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, Nonlinear Anal. 72(2010) 4438-4451. 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, Z. Angew. Math. Phys. 64(6)(2013) 1661-1678. Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

X. Hu, 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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal. 42 (2010), 2539-2553. Google Scholar

[13]

S. Jiang, Q. Ju, F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity 25 (2012), no. 5, 1351-1365. 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, Adv. Math. 259 (2014), 384-420. Google Scholar

[15]

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

[16]

A. Suen, D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics, Arch. Ration. Mech. Anal. 205 (2012), no. 1, 27-58. Google Scholar

[17]

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

[18]

Y. Yang, X. Gu and C. Dou, Global well-posedness of strong solutions to the magnetohydrodynamic equations of compressible flows, Nonlinear Anal. 95 (2014), 23-37. Google Scholar

show all references

References:
[1]

Q. Chen, Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal. 72(2010) 4438-4451. 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, Z. Angew. Math. Phys. 64(6)(2013) 1661-1678. Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

X. Hu, 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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal. 42 (2010), 2539-2553. Google Scholar

[13]

S. Jiang, Q. Ju, F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity 25 (2012), no. 5, 1351-1365. 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, Adv. Math. 259 (2014), 384-420. Google Scholar

[15]

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

[16]

A. Suen, D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics, Arch. Ration. Mech. Anal. 205 (2012), no. 1, 27-58. Google Scholar

[17]

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

[18]

Y. Yang, X. Gu and C. Dou, Global well-posedness of strong solutions to the magnetohydrodynamic equations of compressible flows, Nonlinear Anal. 95 (2014), 23-37. Google Scholar

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