American Institute of Mathematical Sciences

Advanced
May  2012, 11(3): 1167-1183. doi: 10.3934/cpaa.2012.11.1167

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

Ming Lu 1, , Yi Du 2, , Zheng-An Yao 3, and  Zujin Zhang 1,

1. 

Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275, China, China
2. 

Department of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
3. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275

Received  December 2010 Revised  April 2011 Published  December 2011

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: strong solutions, 3D compressible magnetohydrodynamic equations., Blow-up criterion.
Mathematics Subject Classification: Primary: 35Q35, 35Q80; Secondary: 76N1.
Citation: 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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. Duvaut and J. L. Lions, Inequation en theremoelasticite et magnetohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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]  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. Duvaut and J. L. Lions, Inequation en theremoelasticite et magnetohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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]  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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