September  2013, 6(3): 481-503. doi: 10.3934/krm.2013.6.481

Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations

1. 

The Graduate School of China Academy of Engineering Physics, Beijing 100088, China

2. 

Institute of Applied Physics & Computational Math., Beijing 100088

Received  January 2013 Revised  April 2013 Published  May 2013

We are concerned with global existence and large-time behavior of solutions to the isentropic electric-magnetohydrodynamic equations in a bounded domain $\Omega\subseteq\mathbb{R}^{N}$, $N=2,\ 3$. We establish the existence and large-time behavior of global weak solutions through a three-level approximation, energy estimates on condition that the adiabatic constant satisfies $\gamma>3/2$.
Citation: Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
References:
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D. Bian and B. Guo, Well-posedness in critical spaces for the full compressible MHD equations, to appear in Acta Math. Sci. Ser. B, 2013. Google Scholar

[2]

D. Bian and B. Guo, Blow-up of smooth solutions to the compressible MHD equations, to appear in Appl. Anal., 2013. doi: 10.1080/00036811.2013.766324.  Google Scholar

[3]

D. Bian and B. Yuan, Local well-posedness in critical spaces for compressible MHD equations, submitted, (2010), 1-30. Google Scholar

[4]

D. Bian and B. Yuan, Well-posedness in super critical Besov spaces for compressible MHD equations, Int. J. Dynamical Systems and Differential Equations, 3 (2011), 383-399. doi: 10.1504/IJDSDE.2011.041882.  Google Scholar

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[6]

G.-Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632. doi: 10.1007/s00033-003-1017-z.  Google Scholar

[7]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019.  Google Scholar

[8]

S. Ding, H. Wen, L. Yao and C. Zhu, Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum, SIAM J. Math. Anal., 44 (2012), 1257-1278. doi: 10.1137/110836663.  Google Scholar

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B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629. doi: 10.1007/s00220-006-0052-y.  Google Scholar

[10]

J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Commun. Math. Phys., 270 (2007), 691-708. doi: 10.1007/s00220-006-0167-1.  Google Scholar

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E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.  Google Scholar

[12]

E. Feireisl and H. Petzeltová, Large-time behavior of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96. doi: 10.1007/s002050050181.  Google Scholar

[13]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.  Google Scholar

[14]

H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128. doi: 10.1137/S0036141093247366.  Google Scholar

[15]

B. Guo and J. Zhang, Global existence of solution for thermally radiative magnetohydrodynamic equations with the displacement current, J. Math. Phys., 54 (2013), 18 pp. doi: 10.1063/1.4776205.  Google Scholar

[16]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804. doi: 10.1007/s00033-005-4057-8.  Google Scholar

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X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Rational Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.  Google Scholar

[18]

S. Jiang and Q. Jiu, Global existence of solutions to the high-dimensional compressble insentropic Navier-Stokes equations with large initial data, internal notes, 2006. Google Scholar

[19]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary-value problems for the one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282. doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

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S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387. doi: 10.3792/pjaa.58.384.  Google Scholar

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L. D. Laudau and E. M. Lifshitz, "Electrodynamics of Continuous Media," Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

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X. Li, N. Su and D. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyper. Differential Equations, 8 (2011), 415-436. doi: 10.1142/S0219891611002457.  Google Scholar

[23]

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[24]

P.-L. Lion, "Mathematics Topic 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

[25]

T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Memoirs of the American Mathematical Society, 125 (1997).  Google Scholar

[26]

T.-P. Liu, Z. Xin and T. Yang, Vacuum states of compressible flows, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. doi: 10.3934/dcds.1998.4.1.  Google Scholar

[27]

T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM. J. Math. Anal., 31 (1999), 1175-1191. doi: 10.1137/S0036141097331044.  Google Scholar

[28]

H. Ma, S. Ukai and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293. doi: 10.1016/j.jde.2009.11.031.  Google Scholar

[29]

P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. doi: 10.1088/0951-7715/4/2/013.  Google Scholar

[30]

X. Pu and B. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, to appear in Z. Angew. Math. Phys., (2012). doi: 10.1007/s00033-012-0245-5.  Google Scholar

[31]

S. N. Shore, "An Introduction to Atrophysical Hydrodynamics," Academic Press, New York, 1992. Google Scholar

[32]

D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441. doi: 10.1137/S0036139902409284.  Google Scholar

[33]

J. Wu, Generalized MHD equations, J. Differemtial Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar

[34]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.  Google Scholar

[35]

J. Zhang and F. Xie, Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics, J. Differential Equations, 245 (2008), 1853-1882. doi: 10.1016/j.jde.2008.07.010.  Google Scholar

[36]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[37]

Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal., 72 (2010), 3643-3648. doi: 10.1016/j.na.2009.12.045.  Google Scholar

[38]

Z. Xin, Blow up 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>3.0.CO;2-C.  Google Scholar

[39]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, to appear in Commun. Math. Phys., (2012). doi: 10.1007/s00220-012-1610-0.  Google Scholar

show all references

References:
[1]

D. Bian and B. Guo, Well-posedness in critical spaces for the full compressible MHD equations, to appear in Acta Math. Sci. Ser. B, 2013. Google Scholar

[2]

D. Bian and B. Guo, Blow-up of smooth solutions to the compressible MHD equations, to appear in Appl. Anal., 2013. doi: 10.1080/00036811.2013.766324.  Google Scholar

[3]

D. Bian and B. Yuan, Local well-posedness in critical spaces for compressible MHD equations, submitted, (2010), 1-30. Google Scholar

[4]

D. Bian and B. Yuan, Well-posedness in super critical Besov spaces for compressible MHD equations, Int. J. Dynamical Systems and Differential Equations, 3 (2011), 383-399. doi: 10.1504/IJDSDE.2011.041882.  Google Scholar

[5]

G.-Q. Chen and D. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differemtial Equations, 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111.  Google Scholar

[6]

G.-Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632. doi: 10.1007/s00033-003-1017-z.  Google Scholar

[7]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019.  Google Scholar

[8]

S. Ding, H. Wen, L. Yao and C. Zhu, Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum, SIAM J. Math. Anal., 44 (2012), 1257-1278. doi: 10.1137/110836663.  Google Scholar

[9]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629. doi: 10.1007/s00220-006-0052-y.  Google Scholar

[10]

J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Commun. Math. Phys., 270 (2007), 691-708. doi: 10.1007/s00220-006-0167-1.  Google Scholar

[11]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.  Google Scholar

[12]

E. Feireisl and H. Petzeltová, Large-time behavior of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96. doi: 10.1007/s002050050181.  Google Scholar

[13]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.  Google Scholar

[14]

H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128. doi: 10.1137/S0036141093247366.  Google Scholar

[15]

B. Guo and J. Zhang, Global existence of solution for thermally radiative magnetohydrodynamic equations with the displacement current, J. Math. Phys., 54 (2013), 18 pp. doi: 10.1063/1.4776205.  Google Scholar

[16]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804. doi: 10.1007/s00033-005-4057-8.  Google Scholar

[17]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Rational Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.  Google Scholar

[18]

S. Jiang and Q. Jiu, Global existence of solutions to the high-dimensional compressble insentropic Navier-Stokes equations with large initial data, internal notes, 2006. Google Scholar

[19]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary-value problems for the one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282. doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[20]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387. doi: 10.3792/pjaa.58.384.  Google Scholar

[21]

L. D. Laudau and E. M. Lifshitz, "Electrodynamics of Continuous Media," Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[22]

X. Li, N. Su and D. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyper. Differential Equations, 8 (2011), 415-436. doi: 10.1142/S0219891611002457.  Google Scholar

[23]

P.-L. Lion, "Mathematics Topic in Fluid Mechanics. Vol. 1. Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[24]

P.-L. Lion, "Mathematics Topic 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

[25]

T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Memoirs of the American Mathematical Society, 125 (1997).  Google Scholar

[26]

T.-P. Liu, Z. Xin and T. Yang, Vacuum states of compressible flows, Discrete Contin. Dyn. Syst., 4 (1998), 1-32. doi: 10.3934/dcds.1998.4.1.  Google Scholar

[27]

T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM. J. Math. Anal., 31 (1999), 1175-1191. doi: 10.1137/S0036141097331044.  Google Scholar

[28]

H. Ma, S. Ukai and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293. doi: 10.1016/j.jde.2009.11.031.  Google Scholar

[29]

P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. doi: 10.1088/0951-7715/4/2/013.  Google Scholar

[30]

X. Pu and B. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, to appear in Z. Angew. Math. Phys., (2012). doi: 10.1007/s00033-012-0245-5.  Google Scholar

[31]

S. N. Shore, "An Introduction to Atrophysical Hydrodynamics," Academic Press, New York, 1992. Google Scholar

[32]

D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441. doi: 10.1137/S0036139902409284.  Google Scholar

[33]

J. Wu, Generalized MHD equations, J. Differemtial Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar

[34]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.  Google Scholar

[35]

J. Zhang and F. Xie, Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics, J. Differential Equations, 245 (2008), 1853-1882. doi: 10.1016/j.jde.2008.07.010.  Google Scholar

[36]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[37]

Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal., 72 (2010), 3643-3648. doi: 10.1016/j.na.2009.12.045.  Google Scholar

[38]

Z. Xin, Blow up 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>3.0.CO;2-C.  Google Scholar

[39]

Z. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, to appear in Commun. Math. Phys., (2012). doi: 10.1007/s00220-012-1610-0.  Google Scholar

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