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:
[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).

[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.

[3]

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

[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. doi: 10.1504/IJDSDE.2011.041882.

[5]

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

[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. doi: 10.1007/s00033-003-1017-z.

[7]

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

[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. doi: 10.1137/110836663.

[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. doi: 10.1007/s00220-006-0052-y.

[10]

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

[11]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 26 (2004).

[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. doi: 10.1007/s002050050181.

[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. doi: 10.1007/PL00000976.

[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. doi: 10.1137/S0036141093247366.

[15]

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

[16]

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

[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. doi: 10.1007/s00205-010-0295-9.

[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).

[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. doi: 10.1016/0021-8928(77)90011-9.

[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. doi: 10.3792/pjaa.58.384.

[21]

L. D. Laudau and E. M. Lifshitz, "Electrodynamics of Continuous Media,", Course of Theoretical Physics, (1960).

[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. doi: 10.1142/S0219891611002457.

[23]

P.-L. Lion, "Mathematics Topic in Fluid Mechanics. Vol. 1. Incompressible Models,", Oxford Lecture Series in Mathematics and its Applications, 3 (1996).

[24]

P.-L. Lion, "Mathematics Topic in Fluid Mechanics. Vol. 2, Compressible Models,", Oxford Lecture Series in Mathematics and its Applications, 10 (1998).

[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).

[26]

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

[27]

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

[28]

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

[29]

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

[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.

[31]

S. N. Shore, "An Introduction to Atrophysical Hydrodynamics,", Academic Press, (1992).

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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. doi: 10.1016/j.na.2009.12.045.

[38]

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

[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.

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).

[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.

[3]

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

[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. doi: 10.1504/IJDSDE.2011.041882.

[5]

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

[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. doi: 10.1007/s00033-003-1017-z.

[7]

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

[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. doi: 10.1137/110836663.

[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. doi: 10.1007/s00220-006-0052-y.

[10]

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

[11]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 26 (2004).

[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. doi: 10.1007/s002050050181.

[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. doi: 10.1007/PL00000976.

[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. doi: 10.1137/S0036141093247366.

[15]

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

[16]

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

[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. doi: 10.1007/s00205-010-0295-9.

[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).

[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. doi: 10.1016/0021-8928(77)90011-9.

[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. doi: 10.3792/pjaa.58.384.

[21]

L. D. Laudau and E. M. Lifshitz, "Electrodynamics of Continuous Media,", Course of Theoretical Physics, (1960).

[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. doi: 10.1142/S0219891611002457.

[23]

P.-L. Lion, "Mathematics Topic in Fluid Mechanics. Vol. 1. Incompressible Models,", Oxford Lecture Series in Mathematics and its Applications, 3 (1996).

[24]

P.-L. Lion, "Mathematics Topic in Fluid Mechanics. Vol. 2, Compressible Models,", Oxford Lecture Series in Mathematics and its Applications, 10 (1998).

[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).

[26]

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

[27]

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

[28]

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

[29]

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

[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.

[31]

S. N. Shore, "An Introduction to Atrophysical Hydrodynamics,", Academic Press, (1992).

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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. doi: 10.1016/j.na.2009.12.045.

[38]

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

[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.

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