American Institute of Mathematical Sciences

Advanced
October  2005, 12(5): 881-886. doi: 10.3934/dcds.2005.12.881

Remarks on regularities for the 3D MHD equations

Yong Zhou 1,

1. 

Department of Mathematics, East China Normal University, Shanghai 200062, China

Received  January 2004 Revised  November 2004 Published  February 2005

In this paper we consider the regularity criteria for the solution to the 3D MHD equations. It is proved that if the gradient of the velocity field belongs to $L^{\alpha,\gamma}$ with $2/\alpha+3/\gamma \leq 2$ or the velocity field belongs to $L^{\alpha,\gamma}$ with $2/\alpha+3/\gamma \leq 1$ on $[0,T]$, then the solution remains smooth on $[0,T]$. The significance is that there are no restriction on the magnetic field. Moreover, the norms $||\nabla u||_{L^{\alpha,\gamma}}$ and $\|\|u\|\|_{L^{\alpha,\gamma}}$ are scaling dimension zero for $2/\alpha+3/\gamma=2$ and $2/\alpha+3/\gamma=1$ respectively.
Keywords: regularity criterion, MHD equations, a priori estimates..
Mathematics Subject Classification: 35Q35, 35B65, 76D0.
Citation: Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881
[1]

Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
[2]

Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations. Electronic Research Archive, 2020, 28 (1) : 183-193. doi: 10.3934/era.2020012
[3]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[4]

Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721
[5]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167
[6]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[7]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
[8]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395
[9]

Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543
[10]

Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335
[11]

Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. Kinetic & Related Models, 2012, 5 (3) : 505-516. doi: 10.3934/krm.2012.5.505
[12]

Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
[13]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258
[14]

Zhengguang Guo, Sadek Gala. Regularity criterion of the Newton-Boussinesq equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 443-451. doi: 10.3934/cpaa.2012.11.443
[15]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
[16]

Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061
[17]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
[18]

Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic & Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291
[19]

Jishan Fan, Tohru Ozawa. Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model. Kinetic & Related Models, 2009, 2 (2) : 293-305. doi: 10.3934/krm.2009.2.293
[20]

Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (141)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences