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 & 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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, With Applications to the Equations of Magnetohydrodynamics," Ph.D. Thesis, Kyoto University, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Cabannes, "Theoretical Magneto-Fluid Dynamics," Academic Press, New York, London, 1970. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, With Applications to the Equations of Magnetohydrodynamics," Ph.D. Thesis, Kyoto University, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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