A blow-up criterion for the 3D compressible MHD equations

Ming Lu; Yi Du; Zheng-An Yao; Zujin Zhang

doi:10.3934/cpaa.2012.11.1167

Article Contents

2012, Volume 11, Issue 3: 1167-1183. Doi: 10.3934/cpaa.2012.11.1167

This issue Previous Article One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane Next Article Analysis of a contact problem for electro-elastic-visco-plastic materials

A blow-up criterion for the 3D compressible MHD equations

1.
Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275
2.
Department of Mathematical Sciences, South China Normal University, Guangzhou, 510631
3.
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275

Received: December 2010

Revised: April 2011

Published: May 2012

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we study the 3D compressible magnetohydrodynamic equations. We extend the well-known Serrin's blow-up criterion(see [32]) for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our case.

Keywords:

Blow-up criterion,
strong solutions,
3D compressible magnetohydrodynamic equations.

Mathematics Subject Classification: Primary: 35Q35, 35Q80; Secondary: 76N10.

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References

[1]	J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys, 94 (1984), 61-66.doi: 10.1007/BF01212349.
[2]	R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.doi: 10.1007/s002200050067.
[3]	Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 3 (2007), 861-872.doi: 10.1007/s00220-007-0319-y.
[4]	B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.doi: 10.1007/s00220-006-0052-y.
[5]	Y. Du, Y. Liu and Z. Yao, Remarks on the blow-up criteria for three-dimensional ideal magnetohydrodynamics equations, J. Math. Phys., 50 (2009), 023507.doi: 10.1063/1.3075570.
[6]	G. Duvaut and J. L. Lions, Inequation en theremoelasticite et magnetohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.doi: 10.1007/BF00250512.
[7]	J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal. TMA, 69 (2008), 3637-3660.doi: 10.1016/j.na.2007.10.005.
[8]	J. Fan and S. Jiang, Blow-up criteria for the Navier-Stokes equations of compressible fluids, J. Hyper. Diff. Eqns., 5 (2008), 167-185.doi: 10.1142/S0219891608001386.
[9]	J. Fan, S. Jiang and Y. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. Inst. H. Poincar'e Anal. Non Lin'eaire, 27 (2010), 337-350.doi: 10.1016/j.anihpc.2009.09.012.
[10]	C. Foias and R. Temam, Gevrey class regulairity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.doi: 10.1016/0022-1236(89)90015-3.
[11]	C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differ. Equations, 213 (2005), 235-254.doi: 10.1016/j.jde.2004.07.002.
[12]	C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.doi: 10.1016/j.jfa.2005.06.009.
[13]	C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetodynamic equations, J. Differ. Equations, 238 (2007), 1-17.doi: 10.1016/j.jde.2007.03.023.
[14]	D. Hoff, Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech., 7 (2005), 315-338.doi: 10.1007/s00021-004-0123-9.
[15]	X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.doi: 10.1007/s00220-008-0497-2.
[16]	X. Hu and D. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198.doi: 10.1016/j.jde.2008.07.019.
[17]	X. Hu and D. Wang, Low Mach number limit of viscous compressible magnetohydro-dynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.doi: 10.1137/080723983.
[18]	X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.doi: 10.1007/s00205-010-0295-9.
[19]	X. Huang and Z. Xin, A Blow-up criterion for the compressible Navier-Stokes equations, Sci. China Math., 53 (2010), 671–686,doi: 10.1007/s11425-010-0042-6.
[20]	X. Huang, J. Li and Z. Xin, Blow-up criterion for viscous barotropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35.doi: 10.1007/s00220-010-1148-y.
[21]	M. Lu, Y. Du and Z. Yao, Blow-up criterion for compressible MHD equations, J. Math. Anal. Appl., 379 (2011), 425-438.doi: 10.1016/j.jmaa.2011.01.043.
[22]	M. Lu, Y. Du and Z. Yao, Blow-up phenomena for the 3D compressible MHD equations, Discrete Contin. Dyn. Syst-A., To be published (2012).
[23]	O. Rozanova, Blow-up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity, J. Diff. Eqns., 245 (2008), 1762-1774.doi: 10.1016/j.jde.2008.07.007.
[24]	O. Rozanova, Blow-up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy, Proc. Sympos. Appl. Math.,67 (2009), 911-917.arXiv:0811.4359v1 [math.AP]
[25]	M. Sermange and R. Teman, Some mathematical questions related to the MHD equations, Comm. Pure Appl., 36 (1983), 635-664.doi: 10.1002/cpa.3160360506.
[26]	Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47,.doi: 10.1016/j.matpur.2010.08.001.
[27]	Z. Tan and Y. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.doi: 10.1016/j.na.2009.05.012.
[28]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995.
[29]	T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.doi: 10.1007/BF03167068.
[30]	A. I. Volpert and S. I. Hudjaev, The Cauchy problem for composite systems of nonlinear differential equations, Mat. Sbornik, 87 (1972), 504-528.doi: 10.1070/SM1972v016n04ABEH001438.
[31]	J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.doi: 10.1007/s00332-002-0486-0.
[32]	Z. Xin, Blow-up of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.
[33]	Z. Zhang, Remarks on the regularity criteria for generalized MHD equations, J. Math. Anal. Appl., 375 (2011), 799–802.doi: 10.1016/j.jmaa.2010.10.017.
[34]	Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 5 (2005), 881-886.doi: 10.3934/dcds.2005.12.881.
[35]	Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 10 (2006), 1174-1180.doi: 10.1016/j.ijnonlinmec.2006.12.001.
[36]	Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (2007), 491-505.doi: 10.1016/j.anihpc.2006.03.014.