# American Institute of Mathematical Sciences

August  2013, 33(8): 3791-3805. doi: 10.3934/dcds.2013.33.3791

## A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density

 1 Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419, United States

Received  June 2012 Revised  October 2012 Published  January 2013

We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\mathbb{R}^3$. We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum ($\rho= 0$) and the concentration of mass ($\rho=\infty$) and if the magnetic field is bounded above in terms of $L^\infty$-norm, then a local strong solution can be continued globally in time.
Citation: Anthony Suen. A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3791-3805. doi: 10.3934/dcds.2013.33.3791
##### References:
 [1] H. Cabannes, "Theoretical Magneto-Fluid Dynamics," Academic Press, New York, London, 1970. [2] J. Fan, S. Jiang and Y. Ou, A blow-up criterion for three-dimensional compressible viscous heat-conductive flows, Annales de l'Institut Henri Poincaŕe Analysis Non Linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012. [3] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J. Diff. Eqns., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111. [4] 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. [5] D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760. doi: 10.1137/040618059. [6] P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. [7] M. Lu, Y. Du and Z. Yai, Blow-up phenomena for the 3D compressible MHD equations, Discrete and Continuous Dynamical Systems, 32 (2012), 1835-1855. doi: 10.3934/dcds.2012.32.1835. [8] 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. [9] 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. [10] S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, With Applications to the Equations of Magnetohydrodynamics," Ph.D. Thesis, Kyoto University, 1983. [11] 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. [12] R. Sart, Existence of finite energy weak solutions for the equations MHD of compressible fluids, Appl. Anal., 88 (2009), 357-379. doi: 10.1080/00036810802713933. [13] E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970. [14] A. Suen and D. Hoff, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics, Arch. Rational Mechanics Ana., 205 (2012), 27-58. doi: 10.1007/s00205-012-0498-3. [15] 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. [16] Z. Xin, Blowup 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>3.3.CO;2-K. [17] W. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] H. Cabannes, "Theoretical Magneto-Fluid Dynamics," Academic Press, New York, London, 1970. [2] J. Fan, S. Jiang and Y. Ou, A blow-up criterion for three-dimensional compressible viscous heat-conductive flows, Annales de l'Institut Henri Poincaŕe Analysis Non Linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012. [3] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J. Diff. Eqns., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111. [4] 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. [5] D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760. doi: 10.1137/040618059. [6] P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. [7] M. Lu, Y. Du and Z. Yai, Blow-up phenomena for the 3D compressible MHD equations, Discrete and Continuous Dynamical Systems, 32 (2012), 1835-1855. doi: 10.3934/dcds.2012.32.1835. [8] 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. [9] 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. [10] S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, With Applications to the Equations of Magnetohydrodynamics," Ph.D. Thesis, Kyoto University, 1983. [11] 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. [12] R. Sart, Existence of finite energy weak solutions for the equations MHD of compressible fluids, Appl. Anal., 88 (2009), 357-379. doi: 10.1080/00036810802713933. [13] E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970. [14] A. Suen and D. Hoff, Global low-energy weak solutions of the equations of 3D compressible magnetohydrodynamics, Arch. Rational Mechanics Ana., 205 (2012), 27-58. doi: 10.1007/s00205-012-0498-3. [15] 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. [16] Z. Xin, Blowup 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>3.3.CO;2-K. [17] W. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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