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