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1 J. Differential Equations 48 (1) 6 74 Contents lists available at ScienceDirect Journal of Differential Equations Two regularity criteria for the D MHD equations Chongsheng Cao a, Jiahong Wu b, a Department of Mathematics, Florida International University, Miami, FL 199, USA b Department of Mathematics, Oklahoma State University, Stillwater, OK 7478, USA article info abstract Article history: Received 15 August 9 Revised 14 September 9 Availableonline1October9 MSC: 5B45 5B65 76W5 This work establishes two regularity criteria for the D incompressible MHD equations. The first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction. 9 Elsevier Inc. All rights reserved. Keywords: D MHD equations Regularity criteria 1. Introduction This paper is concerned with the global regularity of solutions to the D incompressible magnetohydrodynamical (MHD) equations u t + u u = ν u p + b b, x R, t >, (1.1) b t + u b = η b + b u, x R, t >, (1.) u =, x R, t >, (1.) b =, x R, t >, (1.4) * Corresponding author. addresses: caoc@fiu.edu (C. Cao), jiahong@math.okstate.edu (J. Wu). -96/$ see front matter 9 Elsevier Inc. All rights reserved. doi:1.116/j.jde.9.9.

2 64 C. Cao, J. Wu / J. Differential Equations 48 (1) 6 74 where u is the fluid velocity, b the magnetic field, p the pressure, ν the viscosity and η the magnetic diffusivity. Without loss of generality, we set ν = η = 1 in the rest of the paper. The MHD equations govern the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas. (1.1) reflects the conservation of momentum, (1.) is the induction equation and (1.) specifies the conservation of mass. Besides their physical applications, the MHD equations are also mathematically significant. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been obtained (see, e.g., [,7,11, 1,16 19,,6 8,4 6,8 4,4,45,47 5,55]). Attention here is focused on the global regularity of solutions to the initial-value problem (IVP) of (1.1), (1.), (1.) and (1.4) with a given initial data u(x, ) = u (x), b(x, ) = b (x), x R. (1.5) It is currently unknown whether solutions of this IVP can develop finite time singularities even if (u, b ) is sufficiently smooth. This work presents new regularity criteria under which the regularity of the solution is preserved for all time. The global regularity issue has been thoroughly investigated for the D Navier Stokes equations and many important regularity criteria have been established (see, e.g., [ 6,8 1,1,14,15,,1,5,9,7,41,4,44,46,5,54]). Some of these criteria can be extended to the D MHD equations by making assumptions on both u and b (see, e.g., [7,47]). Realizing the dominant role played by the velocity field in the regularity issue, He and Xin were able to derive criteria in terms of the velocity field u alone [7,8]. They showed that, if u satisfies T u(, t) β α dt < with α + = and1<β, (1.6) β then the solution (u, b) is regular on [, T ]. This assumption was weakened in [51] with L α -norm replaced by norms in Besov spaces and further improved by Chen, Miao and Zhang in [17]. As pointed out in [7], the regularity criteria in terms of the velocity field alone are consistent with the numerical simulations in [4] and with the observations of space and laboratory plasmas in [4]. This paper presents two new regularity criteria. The first one assumes and the second requires the pressure satisfy T uz (, t) β α dt < with α and α + 1 (1.7) β T pz (τ ) β 1 α dτ < with α 7 and α + β 7 4. (1.8) That is, any solution (u, b) of the D MHD equations is regular if the derivative of u in one direction, say along the z-axis, is bounded in L β ([, T ]; L α ) with (α,β) satisfying (1.7) or if the derivative of p in one direction satisfies (1.8). The proof of the first criterion is accomplished through two stages with the first controlling the time integrals of u z and b z in terms of the L β ([, T ]; L α )-norm of u z and the second bounding u and b by the time integrals of u z and b z.the details are presented in the second section. The criterion in terms of p z anditsproofareprovidedin the third section. We will use the following elementary inequalities: φ γ C φ x 1 φ y 1 φ z 1 μ, (1.9)

3 C. Cao, J. Wu / J. Differential Equations 48 (1) where the parameters μ, and γ satisfy 1 μ,<, 1 μ + > 1 and 1+ γ = 1 μ + and φ r C(r) φ 6 r r φ x r r φ y r r φ z r r, r 6. (1.1) These inequalities may be found in [1,,]. For the convenience of the readers, the proofs of these inequalities are provided in Appendix A. Throughout the rest of this paper the L p -normofafunction f is denoted by f p,theh s -norm by f H s and the norm in the Sobolev space W s,p by f W s,p.. Criterion in terms of u z This section establishes the regularity criteria in terms of u z. Theorem.1. Assume (u, b ) H, u = and b =.Let(u, b) be the corresponding solution of the D MHD equations (1.1), (1.), (1.) and (1.4). If u satisfies M(T ) T uz (, t) β α dt < with α and α + 1 (.1) β for some T >,then(u, b) can be extended to the time interval [, T + ɛ) for some ɛ >. The proof of this theorem is divided into two major parts. The first part establishes bounds for u z, b z and the time integrals of u z and b z while the second controls u and b in terms of the time integrals of u z and b z..1. Bounds for u z and b z This subsection bounds u z and b z in terms of M in (.1). Proposition.. Assume (u, b ) H, u = and b =.Let(u, b) be the corresponding solution of the D MHD equations (1.1), (1.), (1.) and (1.4). Suppose (.1) holds. Then, for any t T, uz (t) + bz (t) Ce ( u + b ) e M(t)[( uz () + bz () ) α + C ( u + b + M(t))] α (.) and t ( u z (τ ) + b z (τ ) ) ( ) dτ F M(t) <, (.) where F (M(t)) is an explicit function of M(t).

4 66 C. Cao, J. Wu / J. Differential Equations 48 (1) 6 74 Proof. It is easy to see that (u, b) satisfies u(t) + b(t) + t ( u(τ ) + b(τ ) ) dτ u + b. (.4) Adding the inner products of u z with z of (1.1) and of b z with z of (1.), we obtain, after integration by parts, 1 d( u z + b z ) + u z dt + b z [(uz ] = u) u z (b z b) u z + (u z b) b z (b z u) b z dxdy dz I 1 + I + I + I 4. To bound I 1, we integrate by parts and apply Hölder s inequality to obtain I 1 = (u z u z ) u C u z u z r u α, where we have omitted dxdydz in the integral for notational convenience and r satisfies r 6, 1 r + 1 α = 1. (.5) Applying the Sobolev inequality and bounding u α by (1.9), we find u z r C u z 1 ( 1 1 r ) u z ( 1 1 r ) By Young s inequality, I 1 C u z 1+( 1 1 r ) u z 1 ( 1 1 r ) u z 1 α u. I u z + C u z u z q α u q with q = 9( 1 1 r ) = (1 α 1 ). (.6) When α, we have q and another application of Young s inequality implies I u z + C u z ( uz γ α + u ),

5 C. Cao, J. Wu / J. Differential Equations 48 (1) where γ q 1 q = 1 α or α + γ = 1. We now bound I. By Hölder s, Sobolev s and Young s inequalities, where I C b u z α b z α α C b u z α b z 1 α α b z 1 4 b z α α + C b u z α α 1 4 b z + C( b + u z γ α γ = α α α or α + γ = 1. b z α 6 α ) bz α 6 α, I can be bounded exactly as I.ToboundI 4, we integrate by parts and apply Hölder s inequality, I 4 = [(bz u) b z ] = [(bz b z ) u ] b z b z r u α, where 1 r + 1 α = 1. Following the steps as in the bound of I 1,wehave I b z + C( u z γ α + u ) bz. Combining the estimates for I 1, I, I and I 4,wefind d( u z + b z ) + u z dt + b z ( C u z γ ) ( α + u )( uz + b z + C b + u z γ ) α bz α 6 α. (.7) (.) and (.) then follow from (.4), (.7) and Gronwall s inequality... Bounds for u and b This subsection establishes bounds for u and b. Proposition.. Assume (u, b ) H, u = and b =.Let(u, b) be the corresponding solution of the D MHD equations (1.1), (1.), (1.) and (1.4). Suppose (.1) holds. Then, for any t T, u(t) + b(t) t + ( u(τ ) + b(τ ) ) ( ) dτ G M(t) <, where G(M(t)) denotes an explicit function of M(t).

6 68 C. Cao, J. Wu / J. Differential Equations 48 (1) 6 74 Proof. Adding the inner products of (1.1) with u and of (1.) with b and integrating by parts, we have 1 d dt ( ) u + b + u + b = u u u + b b u u b b + b u b. By further integrating by parts, we obtain u u u + b b u u b b + b u b u + u b. By (1.1), where h ( x, y ).ByYoung sinequality, u C( u 1 h u 1 u z 1 ) 6, Similarly, u 1 4 h u + C u u z 1 4 h u + C( u + u z ) u. Therefore, u b 1 4 h u + 1 h b + C( u + u z + b z ) b. d ( ) u dt + b + u + b C( )( u + u z + b z u ) + b. Gronwall s inequality coupled with Proposition. then yields the desired bounds.. Criterion in terms of p z This section presents the regularity criterion with an assumption on p z. Theorem.1. Assume the initial data (u, b ) H 1 L 4, u = and b =. Let(u, b) be the corresponding solution of the D MHD equations (1.1), (1.), (1.) and (1.4). If the pressure p associated with the solution satisfies T pz (τ ) β 1 α dτ < with α 7 and α + β 7 4 (.1) for some T >,then(u, b) remains regular on [, T ], namely (u, b) C([, T ]; H 1 L 4 ). Since higher-order Sobolev norms of (u, b) can be controlled by its H 1 -norm (see e.g. [45]), a special consequence of this theorem is that (.1) yields the global regularity of classical solutions. To prove this theorem, we establish the L 4 -bound of (u, b) and the desired regularity then follows from the standard Serrin type criteria on the D MHD equations [48].

7 C. Cao, J. Wu / J. Differential Equations 48 (1) Proposition.. Assume the initial data (u, b ) H 1 L 4, u = and b =. Let(u, b) be the corresponding solution of the D MHD equations (1.1), (1.), (1.) and (1.4). If the pressure p satisfies (.1), then (u, b) obeys the bound t w w ( w + + w ) dτ + 4 t ( w + w + + w w ) dxdy dzdτ < for any t T,where w ± = u ± b. Proof. We first convert the MHD equations into a symmetric form. Adding and subtracting (1.1) and (1.), we find that w + and w satisfy t w + + w w + = w + p, (.) t w + w + w = w p, (.) w + =, w =. (.4) Adding the inner products of (.) with w + w + and of (.) with w w and integrating by parts, we find 1 d ( w dt + w 4) 4 1 ( + w + + w ) ( w + + w + + w w ) = J 1 + J, (.5) where J 1 = pw + w +, J = pw w. By Hölder s inequality, J 1 C p 4 w + 4 w +. We choose such that 1 α + = 7 4 or (1 α 1 ) = 4. It then follows from (1.9) that p 4 C p z 1 α p.

8 7 C. Cao, J. Wu / J. Differential Equations 48 (1) 6 74 To further bound p, we take the divergence of (.) to obtain By Hölder s inequality, Furthermore, by Sobolev s inequality, w = w 1 p = (w w + ). p C w w +. C w 7 4 w 9 4 = C w w 9 4, where we have used the fact that 1 7 α and thus 1 7. Therefore, p 4 C p z 1 α w + w w and thus J 1 C p z 1 α w + w w w + 4 w +. By Young s inequality, J 1 1 w w 8 + C p z α Further applications of Young s inequality imply Since (1 7) ( ) (4 ) 8 4(1 7) (4 ) w + (4 ) w 4 w + 4 (4 ) p z α w + (4 ) 1 7 p z 4 α + w +, w 4(1 7) (4 ) 4 w w + 4 (1 7) 4 + w ( ) 4. < 4, we obtain without loss of generality that 4 4. J 1 1 w w 8 + C( 1 7 p z 4 α + w + )( w w 4) 4. (.6) Similarly, we have J = pw w 1 w w 8 + C( 1 7 p z 4 α + w )( w w 4) 4. (.7) Inserting (.6) and (.7) in (.5) and applying Gronwall s inequality, we obtain the desired result. Acknowledgments Cao is partially supported by the NSF grant DMS 798 and a FIU foundation. Wu is partially supported by the NSF grant DMS 9791 and the AT & T Foundation at OSU. We thank Professor B. Yuan for careful reading of this manuscript and for discussions.

9 C. Cao, J. Wu / J. Differential Equations 48 (1) Appendix A This appendix provides the proofs of the inequalities (1.9) and (1.1). For the convenience of future references, we write these inequalities as lemmas. Lemma A.1. Let μ, and γ be three parameters that satisfy 1 μ,<, 1 μ + > 1 and 1 + γ = 1 μ +. Assume φ H 1 (R ), φ x,φ y L (R ) and φ z L μ (R ). Then, there exists a constant C = C(μ,)such that φ γ C φ x 1 φ y 1 φ z 1 μ. (A.1) Especially,when =, there exists a constant C = C(μ) such that which holds for any φ H 1 (R ) and φ z L μ (R ) with 1 μ <. φ μ C φ x 1 φ y 1 φ z 1 μ, (A.) Proof. Clearly, φ(x, y, z) x 1+(1 1 )γ C φ(t, y, z) (1 1 )γ t φ(t, y, z) dt, φ(x, y, z) 1+(1 1 )γ C y φ(x, t, z) (1 1 )γ t φ(x, t, z) dt, φ(x, y, z) z 1+(1 μ 1 )γ C φ(x, y, t) (1 μ 1 )γ t φ(x, y, t) dt. (A.) (A.4) (A.5) Therefore, φ(x, y, z) γ C [ φ(x, y, z) (1 1 )γ x φ(x, y, z) ] 1 dx [ φ(x, y, z) (1 1 )γ y φ(x, y, z) ] 1 dy [ φ(x, y, z) (1 μ 1 )γ z φ(x, y, z) ] 1 dz. Integrating with respect to x and applying Hölder s inequality, we have

10 7 C. Cao, J. Wu / J. Differential Equations 48 (1) 6 74 φ(x, y, z) γ dx [ φ(x, y, z) (1 1 )γ x φ(x, y, z) ] 1 dx [ [ φ(x, y, z) (1 1 )γ y φ(x, y, z) ] 1 dxdy φ(x, y, z) (1 μ 1 )γ z φ(x, y, z) ] 1 dxdz. Further integration with respect to y and z yields φ(x, y, z) [ γ dxdy dz φ(x, y, z) (1 1 )γ x φ(x, y, z) ] 1 dxdy dz R R [ φ(x, y, z) (1 1 )γ y φ(x, y, z) ] 1 dxdy dz R [ R φ(x, y, z) (1 μ 1 )γ z φ(x, y, z) ] 1 dxdy dz. If Hölder s inequality is applied again, we have φ γ γ C φ (1 1 ) γ γ x φ 1 φ (1 1 ) γ γ y φ 1 φ (1 1 μ ) γ γ z φ 1 μ, which leads to (A.1). Lemma A.. Let q 6 and assume φ H 1 (R ). Then, there exists a constant C = C(q) such that 6 q q φ q C φ q q q q q q x φ y φ z φ. (A.6) 6 q q Proof. This inequality can be obtained by interpolating the trivial inequality φ q φ φ and (A.) with μ =, namely q 6 References φ 6 C φ x 1 φ y 1 φ z 1. [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, [] R. Agapito, M. Schonbek, Non-uniform decay of MHD equations with and without magnetic diffusion, Comm. Partial Differential Equations (7) [] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the -D Euler equations, Comm. Math. Phys. 94 (1984) [4] H. Beirão da Veiga, A new regularity class for the Navier Stokes equations in R n, Chinese Ann. Math. 16 (1995) [5] H. Beirão da Veiga, On the smoothness of a class of weak solutions to the Navier Stokes equations, J. Math. Fluid Mech. () 15.

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