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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 |
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. |
show all references
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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