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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

Anthony Suen 1,

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.
Keywords: compressible magnetohydrodynamics., Blow-up criteria.
Mathematics Subject Classification: 76W05, 35Q3.
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]

Academic Press, New York, London, 1970. Google Scholar

[2]

Annales de l'Institut Henri Poincaŕe Analysis Non Linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012.  Google Scholar

[3]

J. Diff. Eqns., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[4]

J. Math. Fluid Mech., 7 (2005), 315-338. doi: 10.1007/s00021-004-0123-9.  Google Scholar

[5]

SIAM J. Math. Anal., 37 (2006), 1742-1760. doi: 10.1137/040618059.  Google Scholar

[6]

Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[7]

Discrete and Continuous Dynamical Systems, 32 (2012), 1835-1855. doi: 10.3934/dcds.2012.32.1835.  Google Scholar

[8]

Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.  Google Scholar

[9]

Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.  Google Scholar

[10]

Ph.D. Thesis, Kyoto University, 1983. Google Scholar

[11]

J. Diff. Eqns., 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[12]

Appl. Anal., 88 (2009), 357-379. doi: 10.1080/00036810802713933.  Google Scholar

[13]

Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970.  Google Scholar

[14]

Arch. Rational Mechanics Ana., 205 (2012), 27-58. doi: 10.1007/s00205-012-0498-3.  Google Scholar

[15]

J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[16]

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]

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]

Academic Press, New York, London, 1970. Google Scholar

[2]

Annales de l'Institut Henri Poincaŕe Analysis Non Linéaire, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012.  Google Scholar

[3]

J. Diff. Eqns., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[4]

J. Math. Fluid Mech., 7 (2005), 315-338. doi: 10.1007/s00021-004-0123-9.  Google Scholar

[5]

SIAM J. Math. Anal., 37 (2006), 1742-1760. doi: 10.1137/040618059.  Google Scholar

[6]

Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[7]

Discrete and Continuous Dynamical Systems, 32 (2012), 1835-1855. doi: 10.3934/dcds.2012.32.1835.  Google Scholar

[8]

Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.  Google Scholar

[9]

Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.  Google Scholar

[10]

Ph.D. Thesis, Kyoto University, 1983. Google Scholar

[11]

J. Diff. Eqns., 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[12]

Appl. Anal., 88 (2009), 357-379. doi: 10.1080/00036810802713933.  Google Scholar

[13]

Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970.  Google Scholar

[14]

Arch. Rational Mechanics Ana., 205 (2012), 27-58. doi: 10.1007/s00205-012-0498-3.  Google Scholar

[15]

J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[16]

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]

Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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