2015, 2015(special): 387-394. doi: 10.3934/proc.2015.0387

Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037

2. 

Department of Mathematics, Nanjing University, Nanjing 210093

3. 

Department of Mathematics, Inha University, Incheon 402-751

Received  July 2014 Revised  January 2015 Published  November 2015

In this paper we establish the global existence of strong solutions to the three-dimensional compressible magnetohydrodynamic equations in a bounded domain with small initial data. Moreover, we study the low Mach number limit to the corresponding problem.
Citation: Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387
References:
[1]

Q. Chen, Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations,, \emph{Nonlinear Anal.} 72(2010) 4438-4451., (2010), 4438. Google Scholar

[2]

C. Dou, S. Jiang and Q. Ju, Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary,, \emph{Z. Angew. Math. Phys.} 64(6)(2013) 1661-1678., (2013), 1661. Google Scholar

[3]

C. Dou, Q. Ju, Low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain for all time,, \emph{Commun. Math. Sci.} 12(4)(2014) 661-679., (2014), 661. Google Scholar

[4]

B. Ducomet, E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, \emph{Commun. Math. Phys.} 266(2006) 595-629., (2006), 595. Google Scholar

[5]

J. Fan, W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum,, \emph{Nonlinear Anal.}: RWA 10(2009) 392-409., (2009), 392. Google Scholar

[6]

J. Fan, W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, \emph{Nonlinear Anal.} 69(2008) 3637-3660., (2008), 3637. Google Scholar

[7]

J. Fan, H. Gao and B. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient,, \emph{Math. Methods Appl. Sci.} 34 (2011) 2181-2188., (2011), 2181. Google Scholar

[8]

X. Hu, D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows,, \emph{Arch. Ration. Mech. Anal.} 197(2010) 203-238., (2010), 203. Google Scholar

[9]

X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flow,, \emph{Commun. Math. Phys.} 283(2008) 255-284., (2008), 255. Google Scholar

[10]

X. Hu, D. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows,, SIAM \emph{J. Math. Anal.} 41(2009) 1272-1294., (2009), 1272. Google Scholar

[11]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions,, \emph{Commun. Math. Phys.} 297(2010) 371-400., (2010), 371. Google Scholar

[12]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients,, SIAM \emph{J. Math. Anal.} 42 (2010), (2010), 2539. Google Scholar

[13]

S. Jiang, Q. Ju, F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations,, \emph{ Nonlinearity} 25 (2012), (2012), 1351. Google Scholar

[14]

S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data,, \emph{Adv. Math.} 259 (2014), (2014), 384. Google Scholar

[15]

F. Li, H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations,, \emph{Proc. Royal Soc. Edinburgh} 141A(2011) 109-126., (2011), 109. Google Scholar

[16]

A. Suen, D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics,, \emph{Arch. Ration. Mech. Anal.} 205 (2012), (2012), 27. Google Scholar

[17]

A. I. Vol'pert, S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations,, \emph{Math. USSR.-Sb.} 16(1972) 517-544., (1972), 517. Google Scholar

[18]

Y. Yang, X. Gu and C. Dou, Global well-posedness of strong solutions to the magnetohydrodynamic equations of compressible flows,, \emph{Nonlinear Anal.} 95 (2014), (2014), 23. Google Scholar

show all references

References:
[1]

Q. Chen, Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations,, \emph{Nonlinear Anal.} 72(2010) 4438-4451., (2010), 4438. Google Scholar

[2]

C. Dou, S. Jiang and Q. Ju, Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary,, \emph{Z. Angew. Math. Phys.} 64(6)(2013) 1661-1678., (2013), 1661. Google Scholar

[3]

C. Dou, Q. Ju, Low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain for all time,, \emph{Commun. Math. Sci.} 12(4)(2014) 661-679., (2014), 661. Google Scholar

[4]

B. Ducomet, E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, \emph{Commun. Math. Phys.} 266(2006) 595-629., (2006), 595. Google Scholar

[5]

J. Fan, W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum,, \emph{Nonlinear Anal.}: RWA 10(2009) 392-409., (2009), 392. Google Scholar

[6]

J. Fan, W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, \emph{Nonlinear Anal.} 69(2008) 3637-3660., (2008), 3637. Google Scholar

[7]

J. Fan, H. Gao and B. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient,, \emph{Math. Methods Appl. Sci.} 34 (2011) 2181-2188., (2011), 2181. Google Scholar

[8]

X. Hu, D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows,, \emph{Arch. Ration. Mech. Anal.} 197(2010) 203-238., (2010), 203. Google Scholar

[9]

X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flow,, \emph{Commun. Math. Phys.} 283(2008) 255-284., (2008), 255. Google Scholar

[10]

X. Hu, D. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows,, SIAM \emph{J. Math. Anal.} 41(2009) 1272-1294., (2009), 1272. Google Scholar

[11]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions,, \emph{Commun. Math. Phys.} 297(2010) 371-400., (2010), 371. Google Scholar

[12]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients,, SIAM \emph{J. Math. Anal.} 42 (2010), (2010), 2539. Google Scholar

[13]

S. Jiang, Q. Ju, F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations,, \emph{ Nonlinearity} 25 (2012), (2012), 1351. Google Scholar

[14]

S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data,, \emph{Adv. Math.} 259 (2014), (2014), 384. Google Scholar

[15]

F. Li, H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations,, \emph{Proc. Royal Soc. Edinburgh} 141A(2011) 109-126., (2011), 109. Google Scholar

[16]

A. Suen, D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics,, \emph{Arch. Ration. Mech. Anal.} 205 (2012), (2012), 27. Google Scholar

[17]

A. I. Vol'pert, S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations,, \emph{Math. USSR.-Sb.} 16(1972) 517-544., (1972), 517. Google Scholar

[18]

Y. Yang, X. Gu and C. Dou, Global well-posedness of strong solutions to the magnetohydrodynamic equations of compressible flows,, \emph{Nonlinear Anal.} 95 (2014), (2014), 23. Google Scholar

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