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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 - A, 2013, 33 (8) : 3791-3805. doi: 10.3934/dcds.2013.33.3791
References:
[1]

H. Cabannes, "Theoretical Magneto-Fluid Dynamics,", Academic Press, (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. 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. 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. 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. 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 (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. 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. 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. 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, (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. 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. doi: 10.1080/00036810802713933. Google Scholar

[13]

E. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (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. 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. 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. 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 (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:
[1]

H. Cabannes, "Theoretical Magneto-Fluid Dynamics,", Academic Press, (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. 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. 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. 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. 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 (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. 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. 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. 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, (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. 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. doi: 10.1080/00036810802713933. Google Scholar

[13]

E. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (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. 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. 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. 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 (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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