide a second proof of the global well-posedness in case the diffusion is in the form of a full aplacian, and the dissipation is in the form of a fractional aplacian with an arbitrary small power. In contrast to the first proof of this result in the literature that took advantage of the property of a heat kernel, the main tools in this manuscript consist of Besov space techniques, in particular fractional chain rule, which has been proven to possess potentials to lead to resolutions of difficult problems, in particular of fluid dynamics partial differential equations. Keywords: Besov space; fractional aplacians; global regularity; magnetohydrodynamics system; Navier-Stokes equations.. Introduction One of the most difficult outstanding open problems in mathematical analysis questions whether or not, given an initial data u that is sufficiently smooth, the following Navier-Stokes equations (NSE) of fluid mechanics in case N = 3 admits a unique solution for all time: u + (u )u + π ν u =, t (a) u =, u(x, ) = u (x), (b) where u (u,..., u N )(x, t), π(x, t) represent the velocity, pressure fields respectively, while ν the viscosity coefficient. Due to the scaling property that u λ (x, t) λ u(λx, λ t), λ R +, solves the NSE if u(x, t) does, for simplicity hereafter we assume ν = ; moreover, let us also write t = t, = i, i N, where x i x = (x,..., x N ), and f = f(x)dx. R N MSC : 35B65, 35Q35, 35Q6

2 KAZUO YAMAZAKI The magnetohydrodynamics (MHD) system describes the motion of electrically conducting fluids, has broad applications in applied sciences such as astrophysics, geophysics and plasma physics, and has been studied extensively ever since the pioneering work of [, 6]. We first introduce a fractional aplacian Λ ( ) defined through Fourier transform of Λ r f(ξ) ξ r ˆf(ξ), r R, and now the generalized MHD system, which is a coupling of the NSE (a)-(b) with Maxwell s equations of electromagnetism: t u + (u )u + π + Λ α u = (b )b, t b + (u )b + Λ β b = (b )u, (a) (b) u = b =, (u, b)(x, ) = (u, b )(x), (c) where b = (b,..., b N ) represents the magnetic field, α, β [, ]; let us call the system (a)-(c) at α = β = the classical MHD system. We remark that at α =, the equation (a) is exactly (a) forced by the term (b )b. It is obvious that if one accomplishes in proving that given an arbitrary initial data (u, b ) sufficiently smooth, there exists a unique smooth solution to the MHD system (a)-(c), then taking b deduces by uniqueness, the smooth solution to the NSE (a)-(b); therefore, it may be argued that the proof of the regularity of the solution is harder for the classical MHD system (a)-(c) than the NSE (a)-(b). For the classical MHD system, the existence of the unique strong solution globally in two-dimensional (d) case and locally in three-dimensional (3d) case, as well as the existence of a weak solution globally in both dimensions were shown in []. For the more general case with fractional aplacians, it was shown in [7] that if α + N 4, β + N 4, then the global regularity result may be attained in R N, N ; in particular, in the case N =, this requires α = β =. Taking -inner products of (a)-(b) with (u, b) respectively, and taking advantage of (c) lead to t t u(t) + b(t) + Λ α u dτ + Λ β b dτ = u + b (3) for all t [, T ], the interval over which the solution exists; this represents the conservation of energy and cumulative energy dissipation and diffusion. This threshold of + N may be seen as the endpoint of the energy criticality such that if either 4 power, α or β, lies below, then the dissipation and diffusion are no longer strong enough to suppress the non-linear terms unless the better bound beyond those in (3) are discovered (see logarithmic improvements in [3, 6, 9, 3]). A remarkable feature of the solution to the d NSE due to the incompressibility condition (b) is that only in this dimension, the better bound in fact exists, even with zero dissipation, the case in which (a)-(b) recovers the Euler equations. Indeed, upon applying a curl operator on (a) without the dissipation, we see that the vorticity w u evolves in time over the transport equation of t w + (u )w =

3 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 3 so that taking p -estimate, p [, ), using that (u )w w p w = (u ) w p = p due to (b), deduces hence, writing p t w p p p t w p p = ; (4) = w p t w p p, dividing by w p and thereafter tak- p ing the limit p shows that -norm of w is bounded, which is almost as good as that of u ([38]). It is a natural question to ask whether such a favorable formulation of the vorticity equation may be utilized to improve the results in [, 7] that stated that the global well-posedness of the generalized MHD system (a)-(c) requires α, β. Here, in contrast to (4), the additional difficulty is that upon the p -estimate of w where w u and u solves (a), p [, ), we are faced with p t w p + Λ α w w p w = (b )j w p w (5) (see (a)) and hence the task of having to estimate (b )j w p w, j b, while also making use of the fractional dissipative term Λ α w w p w if necessary. This explains precisely why many realized that if β is sufficiently large so that (b )j within (b )j w p w has an adequate bound or α is sufficiently large so that Λ α w w p w can provide adequate support to help estimate (b )j w p w, then the regularity of the solution (u, b) to the system (a)-(c) may be attained. We now review prominent results by those who made significant contribution based on this observation. For the case α = ; i.e. no dissipation term, Tran, Yu and Zhai in [5] showed that β > suffices to prove the regularity of the solution for all time. Jiu and Zhao in [3] and the author [3] independently improved to β > 3. Thereafter, Cao, Wu and Yuan in [5] showed that β > in fact suffices via Besov space techniques approach; subsequently, Jiu and Zhao in [4] also obtained the same result by a completely different approach from [5]. On the other hand, for the case β =, Tran, Yu and Zhai in [5] showed that α suffices. Subsequently, Yuan and Bai in [37], as well as the author in [3], independently improved this result to α > 3. Thereafter, Ye and Xu in [36] improved to α 4. Finally, making use of the property of a heat kernel, the authors in [] proved that α > suffices. Therefore, coming from both directions, α = fixed and reducing the powers of β, or β = fixed and reducing the powers of α, we have come to the crossroad at which the only case left is α =, β =, which is the extension of the classical result from [38] (see also [6]). Numerical analysis indicates that the regularity is more likely than the blow-up (e.g. [4]), and various regularity criteria have been obtained (e.g. [, 35]); however, this problem has remained open since, seemingly asking for a new idea. It is worth emphasizing that the resolution of this problem should lead immediately to analogous results for various related systems of equations such as magnetic Bénard problem, and possibly magneto-micropolar

4 4 KAZUO YAMAZAKI fluid system ([33, 34]). The purpose of this manuscript is to provide a second proof of the global well-posedness of the system (a)-(c) in case α >, β = as follows. Theorem.. et α >, β =. For every (u, b ) H s (R ), s 3, there exists a unique solution pair (u, b) to the generalized MHD system (a)-(c) such that u C([, ); H s (R )) ([, ); H s+α (R )), b C([, ); H s (R )) ([, ); H s+ (R )). Remark.. The proof follows the approach of [5] that considered β > and zero dissipation in (a)-(c). et us already point out the distinctive difference between their iteration schemes and ours. The authors in [5] obtained the q -bound of w and q -bound of j (see Proposition 3. [5]), and then T Bδ q,-bound of j (see Proposition 4. [5]), specifically T j B δ q, dτ c, where q β, q < δ < β, (6) which consequently leads in particular to the bound of T j dτ, q < 4 3β if β 4 3 (see Proposition 5. [5]). This improvement through the T Bδ q,-bound allows one to go back and attain q -estimate of w, j and repeat. It is clear that one cannot readily extend this strategy to the case β =, because if β =, then we require q = in (6) which disallows us to find any δ such that < δ < β, and q similarly we will not be able to find any r [, ) in (7) if β =. Moreover, 4 3β for the authors in [5] to make the crucial improvement through this T Bδ q,-bound, it seems β > is a crucial assumption which is absent in our hypothesis. However, we can make use of α > in the q -estimate of j immediately. In other words, our iteration cycle will take place as follows: q -bound of w (see Proposition 3.), q -bound of j where q = q (7) (see Proposition 3.3), the α T Bδ q,-bound of j (see Proposition 3.4) which leads to q -bound of w (see Proposition 3.5) and repeat thereafter. Here, upon making careful use of α >, in contrast to the approach of [5], it will be crucial to rely on the fractional chain rule emma.3 (see the proof of Proposition 3.5, in particular (49), (5), (5)). In the next section, we set up notations and state key lemmas. The proof of the local existence result is standard; we refer to e.g. [] where by using mollifiers the local existence proof is shown in the case of the NSE and the Euler equations. The initial regularity space in the statement of Theorem. may be generalized in many ways; we choose to focus on the a priori estimates in this manuscript.. Preliminaries et us use the notations A 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. The following lower bound on the fractional aplacian has found many applications:

5 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 5 emma.. ([] and emma 3.3 [5]) et r [, ], and f, Λ r f p (R ), p. Then Λ r f p p f p fλ r f. We use the following well-known inequalities: emma.. (e.g. Theorem 3.. [7]) Suppose f satisfies f =, f p (R ), p (, ). Then f p c p p f p. emma.3. (e.g. emma A [8], Proposition 3. [9] ) et F be C mapping such that F () = and F (τf + ( τ)g) h(τ) G(f) + G(g) where G C, G >, h ([, ]). Then for δ (, ), p, p (, ), p (, ] satisfying p = p + p, F f Ẇ δ,p G f p f Ẇ δ,p. emma.4. (e.g. emma A. [7], see also [7, ]) et f W δ,p (R ) q (R ), g W δ,p (R ) q (R ), δ, < p k <, < q k, + =, k =,. Then p k q k p fg Ẇ δ,p ( f Ẇ δ,p g q + f g Ẇ δ,p ). emma.5. ([4]; see also the Appendix [3] for proof)) et f H s (R ), s >, satisfy f =, f (R ). Then there exists a constant c such that f ( f + f log ( + f H s) + ). emma.6. ([9]) et f, g be smooth such that f p (R ), Λ s g p (R ), Λ s f p3 (R ), g p4 (R ), p (, ), p = p + p = p 3 + p 4, p, p 3 s >. Then Λ s (fg) fλ s g p ( f p Λ s g p + Λ s f p 3 g p 4 ). (, ), et us recall the notion of Besov spaces (cf. [, 7]). We denote by S(R ) the Schwartz class functions and S (R ), its dual. We define S to be the subspace of S in the following sense: S {φ S, φ(x)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 space of polynomials. A k {ξ R : k < ξ < k+ }. It is well-known that there exists a sequence {Φ k } S(R ) such that supp ˆΦ k A k, ˆΦk (ξ) = ˆΦ ( k ξ) or Φ k (x) = k Φ ( k x) and { if ξ R ˆΦ \ {}, k (ξ) = if ξ =. k=

6 6 KAZUO YAMAZAKI Consequently, k= Φ k f = f for any f S. We set Ψ C (R ) be such that = ˆΨ(ξ) + ˆΦ k (ξ), Ψ f + Φ k f = f, k= k= 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, p, q [, ], the inhomogeneous Besov space where f B s p,q For any s R, < p <, Bp,q s {f S : f B s p,q < }, { ( k= (ks k f p) q ) q, if q <, sup k< ks k f p if q =. B s p,min{p,} W s,p B s p,max{p,} (8) (pg. 5 [3], Theorems.4 and.4 in []). The following lemmas will be useful in obtaining upper and lower estimates: emma.7. (cf. [7]) Bernstein s inequality: et f p (R ) with p q and < r < R. Then for all k Z + {}, and λ >, there exists a constant C k such that { sup r =k r f q C k λ k+( p q ) f p if supp ˆf {ξ : ξ λr}, C k λk f p sup r =k r f p C k λ k f p if supp ˆf {ξ : λr ξ λr}, and if we replace derivative r by the fractional aplacian, the inequalities remain valid only with trivial modifications. emma.8. (cf. [8, 8]) et r and p = or r and < p <. Then for any k Z, f S, rk k f p p r,p k f p k fλ r k f. Moreover, Bony s paraproduct decomposition (cf. [7]) will be used frequently: where T f g k S k f k g, fg = T f g + T g f + R(f, g) R(f, g) k,k : k k k f k g, S k l:l k l. (9) Now let us state the basic energy conservation of the system (a)-(c) as in (3): sup t [,T ] ( u + b ) (t) + T Λ α u + b dτ. ()

7 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 7 The following proposition was observed in many previous works (e.g. Proposition 3. [3]): Proposition.9. et α, β =. Suppose (u, b ) H s (R ), s 3. Then its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] has the following bounds: sup t [,T ] ( w + j ) (t) + T Λ α w + Λj dτ. For completeness we leave the proof in the Appendix. Now in order to clearly explain the plan of the proof of Theorem., let us discuss the regularity criteria that has been observed in the previous work (e.g. Proposition 3.4 [3]); the proof of Theorem. would be complete once we attain the following bound: Proposition.. et α >, β =. Suppose (u, b ) H s (R ), s 3 and its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] satisfies Then T w α dτ. () u C([, T ]; H s (R )) ([, T ]; H s+α (R )), b C([, T ]; H s (R )) ([, T ]; H s+ (R )). For completeness, we sketch its proof in the Appendix. 3. Proof of Theorem. et us assume α (, 3 ); we explain the reason for this restriction at the Remark 3.. Applying the curl operator on (a)-(b), we obtain tw + Λ α w = (u )w + (b )j, tj + Λ j = (u )j + (b )w + [ b ( u + u ) u ( b + b )]. (a) (b) The following proposition is similar to but slightly better than Proposition 3. [3]: Proposition 3.. et α (, 3 ), β =. Suppose (u, b ) H s (R ), s 3. Then its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] has the following bounds: T Λ α j(t) + sup t [,T ] Λ +α j dτ. For completeness we leave this proof in the Appendix as well. The next proposition is also similar to but slightly better than Proposition 3.3 [3]. Proposition 3.. et α (, 3 ), β =. Suppose (u, b ) H s (R ), s 3. Then its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] has the following bounds: for q [, ( + α) α ], sup w(t) q q + t [,T ] T w q q α dτ q.

8 8 KAZUO YAMAZAKI Remark 3.. The reason for the restriction of α (, 3 ) is because if α 3, then by Proposition. and Proposition 3., the proof of Theorem. is complete and ( + α) no iteration scheme is necessary because ( α) α if and only if α 3. ( + α) Proof. et us assume ( + α) q α below because the case q [, (+ α)) can be interpolated once we obtain higher p -bounds. We multiply (a) by w q w, integrate in space, use divergence-free property of u from (c) to obtain (5) with p replaced by q. We use emma., the Sobolev embedding of Ḣ α (R ) α (R ) to obtain the lower bound on the dissipation as c(q ) w q Λ α w q q Λ α w w q w q α for a constant c(q ) that depends on q so that we can estimate q t w q q + c(q ) w q q α b j q +α w q q w q α b α +α Λ α j +α j q w q +α q w q α c(q q ) w q q q + c j q w ( q q )(q ) q α +α by Hölder s inequalities, Gagliardo-Nirenberg ( inequality, (), ) Proposition 3. and ( + α) Young s inequality. Now for any q ( + α),, we may estimate α (+α) ( α)q j q j q α +α by Gagliardo-Nirenberg inequality while (3) Λ +α j (+α) ( α)q q α q + Λ +α j (4) j q +α = { j α Λ+α j if q = (+α) α, j + Λ +α j if q = ( + α), (5) by Sobolev embedding of Ḣ α (R ) α (R ) and Proposition.9. Therefore, applying (4) and (5) in (3), subtracting c(q) w q q from both sides, Young s α inequalities lead to t w q q q + c(q ) w q q q α ( + Λ +α j )( + w q q ). Applications of Gronwall s inequality and Proposition 3. complete the proof of Proposition 3.. We now show that the bound on of j for q = q α. T w q q α dτ directly leads to an q -bound

9 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 9 Proposition 3.3. et α (, 3 ), β =. Suppose (u, b ) H s (R ), s 3. Then its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] has the following bound: for q [, ( + α) ( α) ], sup j(t) q q. t [,T ] Proof. We consider for simplicity of the proof, q = (+α) ( α) as the lower q -bounds may be attained via interpolation. We multiply (b) by j q j, integrate in space to obtain due to the incompressibility of u from (c), t j q q q + j q jλ j (6) = j q j ((b )w + [ b ( u + u ) u ( b + b )]). By emma., we have on the diffusive term, c(q ) Λ j q j q jλ j (7) whereas we estimate by integration by parts j q j(b )w =(q ) wb j j q = (q ) wb ( j q ) q sgn(j) j q {j } q b w j q j q q b α +α Λ α j +α w j q j q c(q ) 4 Λ j q + c w ( j q + ) where sgn(j) = j j, by Hölder s inequalities, Gagliardo-Nirenberg inequality, (), Proposition 3., Young s inequalities and Plancherel theorem. Moreover, we estimate j q j[ b ( u + u ) u ( b + b )] j q b u j q q q b j q w Λ j q q j q w c(q ) 4 Λ j q + c j q w q q by Hölder s inequality, Gagliardo-Nirenberg inequality, emma., and Young s inequality. Applying (7), (8), (9) in (6) and subtracting c(q) j q from both sides give q t j q + c(q ) Λ j q ( j q q q + )( w + w ). (8) (9)

10 KAZUO YAMAZAKI Now Gronwall s inequality and Proposition 3. at q = (+α) α of Proposition 3.3. complete the proof The next proposition follows the work of [5] closely; however, our proof seems slightly more straight-forward (see e.g. (9), (43)). Proposition 3.4. et α (, 3 ), β =. Suppose (u, b ) H s (R ), s 3. Then its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] has the following bound: for α < q ( + α) ( α), q ( α) < δ <, T j B δ q,dτ q. Proof. We fix k 3, apply k on (b), multiply by k j q k j and integrate in space to obtain t k j q q + k j q k jλ k j = k j q k j k ( (u )j) + k j q k j k ((b )w) + k j q k j k ( b u ) + k j q k j k ( b u ) () + k j q k j k ( u b ) + k j q k j k ( u b ) 6 II i. i= For the diffusive term, by emma.8 we obtain its lower bound c k k j q k j q k jλ k j. () Now we estimate the nonlinear terms. We apply Bony s product decomposition (9), subtract and add to rewrite II from () as II = k j q k j [ k, S l u ] l j k j q k j k j q k j k j q k j k j q k j l: k l l: k l l: k l l: k l (S l u S k u) k l j S k u k l j l:l k,l : l l k ( l u S l j) k ( l u l j) 5 II i. i= ()

11 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM The advantage of this is that II 3 = due to (c): II 3 = k j q k j S k u k l j =. (3) Besides, we first estimate II = k j q k j l: k l l: k l l: k l [ k, S l u ] l j k j q [ k, S l u ] l j k j q k( ) S k u k j xφ by (), Hölder s inequalities, the fact that for all l such that k l, we may replace l by k modifying constants, and standard commutator estimate (e.g. emma. [9]). We may now continue this estimate as follows: II k j q k( ) S k u k j xφ k j q k u k(+( q )) k j k j q w k( q ) j q k j q k( q ) w (4) by Bernstein s inequalities, emma., (8) and Proposition 3.3. Similarly, II k j q S l u S k u k l j l: k l k j q S k u S k u k j k j q k u k(+( q )) k j k j q k u k( q ) j q k j q k( q ) w by (), Hölder s inequalities, (9), Bernstein s inequalities using that k 3, (8) and Proposition 3.3. Next, II 4 k j q l u S l j l: k l k j q k u S k j k j q k k u k j q w k( q ) m:m k m:m k m j (m k)(+ q ) m j k j q w k( q ) k (+ q ) l j B q, k j q w k( q ) j B q,q k j q w k( q ) j q k j q k( q ) w by (), Hölder s inequalities, Bernstein s inequality with the fact that k 3, (9), Young s inequality for convolution, (8), and Proposition 3.3. Finally, in particular (5) (6)

12 KAZUO YAMAZAKI making use of (c) we estimate II 5 k j q k j q k j q k( q ) k j q k( q ) k j q k( q ) k j q k( q ) k (div( l u l j)) k l l u l( q ) l j (k l)( q ) l w l j (k l)( q ) l j w B q, (k l)( q ) l j w B q,q (k l)( q ) l j w k j q k( q ) j w q k j q k( q ) w (7) by (), Hölder s inequalities, Bernstein s inequalities using k 3, (8), Young s inequalities for convolution, and Proposition 3.3. Therefore, considering (3), (4), (5), (6), (7) in (), we have II q Next, we rewrite II from () as k j q k( q ) w. (8) II = = k j q k j k ((b )w) k j q k j k ( l: k l + S l b l w + l:l k,l : l l l: k l l b S l w l b l w) 3 i= II i (9) by Bony s paraproduct decomposition. Below we will estimate (9) in a more straight-forward manner than the computations in [5] (cf.,, 3, 4, 5 in the proof of Proposition 4. [5]). Firstly, we estimate II k j q l: k l k (S l b l w) k j q S k b k w k j q b k k w k j q b α +α Λ +α b +α k w k j q k w (3)

13 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 3 by (9), Hölder s inequality, Bernstein s inequality and Gagliardo-Nirenberg inequalities, (8), () and Proposition 3.. Next, II k j q k j q l: k l l: k l k div( l bs l w) k l b S l w k j q k( q ) k b w k j q k( q ) j w q k j q k( q ) w by (9), Hölder s inequality, (c), Bernstein s inequalities with the fact that k 3, (8) and Proposition 3.3. Finally, II 3 k j q k j q k j q k j q k j q k( q ) k div( l b l w) k l b l w k l b w B q, k l b w B q,q (k l)( q ) l j w k j q k( q ) j w q k j q k( q ) w by (9), Hölder s inequalities, (c), Bernstein s inequalities with the fact that k 3, (8), Young s inequalities for convolution, and Proposition 3.3. Thus, considering (3), (3), (3) in (9) gives II q (3) (3) k j q ( k( q ) + k ) w. (33) Next, we work on II 3 from (), noting that identical estimates will work for II 4, II 5, II 6 : we first rewrite II 3 = k j q k j k ( b u ) = k j q k j k ( S l b l u + + l: k l l: k l l:l k,l : l l l b S l u l b l u ) 3 i= II 3i (34)

14 4 KAZUO YAMAZAKI by () and (9). Firstly, II 3 k j q l: k l k (S l b l u ) k j q S k j k w k j q j k( q ) k w q k j q k( q ) w by (34), Hölder s inequality, Bernstein s inequality, and Proposition 3.3. Next, II 3 k j q l b S l u l: k l k j q k b w k j q k( q ) k j w k j q k( q ) j w q k j q k( q ) w by Hölder s inequality, Bernstein s inequality, (8) and Proposition 3.3. Finally, II 33 k j q l b l u k j q k j q k j q l( q ) l b l( q ( α) q ) l w ( α) l q ( α) l j w ( α) l q ( α) l j by (34), Hölder s inequalities, Bernstein s inequality, (8) and Proposition 3.; we also used the crucial hypothesis that q ( α) >. Therefore, we obtain from (35), (36), (37) applied to (34), II 3 q k j q k( q ) w + Considering (), (8), (33), (38) applied to () gives t k j q q q + c k k j q q k j q ( k( q ) + k ) w + Making use of the fact that q t k j q t ( k j e ck t ) q e ck t (35) (36) (37) l q ( α) l j. (38) l q ( α) l j. = k j q t k j, this leads to l q ( α) l j ( k( q ) + k ) w +.

15 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 5 Now integrating in time, we obtain k j(t) q e ckt k j + t e ck (t τ) ( k( q ) + k ) w(τ) + l q ( α) l j(τ) dτ where j b. Taking t -norm, using Minkowski s inequality, taking squares, and applying Minkowski s inequality for convolution, we obtain t k j dτ q e ckt t k j + e ckt t (k( q ) + k ) w(t) + l q ( α) l j(t). t (39) Now we may compute e ckt t k, e ckt t ( k ) 4k ; (4) that we take this Young s inequality for convolution specifically as in (39) is crucial; we will really need this 4k power (see e.g. (45)). Thus, applying (4) to (39) gives for any k 3, t k j dτ q k k j ( k( q ) + k ) w(t) q + 4k t + l q ( α) l j(t). t (4) We multiply (4) by kδ, bound in particular the case k =,,, by = T t k= t sup t [,T ] kδ k j dτ δ j + j + δ j + 4δ j dτ j(t) q T (4)

16 6 KAZUO YAMAZAKI for all t [, T ] by Proposition 3.3, and hence we may sum over k and estimate t j B dτ q δ q T +, k 3 t kδ k j dτ q T + k 3 k(δ ) k j + k(δ ) ( k( q ) + k ) w(t) q t k 3 + t k(δ ) l q ( α) l j(τ) dτ k 3 (43) ct + III + III + III 3 by (4), (4). Again our estimates are more straight-forward here in comparison to the computations in [5] (see M + M + M 3 + M 4 + M 5 + M 6 in the proof of Proposition 4. [5]). Firstly, we estimate III q k(δ ) k j q b B (44) q δ, k by (43). Secondly, we estimate t III q k(δ ) k 3 q k 3 ( k(δ + ( k( q ) + k ) w(τ) dτ q ) + k(δ ) ) where we used (43), the elementary inequality of k 3 t (a + b) 4(a + b ) a, b, w(τ) dτ q and Proposition 3.. Finally, we estimate t III 3 q k(δ ) l( α q ( α) α) l Λ +α j dτ t q k(δ ) k 3 l( α q ( α) α) t q k(δ ) l Λ +α j dτ k 3 l t q Λ +α j dτ q l Λ +α j dτ by (43), Bernstein s inequalities as l k 3, Hölder s inequality, and Proposition 3.. Therefore, applying (44), (45), (46) in (43) leads to t (45) (46) j B dτ q δ q T + III + III + III 3 T + b, B + (47) q δ,

17 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 7 since b H s, s 3. This completes the proof of Proposition 3.4. We now improve the bound of w in Proposition 3.. Proposition 3.5. et α (, 3 ), β =. Suppose (u, b ) H s (R ), s 3. Then its corresponding solution pair (u, b) to the system (a)-(c) in [, T ] has the following bounds: for q (, ( + α) α ( α) ], sup w(t) q + t [,T ] T w q q α dτ q. Proof. As in the proof of Proposition 3., we multiply (a) with w q w, integrate in space to obtain t w q q q + c(q ) Λ α w q (u )w w q w + (b )j w q w (48) = div(bj) w q w div(bj) Ẇ α,q w q Ẇ α, q q by emma., (c) and Hölder s inequality. We remark here that it is crucial to make careful use of the dissipation strength here; in particular, if one uses Hardy- ittlewood-sobolev theorem ([]), and continuity of Riesz transform to deduce div(bj) Ẇ α,q div(bj) q +q α bj Ẇ, q +q α, (49) then we will not be able to estimate this well. The problem is that according to Proposition., we wish to eventually take q α ; however, lim q q + q α α = α where α > is arbitrary small, and the best bound we have on b in terms of regularity is Λ +α j from Proposition 3., which cannot bound j. Instead we estimate as follows: div(bj) Ẇ α,q Λ α (bj) b Λ α j + Λ α b j b α +α +α Λ α j Λ α j + b W,q j α +α Λ +α j Λ α j + ( b + j ) Λ +α j +α +α q +q α Λ α j + Λ +α j +α by continuity of Riesz transform, emma.4, Gagliardo-Nirenberg inequality, (), Proposition 3., Proposition 3.3, Proposition.9, emma.. On the other hand, we set F (x) x q, G(x) x q C and h(x) c ([, ]) so that by emma.3 we deduce (5) w q Ẇ q = F ( w q α, ) Ẇ α, q G( w q ) q q q w q Ḣα q w q Λ α w q. (5)

18 8 KAZUO YAMAZAKI Therefore, applying (5), (5) in (48) we have q t w q + c(q ) Λ α w q ( Λ α j + Λ +α j +α ) w q Λ α w q c(q ) Λα w q + c( + Λ α j + Λ +α j )( w q + ) (5) due to Young s inequality. By Proposition 3. we know T Λ +α j dτ ; thus, it suffices to show the time integrability of Λ α j. Since α > and q > α, we may find δ (max{ α, q ( α) }, ) and estimate by Bernstein s and Hölder s inequalities T T Λ α j dτ Λ α k j dτ k T k( α δ) kδ k j dτ k T k( α δ) kδ k j dτ T k j B δ q,dτ k due to Proposition 3.4 as α δ <. Gronwall s inequality applied on (5), along with (53), completes the proof of Proposition 3.5. (53) Proof of Theorem. Here we explain the iteration scheme. If α 3, as we explained in Remark 3., our proof was complete at Proposition 3. because Proposition. stated that only T w α dτ is needed to complete the proof of Theorem.; we assumed α (, 3 ) so that ( + α) ( α) < α which guaranteed that q in the hypothesis of Proposition 3.5 satisfies q < α. By Proposition 3.5, we now know that now consider only α > such that T w q q α ( + α) ( α) 3 < α, dτ for q = (+α) ( α). Thus, we

19 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM 9 go back to Proposition 3.3, Proposition 3.4, Proposition 3.5 to obtain higher q3 - ( + α) bound, q 3 [, ] and repeat to deduce in particular ( α) 3 T w q k q k α dτ, q k [, ( + α) ( α) k ], k Z+ such that ( + α) ( α) k < α. This implies that because α > has been fixed, for clarity we may denote by λ the first k such that so that consequently and therefore we have the bound of ( + α) ( + α) ( α) λ ( α) k α ( + α) ( α) λ ( + α) > α ( α) λ T w (+α) ( α) λ (+α) It is now straight-forward to deduce by interpolation T w dτ sup w(t) ɛ α t [,T ] ( α) λ dτ. (54) T w ( ɛ) (+α) ( α) λ dτ where ɛ [ ] / [ ] α ( α)λ ( α)λ + ( + α) ( + α) by Proposition.9 and (54); by Proposition., this completes the proof of Theorem.. 4. Appendix 4.. Proof of Proposition.9. Taking -inner products of (a)-(b) with (w, j) respectively and using the incompressibility of u and b in (c), we estimate in sum t( w + j ) + Λα w + Λj = [ b ( u + u ) u ( b + b )]j b 4 u j 4 j j w Λj + c j w where we used Hölder s, Gagliardo-Nirenberg and Young s inequalities and emma.. Subtracting Λj from both sides, applying () and Gronwall s inequality complete the proof of Proposition.9.

20 KAZUO YAMAZAKI 4.. Proof of Proposition.. We take -inner products of (a)-(b) with ( w, j) respectively, sum to estimate t( w + j ) + Λα w + j u w + b j w + u j j + b u j u α w α w + b 4 j 4 w + u j j + j 4 w 4 j where we used integration by parts, Hölder s inequalities. We continue to bound by t( w + j ) + Λα w + j w α Λ α w w + j j j w + u w j j + j j w w j ( Λα w + j ) + c( w + j + )( w α + j + ) due to emma., Sobolev embedding of Ḣα (R ) α (R ), Gagliardo-Nirenberg inequalities, Proposition.9, (), and Young s inequalities. By Proposition.9 and hypothesis, this implies sup t [,T ] ( w + j ) (t) + T Λ α w + j dτ. (55) This implies by Gagliardo Nirenberg inequalities that T sup t [,T ] w + j dτ w α +α T Λ α w +α dτ + sup j t [,T ] T j dτ, (56) by Proposition.9 and (55). This bound immediately leads to higher regularity. We denote for convenience X(t) Λ s u(t) + Λs b(t), apply Λs to (a)-(b), take -inner products with (Λ s u, Λ s b) respectively, use (c) so that u Λ s u Λ s u =, u Λ s b Λ s b =, b Λ s b Λ s u + b Λ s u Λ s b =,

21 TWO DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM to estimate tx(t) + Λ s+α u + Λs+ b = [Λ s ((u )u) u Λ s u] Λ s u + [Λ s ((b )b) b Λ s b] Λ s u [Λ s ((u )b) u Λ s b] Λ s b + [Λ s ((b )u) b Λ s u] Λ s b ( u + b )X(t) ( + w + j ) log ( + X(t))( + X(t)) by emma.5, emma.6 and (). Dividing by + X(t), Gronwall s inequality and (56) completes the proof of Proposition Proof of Proposition 3.. Proof. We take -inner products of (b) with Λ α j to obtain t Λ α j + Λ+α j = ( div(uj) + div(bw)) Λ α j + [ b ( u + u ) u ( b + b )]Λ α j (57) ( Λ α div(uj) + Λ α div(bw) ) Λ +α j + b ( u + u ) u ( b + b ) Λ α j I + I where we used (c), Hölder s inequalities. We estimate them separately: firstly, I ( Λ α div(uj) + Λ α div(bw) ) Λ +α j 8 Λ+α j + c ( Λ α (uj) + Λα (bw) ) 8 Λ+α j + c( Λα u j + u Λα j + Λ α b w + b α α Λα w ) 8 Λ+α j + c( u ( α) u α j ( +α ) Λ +α j ( + u ( α +α ) Λ +α u ( +α ) Λ α j α +α ) + j Λα w + α b ( +α ) Λ +α b ( +α ) Λ α w ) where we used (57), Young s inequalities, emma.4, the Gagliardo-Nirenberg inequalities, and Sobolev embeddings of Ḣ α (R ) α (R ), Ḣα (R ) α (R ). By (), Proposition.9, emma.4 and Young s inequalities we continue to bound

22 KAZUO YAMAZAKI by I 8 Λ+α j + c( Λ +α j +α + Λ α w +α Λ α j + Λ α w + Λα j +α Λ α w ) 4 Λ+α j + c( + Λα w )( + Λα j ). (58) Next, we work on I from (57): I b ( u + u ) u ( b + b ) Λ α j b α u α Λα j α Λ +α j α Λj α Λ α w Λ α j α Λ +α j α 4 Λ+α j + c ( Λj + Λα w ) ( + Λ α j ) (59) by Hölder s inequalities, Gagliardo-Nirenberg inequalities, Sobolev embedding of Ḣ α (R ) α (R ) and Young s inequalities. Considering (58) and (59) in (57), we obtain t Λ α j + Λ+α j Λ+α j + c ( + Λj + Λα w ) ( + Λ α j ). Subtracting Λ+α j from both sides, Gronwall s inequality with Proposition.9 completes the proof of Proposition Acknowledgment The author would like to thank Prof. Jiahong Wu for helpful discussion, and the anonymous referees for valuable comments that improved this manuscript significantly. References [] G. K. Batchelor, On the spontaneous magnetic field in a conducting liquid in turbulent motion, Proc. R. Soc. ond. Ser. A, (95), [] H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaftern, 343, Springer-Verlag Berlin Heidelberg,. [3] J. Bergh, J. öfström, Interpolation Spaces: An Introduction, Springer, Berlin, 976. [4] H. Brezis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (98), [5] C. Cao, J. Wu, B. Yuan, The D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (4), [6] S. Chandrasekhar, The invariant theory of isotropic turbulence in magneto-hydrodynamics, Proc. R. Soc. ond. Ser. A, 4 (95), [7] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press Inc., New York, 998. [8] Q. Chen, C. Miao, Z. Zhang, A new Bernstein s inequality and the D dissipative quasigeostrophic equation, Comm. Math. Phys., 7 (7), [9] F. M. Christ, M. I. Weinstein, Dispersion of small amplitude solutions of generalized Kortewegde Vries equation, J. Functional Analysis, (99), 87 9.

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24 4 KAZUO YAMAZAKI [38] V. Yudovich, Non stationary flows of an ideal incompressible fluid, Zhurnal Vych Matematika, 3 (963), Department of Mathematics, University of Rochester, 7 Hylan Hall, Rochester, NY 467; phone: (585) ; fax: (585) ; kyamazak@ur.rochester.edu