American Institute of Mathematical Sciences

May  2012, 32(5): 1835-1855. doi: 10.3934/dcds.2012.32.1835

Blow-up phenomena 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  August 2011 Published  January 2012

In this paper, we study the three-dimensional(3D) compressible magnetohydrodynamic equations. Firstly, we obtain a blow-up criterion for the local strong solutions in terms of the gradient of the velocity, which is similar to the Beal-Kato-Majda criterion(see [1]) for the ideal incompressible flow. Secondly, we extend the well-known Serrin's blow-up criterion for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our cases.
Citation: Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835
References:
 [1] J. T. Beale, T. Kato and A. Majda., Remarks on the breakdown of smooth solutions for the 3-D 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., 275 (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, 8 pp. [6] G. Duvaut and J. L. Lions, Inequation en theremoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 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 three-dimensional compressible viscous heat-conductive flows, Annales de l'Institut Henri Poincaré Analysis Non Linéaire, 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 magnetohydrodynamic 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 classical solutions to 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, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886. doi: 10.1137/100814639. [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, Z. Yao and Z. Zhang, A blow-up criterion for the 3D compressible MHD equations, Comm. Pure Appl. Math., 11 (2012), 1167-1183. [23] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, 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, in "Hyperbolic Problems: Theory, Numerics and Applications," Proc. Sympos. Appl. Math., 67, Part 2, Amer. Math. Soc., Providence, RI, 2009. Available from: arXiv:0811.4359v1. [25] M. Sermange and R. Teman, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 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," Second edition, 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. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear equations, Mat. Sbornik (N.S.), 87(129) (1972), 504-528. [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., 12 (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., 41 (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, 24 (2007), 491-505.

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References:
 [1] J. T. Beale, T. Kato and A. Majda., Remarks on the breakdown of smooth solutions for the 3-D 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., 275 (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, 8 pp. [6] G. Duvaut and J. L. Lions, Inequation en theremoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 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 three-dimensional compressible viscous heat-conductive flows, Annales de l'Institut Henri Poincaré Analysis Non Linéaire, 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 magnetohydrodynamic 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 classical solutions to 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, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886. doi: 10.1137/100814639. [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, Z. Yao and Z. Zhang, A blow-up criterion for the 3D compressible MHD equations, Comm. Pure Appl. Math., 11 (2012), 1167-1183. [23] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, 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, in "Hyperbolic Problems: Theory, Numerics and Applications," Proc. Sympos. Appl. Math., 67, Part 2, Amer. Math. Soc., Providence, RI, 2009. Available from: arXiv:0811.4359v1. [25] M. Sermange and R. Teman, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 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," Second edition, 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. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear equations, Mat. Sbornik (N.S.), 87(129) (1972), 504-528. [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., 12 (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., 41 (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, 24 (2007), 491-505.
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