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1 SIAM J. MATH. ANAL. Vol. 46, No. 1, pp c 214 Society for Industrial and Applied Mathematics THE 2D INCOMPRESSIBLE MAGNETOHYDRODYNAMICS EQUATIONS WITH ONLY MAGNETIC DIFFUSION CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN Abstract. This paper examines the global (in time) regularity of classical solutions to the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator ( Δ) β. We establish the global regularity for the case when β>1. This result significantly improves previous work which requires β> 3 and brings us closer to the resolution of the well-known global regularity 2 problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely, the case when β =1. Key words. MHD equations, partial dissipation, classical solutions, global regularity AMS subject classifications. 35Q35, 35B35, 35B65, 76D3 DOI / Introduction. This paper focuses on the initial-value problem for the twodimensional (2D) incompressible magnetohydrodynamics (MHD) equations with fractional Laplacian magnetic diffusion, t u + u u = p + b b, x R 2,t>, t b + u b +( Δ) β b = b u, x R 2,t>, (1.1) u =, b =, x R 2,t>, u(x, ) = u (x), b(x, ) = b (x), x R 2, where the fractional Laplacian operator ( Δ) β is defined through the Fourier transform ( Δ) β f(ξ) = ξ 2β f(ξ) with f being the Fourier transform of f, namely, f(ξ) = e ix ξ f(x) dx. R 2 When β = 1, (1.1) reduces to the MHD equations with Laplacian magnetic diffusion, which models many significant phenomena such as the magnetic reconnection in astrophysics and geomagnetic dynamo in geophysics (see, e.g., [2]). What we care about here is the global regularity problem, namely, whether the solutions of (1.1) emanating from sufficiently smooth data (u,b ) remain regular for Received by the editors September 26, 213; accepted for publication (in revised form) November 25, 213; published electronically February 6, Department of Mathematics, Florida International University, Miami, FL (caoc@fiu.edu). This author was partially supported by NSF grant DMS Department of Mathematics, Oklahoma State University, Stillwater, OK 7478, and Department of Mathematics, Chung-Ang University, Seoul , Republic of Korea (jiahong@ math.okstate.edu). This author was partially supported by NSF grant DMS and Henan Polytechnic University. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo City, Henan 454, People s Republic of China (bqyuan@hpu.edu.cn). This author was partially supported by the National Natural Science Foundation of China (117157) and by Innovation Scientists and Technicians Troop Construction Projects of Henan Province ( ). 588

2 MHD EQUATIONS WITH MAGNETIC DIFFUSION 589 all time. The main result of this paper states that (1.1) with any β > 1 always possesses a unique global solution. More precisely, we have the following theorem. Theorem 1.1. Consider (1.1) with β>1. Assume that (u,b ) H s (R 2 ) with s>2, u =, b =,andj = b satisfying j L <. Then (1.1) has a unique global solution (u, b) satisfying, for any T>, (u, b) L ([,T]; H s (R 2 )), j L 1 ([,T]; L (R 2 )), where j = b. We now review some recent work to put our result in a proper content. Due to their prominent roles in modeling many phenomena in astrophysics, geophysics, and plasma physics, the MHD equations have been studied extensively mathematically. Duvaut and Lions constructed a global Leray Hopf weak solution and a local strong solution of the three-dimensional incompressible MHD equations [9]. Sermange and Temam further examined the properties of these solutions [22]. More recent work on the MHD equations develops regularity criteria in terms of the velocity field and deals with the MHD equations with dissipation and magnetic diffusion given by general Fourier multiplier operators such as the fractional Laplacian operators (see, e.g., [6, 7, 1, 11, 13, 14, 23, 25, 26, 27, 28, 3, 31, 32]). Another direction that has generated considerable interest recently is the global regularity problem on the MHD equations with partial dissipation, especially in the 2D case (see, e.g., [3, 4, 12, 15, 16, 17, 29, 33]). Since the global regularity of the 2D MHD equations with both Laplacian dissipation and magnetic diffusion is easy to obtain while the regularity of the completely inviscid MHD equations appears to be out of reach, it is natural to examine the MHD equations with partial dissipation. One particular partial dissipation case is (1.1). When β 1, any solution of (1.1) with (u,b ) H 1 generates a global weak solution (u, b) that remains bound in H 1 for all time (see, e.g., [3, 15]). However, it is not clear if such weak solutions are unique or their regularity can be improved to be in H 2 if the initial data (u,b )is in H 2. The result of this paper obtains the uniqueness and global regularity for the case when β>1. Previous global regularity results require that β> 3 2 (see, e.g., [13, 23, 3, 31]). Our approach here does not appear to be able to extend to the borderline case when β = 1, which is currently being studied by a different method [19]. Progress has also been made on several other partial dissipation cases of the MHD equations. Lin, Xu, and Zhang recently studied the MHD equations with the Laplacian dissipation in the velocity equation but without magnetic diffusion and were able to establish the global existence of small solutions after translating the magnetic field by a constant vector [16, 17, 29]. Cao and Wu examined the anisotrophic 2D MHD equations with horizontal dissipation and vertical magnetic diffusion (or vertical dissipation and horizontal magnetic diffusion) and obtained the global regularity for this partial dissipation case [3]. The anisotrophic MHD equations with horizontal dissipation and horizontal magnetic diffusion were also investigated very recently and progress was also made (see [4]). We now explain our proof of Theorem 1.1. Since the local (in time) well-posedness can be established following a standard approach, our efforts are devoted to proving global a priori bounds for (u, b) in the initial functional setting H s with s>2. (u, b) indeed admits a global H 1 -bound, but direct energy estimates do not appear to easily yield a global H 2 -bound. The difficulty comes from the nonlinear term in the velocity

3 59 CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN equation due to the lack of dissipation. To bypass this difficulty, we first make use of the magnetic diffusion ( Δ) β b with β>1toshow the global bound for ω and j for 2 q 2 2 β. (The range is modified to 2 q< when β = 2.) The magnetic diffusion is further exploited to obtain a global bound for j in the space-time Besov space L 1 t Bq,1, s namely,foranyt>andt T (1.2) j L 1 t Bq,1 s C(T,u,b ) < with 2 q 2 2 β, 2 <s<2β 1. q Roughly speaking, this global bound provides the time integrability of the -norm of j up to (2β 1)-derivative. This global bound is proved through Besov space techniques. Special consequences of this global bound include the time integrability of j L and of j L r for r > q. To gain higher regularity, we go through an iterative process. The bound j L 1 t Lr < allows us to further prove a global bound for ω L r and j L r with r>q, which can be employed to prove (1.2) with q replaced by r. Repeating this process leads to the global bound in (1.2) for any q [2, ), which especially implies, for any t>, (1.3) t j L dτ <. Equation (1.3) can then be further used to obtain a global bound for ω L. The time integrability of j L and the boundedness of ω L t,x are enough to prove a global bound for (u, b) inh s. The rest of this paper is divided into three sections. The second section provides the definition of inhomogeneous Besov spaces and related facts such as Bernstein s inequality. The third section proves the global -bound for (ω, j), while the fourth section establishes the global bound in (1.2). The last section gains higher regularity through an iterative process and proves Theorem Functional spaces. This section provides the definition of Besov spaces and related facts used in the subsequent sections. Materials presented here can be found in several books and many papers (see, e.g., [1, 2, 18, 21, 24]). We start with notation. S denotes the usual Schwarz class and S its dual, the space of tempered distributions. It is a simple fact in analysis that there exist two radially symmetric functions Ψ, Φ S such that ( supp Ψ B, 4 3 Ψ(ξ)+ ), supp Φ A Φ j (ξ) =1, ξ R d, j= (, 3 4, 8 ), 3 where B(,r) denotes the ball centered at the origin with radius r, A(,r 1,r 2 ) denotes the annulus centered at the origin with the inner radius r 1 and the outer r 2,Φ =Φ, and Φ j (x) =2 jd Φ (2 j x)or Φ j (ξ) = Φ (2 j ξ). Then, for any ψ S, Ψ ψ + Φ j ψ = ψ and hence (2.1) Ψ f + j= Φ j f = f j=

4 MHD EQUATIONS WITH MAGNETIC DIFFUSION 591 in S for any f S. To define the inhomogeneous Besov space, we set if j 2, (2.2) Δ j f = Ψ f if j = 1, Φ j f if j =, 1, 2,... Definition 2.1. TheinhomogeneousBesovspaceBp,q s with 1 p, q,and s R consists of f S satisfying f B s p,q 2 js Δ j f L p l q <. Besides the Fourier localization operators Δ j, the partial sum S j is also a useful notation. For an integer j, S j j 1 where Δ k is given by (2.2). For any f S, the Fourier transform of S j f is supported on the ball of radius 2 j. Bernstein s inequalities are useful tools in dealing with Fourier localized functions and these inequalities trade integrability for derivatives. The following proposition provides Bernstein-type inequalities for fractional derivatives. Proposition 2.2. Let α. Let1 p q. (1) If f satisfies Δ k, supp f {ξ R d : ξ K2 j } for some integer j and a constant K>, then (2) If f satisfies ( Δ) α f Lq (R d ) C 1 2 2αj+jd( 1 p 1 q ) f Lp (R d ). supp f {ξ R d : K 1 2 j ξ K 2 2 j } for some integer j and constants <K 1 K 2,then C 1 2 2αj f Lq (R d ) ( Δ) α f Lq (R d ) C 2 2 2αj+jd( 1 p 1 q ) f Lp (R d ), where C 1 and C 2 are constants depending on α, p, andq only. 3. Global -bound for (ω, j). This section proves a global a priori bound for ω and j in the Lebesgue space (R 2 )forq in a suitable range. More precisely, we prove the following proposition. Proposition 3.1. Assume (u,b ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β>1. Then, for any q satisfying (3.1) 2 q 2 2 β (the range of q is modified to 2 q< when β =2), and for any T>and t T, there exists a constant C = C(T,u,b ) such that (3.2) ω(t) C, j(t) C. To prove Proposition 3.1, we first provide two simple bounds. A simple energy estimate yields the global L 2 -bound of (1.1) with β.

5 592 CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN Lemma 3.2. Assume (u,b ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β. Then, for any t, t u(t) 2 L + 2 b(t) 2 L +2 Λ β b(τ) 2 2 L dτ = u 2 2 L + b 2 2 L 2, where Λ= Δ. In addition, by resorting to the equations of ω and j, (3.3) t ω + u ω = b j, t j + u j +( Δ) β j = b ω +2 1 b 1 ( 2 u u 2 ) 2 1 u 1 ( 2 b b 2 ), ω(x, ) = ω (x), ω(x, ) = j (x), the global H 1 -bound on (u, b) can be obtained in a similar fashion as in [3]. Lemma 3.3. Assume (u,b ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β 1. Then, for any T > and t T, there exists a constant C = C(T, (u,b ) H 1) such that (3.4) ω(t) 2 L 2 + j(t) 2 L 2 + t Λ β j(τ) 2 L2 dτ C. ProofofProposition3.1. Multiplying the first equation in (3.3) by ω ω q 2,integrating in space and applying Hölder s inequality, we have 1 d q dt ω q L = b jω ω q 2 ω q 1 q L b q L j. Recall the simple Sobolev inequalities, for β>1, (3.5) b L C b β L Λ β j 1 2 and note that (3.1) ensures that 2(q 1) βq Therefore, 1+β L, j 2 C j 2(q 1) 1 βq L 2 1. By Young s inequality, b L j C ( b 2 L 2 + j 2 L 2 + Λβ j 2 L 2). ω(t) ω + C t ( b 2 L + 2 j 2 L + 2 Λβ j 2 L2) dt Λ β 2(q 1) βq j L, 2 and the bounds in Lemmas 3.2 and 3.3 yield the global bound for ω(t).togeta global bound for j, we first obtain from the equation of j in (3.3) 1 d (3.6) q dt j q L + j j q 2 ( Δ) β j = K q 1 + K 2 + K 3 + K 4 + K 5, where K 1,...,K 5 are given by K 1 = b ωj j q 2, K 2 =2 1 b 1 2 u 1 j j q 2,

6 MHD EQUATIONS WITH MAGNETIC DIFFUSION 593 K 3 =2 1 b 1 1 u 2 j j q 2, K 4 =2 1 u 1 2 b 1 j j q 2, K 5 =2 1 u 1 1 b 2 j j q 2. According to [8], we have the lower bound, for C = C (β,q), (3.7) j j q 2 ( Δ) β j C Λ β ( j q 2 ) 2. By integration by parts and Hölder s inequality, 2(q 1) K 1 = q ωb ( j q q 2 ) j 2 1 C b L ω j q 2 1 L ( j q q 2 ) L 2. By the trivial embedding inequality, for β 1, ( j q 2 ) L 2 j q 2 H β j q 2 L 2 + Λ β ( j q 2 ) L 2 = j q 2 + Λ β ( j q 2 ) L 2, we obtain K 1 C 2 C 2 Λ β ( j q 2 ) 2 + C b L ω j q 1 + C b 2 L ω 2 j q 2 Λ β ( j q 2 ) 2 + C (1 + b L ) b L ( ω q + j q ). K 2 can be easily bounded. In fact, by the Sobolev inequality with β>1, we have f L (R 2 ) C f 1 1 β L 2 (R 2 ) Λβ f 1 β L 2 (R 2 ), (3.8) K b 1 L ω j q 1 C 1 b β L 2 C j 1 1 β L 2 Λ β 1 b 1 1 β L 2 ( ω q + j q ) Λ β j 1 β L 2 ( ω q + j q ), where the simple inequality 1 b 1 L 2 b L 2 = b L 2 = j L 2. Clearly, K 3, K 4,andK 5 admit the same bound as in (3.8). Collecting the estimates, we have 1 d q dt j q L + C Λ β ( j q q 2 ) 2 2 C ( b L + b 2 L + j 1 1 β L 2 Λ β j 1 β L 2 )( ω q + j q ). Thanks to the time integrability of b 2 L (see (3.5) for a bound) and 1 j 1 β L Λ β j 1 β 2 L 2 on any finite time interval and the global bound for ω, this differential inequality yields the global bound for j. This completes the proof of Proposition 3.1.

7 594 CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN 4. Global L 1 t Bs q,1-bound for j. This section establishes a global bound for j in the space-time Besov space L 1 t Bs q,1,whereq satisfies (3.1) and 2 q <s<2β 1. For β>1, this global bound provides a better integrability than the one given in (3.4) and will be exploited to gain higher regularity in the next section. Proposition 4.1. Assume (u,b ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β>1. Then, for any q and s satisfying (4.1) 2 q 2 2 β, 2 q <s<2β 1 (the range of q is modified to 2 q< when β =2), and for any T> and t T, there exists a constant C = C(q, s, T, u,b ) such that (4.2) j L 1 t B s q,1 C. We remark that the proof of this theorem makes use of the global -bound for ω and j and this explains why we need to restrict q to the range in (4.1). It can be seen from the proof that this theorem remains valid for any q 2aslongas ω and j are bounded. We now prove Proposition 4.1. Proof of Proposition 4.1. Let k be an integer. Applying Δ k to the equation of j in (3.3), multiplying by, and integrating in space, we obtain 1 d (4.3) q dt Δ kj q L + Δ q k j Δ k j k 2 ( Δ) β Δ k j = L L 6, where L 1,...,L 6 are given by L 1 = Δ k (u j), L 2 = Δ k (b ω), L 3 =2 Δ k ( 1 b 1 2 u 1 ), L 4 =2 Δ k ( 1 b 1 1 u 2 ), L 5 = 2 Δ k ( 1 u 1 2 b 1 ), L 6 = 2 Δ k ( 1 u 1 1 b 2 ). The term associated the magnetic diffusion admits the lower bound (see [5]) ( Δ) β Δ k j C 1 2 2βk Δ k j q for C 1 = C 1 (β,q). By Bony s paraproducts decomposition, we write L 1 = L 11 + L 12 + L 13 + L 14 + L 15,

8 MHD EQUATIONS WITH MAGNETIC DIFFUSION 595 where L 11 = L 12 = L 13 = L 14 = L 15 = S k u Δ k j, [Δ k,s l 1 u ]Δ l j, (S l 1 u S k u) Δ k Δ l j, Δ k (Δ l u S l 1 j), Δ k (Δ l u Δ l j) with Δ l =Δ l 1 +Δ l +Δ l+1. Thanks to S k u =,wehavel 13 =. ByHölder s inequality, a standard commutator estimate, and Bernstein s inequality, (4.4) L 11 Δ k j q 1 [Δ k,s l 1 u ]Δ l j C Δ k j q 1 L S k 1u q Δ k j L C Δ k j q 1 ω 2k 2 q Δk j. Here we have also applied the simple fact that for fixed k, the summation is for a finite number of l satisfying k l 2 and the estimate for the term with the index l is only a constant multiple of the bound for the term with the index k. In addition, the simple bound S k 1 u u C ω is also used here. It is easily seen that L 12 obeys the same bound as in (4.4). By Hölder s inequality and Bernstein s inequality (both lower bound and upper bound parts), L 14 C Δ k j q 1 Δ ku S k 1 j L 2 (1+ 2 q )m Δ m j C Δ k j q 1 2 k Δ k u C ω Δ k j q 1 2 k m k 1 m k 1 2 (1+ 2 q )m Δ m j. By the divergence-free condition, Δ l u =, L 15 C Δ k j q 1 L 2 k Δ q l u Δ l j L C Δ k j q 1 C ω Δ k j q 1 2 2k q 2 k l Δ l u 2 l 2 q Δl j 2 (k l)(1 2 q ) Δ l j.

9 596 CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN We have thus completed the estimate for L 1. To bound L 2, we also decompose it into five terms, L 2 = L 21 + L 22 + L 23 + L 24 + L 25, where L 21 = L 22 = L 23 = L 24 = L 25 = S k b Δ k ω, [Δ k,s l 1 b ]Δ l ω, (S l 1 b S k b) Δ k Δ l ω, Δ k (Δ l b S l 1 ω), Δ k (Δ l b Δ l ω). The estimates of these terms are similar to those in L 1, but there are some differences, as can be seen from the bounds below. L 2 C2 k 2 q Δk j q 1 L b q ω + C 2k Δ k j q 1 L b q L ω + C 2 k 2 q Δk j q 1 2 (k l)(1 2 q ) Δ l b Δ l ω C 2 k 2 q Δk j q 1 L b q ω + C 2k Δ k j q 1 L b q L ω. To bound L 3, we decompose it into three terms as L 3 = L 31 + L 32 + L 33, where L 31 =2 L 32 =2 L 33 =2 Δ k (S l 1 1 b 1 Δ l 2 u 1 ), Δ k (Δ l 1 b 1 S l 1 2 u 1 ), Δ k (Δ l 1 b 1 Δl 2 u 1 ). These terms can be bounded by L 31 C Δ k j q 1 L S k 1 q 1 b 1 L Δ k 2 u 1 C Δ k j q 1 2 2k q j ω, where we have used the bound u C ω and b C j. Clearly, L 32 admits the same bound. L 33 is bounded by L 33 C Δ k j q 1 L 2 l 2 q q Δl 1 b 1 Δ l 2 u 1 C ω Δ k j q 1 2 l 2 q Δl j.

10 MHD EQUATIONS WITH MAGNETIC DIFFUSION 597 Collecting all the estimates above, we obtain d dt Δ kj + C 1 2 2βk Δ k j RHS(t), where RHS(t) denotes the bound (4.5) RHS(t) C ω 2 k 2 q Δk j + C ω 2 k 2 (1+ 2 q )m Δ m j + C2 k 2 q ω m k 1 2 (k l)(1 2 q ) Δ l j Integrating in time, we have + C2 k 2 q j ω + C 2 k b L ω + C ω 2 l 2 q Δl j. Δ k j(t) e C1 22βkt Δ k j() + t e C1 22βk (t τ ) RHS(τ) dτ. For any fixed t>, we take the L 1 -norm on [,t] and apply Young s inequality to obtain t t Δ k j(τ) dτ C 2 2βk Δ k j() + C 2 2βk RHS(τ) dτ. Multiplying by 2 ks and summing over k =, 1, 2,...,weobtain (4.6) t t j L 1 t Bq,1 s = Δ 1 j(τ) dτ + 2 ks Δ k j(τ) dτ C(t)+C j B s 2β q,1 + k= t 2 k(s 2β) RHS(τ) dτ, k= where we have applied the global bound Δ 1 j(τ) j. For clarity of presentation, we write t 2 k(s 2β) RHS(τ) dτ M 1 + M 2 + M 3 + M 4 + M 5 + M 6, k= where, according to (4.5), t M 1 = C ω L t 2 k(s+ 2 q 2β) Δ k j(τ) dτ, k= t M 2 = C ω L t 2 (m k)(1 s+2β) 2 (s+ 2 q 2β)m Δ m j dτ, k= m k 1 t M 3 = C ω L t 2 k(1+s 2β) 2 ( 1+ 2 q )l Δ l j(τ) dτ, k=

11 598 CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN M 4 = Ct ω L t j L 2 k(s+ 2 t Lq q 2β), k= M 5 = C ω L t b L 1 2 k(s+1 2β), t L k= M 6 = C ω L t 2 k(s 2β) k= 2 l 2 q t Δ l j(τ) dτ. Since 2 q 2β <, we can choose an integer k > such that C ω 2 L t 2k( q 2β) We can split the sum in M 1 into two parts, k t M 1 = C ω L t 2 k(s+ 2 q 2β) Δ k j(τ) dτ k= + C ω L t wherewehaveusedthebound k=k +1 = C(t, u,b ) j L 1 t Bs q,1, t Δ k j(τ) dτ t 2 k(s+ 2 q 2β) Δ k j(τ) dτ t j(τ) dτ C. To deal with M 2,wefirstrealizethat1 s +2β>and(m k)(1 s +2β) <, and we apply Young s inequality for series convolution to obtain t M 2 C ω L t 2 k(s+ 2 q 2β) Δ k j(τ) dτ, k= which obeys the same bound as M 1,namely, M 2 C(t, u,b ) j L 1. t Bs q,1 To bound M 3, we exchange the order of two sums to get t M 3 = C ω L t 2 ( 1+ 2 q )l Δ l j(τ) dτ Since 1 + s 2β <, we have l= 1 l+1 2 k(1+s 2β). k= and thus l+1 2 k(1+s 2β) C k= t M 3 = C ω L t 2 ( 1+ 2 q s)l 2 ls Δ l j(τ) dτ. l= 1

12 MHD EQUATIONS WITH MAGNETIC DIFFUSION 599 Noticing that 1+ 2 q s<, we take a positive integer l such that Then, M 3 is bounded by C ω L t 2( 1+ 2 q s)l < M 3 C(t, u,b ) j L 1. t Bs q,1 Since s +1 2β <, we clearly have M 4 + M 5 Ct ω L t j L t Lq + C ω L t Lq b L 1 t L. M 6 can be bounded in a similar fashion as M 3 and we have, for 2 q <s, M 6 C(t, u,b ) j L 1. t Bs q,1 Inserting the estimates above in (4.6), we have j L 1 t B s q,1 C(t, u,b )+C j B s 2β q, j L 1 t Bs q,1. This completes the proof of Proposition Higher regularity through an iterative process and proof of Theorem 1.1. This section explores some consequences of Proposition 4.1. In particular, we obtain global bounds for j in L 1 t L x and ω in L t,x, which are sufficient for the proof of Theorem 1.1. We now state the proposition for high regularity. Proposition 5.1. Assume (u,b ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β>1. Then, for any T>and t T, there exists a constant C = C(T,u,b ) such that t j(τ) L dτ C, ω(t) L C. The first step is the following integrability result, as a special consequence of Proposition 4.1. Proposition 5.2. Assume (u,b ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β>1. Then, for any r satisfying (5.1) 2 r if β> 4 3, and 2 r< 2 4 3β if β 4 3, and, for any T> and t T, there exists a constant C = C(T,u,b ) such that t j(τ) L r dτ C. We remark that the range for r, namely, (5.1), is bigger than the one for q in (3.1). An immediate consequence is the global bound for ω L r and j L r, as explained in the proof of Proposition 5.1.

13 6 CHONGSHENG CAO, JIAHONG WU, AND BAOQUAN YUAN Proof of Proposition 5.2. By Bernstein s inequality, j L r Δ k j L r 2 k(1+ 2 q 2 r s) 2 ks Δ k j. In the case when β> 4 3,wecanchooseq and s satisfying (4.1), say, q =3ands = 5 3, such that and consequently, for any r [2, ], j L r 1+ 2 q s, 2 ks Δ k j j B s q,1. In the case when β 4 3,wecanchooseq and s satisfying (4.1) and r satisfying (5.1) such that and again 1+ 2 q 2 r s j L r j B s q,1. Proposition 5.2 then follows from Proposition 4.1. We now prove Proposition 5.1. ProofofProposition5.1. Proposition 5.2 allows us to obtain a global bound for ω L r. In fact, it follows from the vorticity equation that 1 d r dt ω r L b r L j L r ω r 1 L. r Due to the Sobolev inequality, for q>2, b L C b q 2 2(q 1) L 2 j q 2q 2 and the time integrability of j L r from Proposition 5.2, we obtain the global bound ω(t) L r ω L r + b L t L x t j L r dτ <. By going through the proof of Proposition 3.1 with q replaced by r, we can show that j(t) L r C. As a consequence of the global bounds for ω(t) L r and j(t) L r,wecanproveproposition 4.1 again with q replaced by r. This iterative process allows us to establish the global bound j L 1 t B s r,1 <

14 MHD EQUATIONS WITH MAGNETIC DIFFUSION 61 for any r [2, ) and 2 r t>, (5.2) In fact, <s<2β 1. As a special consequence, we have, for any t j L dτ <. j L Δ k j L 2 k(1+ 2 r ) Δ k j L r = 2 k(1+ 2 r s) 2 ks Δ k j L r = wherewehavechooser large and s<2β 1 such that 1+ 2 r s. The global bound in (5.2) further allows us to show that 2 ks Δ k j L r j B s r,1, (5.3) ω L <, which follows from the inequality ω(t) L ω L + b L t t L j L dτ. This completes the proof of Proposition 5.1. Finally we prove Theorem 1.1. Proof of Theorem 1.1. The proof of Theorem 1.1 is divided into two main steps. The first step is to construct a local (in time) solution, while the second step extends the local solution into a global one by making use of the global a priori bounds obtained in Proposition 5.1. The construction of a local solution is quite standard and is thus omitted here. The global bounds in Proposition 5.1 are sufficient in proving the global bound This completes the proof of Theorem 1.1. (u, b) H s <. REFERENCES [1] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, New York, 211. [2] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer, New York, [3] C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (211), pp [4] C. Cao, D. Regmi, and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations, 254 (213), pp [5] Q. Chen, C. Miao, and Z. Zhang, A new Bernstein s inequality and the 2D dissipative quasigeostrophic equation, Comm. Math. Phys., 271 (27), pp [6] Q. Chen, C. Miao, and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magnetohydrodynamics equations, Comm. Math. Phys., 275 (27), pp

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