Local strong solutions to the compressible viscous magnetohydrodynamic equations

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July 2016, 21(5): 1617-1633. doi: 10.3934/dcdsb.2016014

Local strong solutions to the compressible viscous magnetohydrodynamic equations

Tong Tang ^1, and Hongjun Gao ^2,

1.	Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098
2.	Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Science, Nanjing Normal University, Nanjing 210023

Received January 2014 Revised March 2014 Published April 2016

Abstract
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In this paper, we consider the compressible magnetohydrodynamic equations with nonnegative thermal conductivity and electric conductivity. The coefficients of the viscosity, heat conductivity and magnetic diffusivity depend on density and temperature. Inspired by the framework of [11], [13] and [15], we use the maximal regularity and contraction mapping argument to prove the existence and uniqueness of local strong solutions with positive initial density in the bounded domain for any dimension.

Keywords: compressible magnetohydrodynamic., strong solutions, Existence.

Mathematics Subject Classification: Primary: 35Q35, 76W05, 76N1.

Citation: Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014

References:

[1]	C. S. Cao and J. H. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.
[2]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.
[3]	R. Denk, M. Hieber and J. Prüss, Optimal $L_p-L_q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.
[4]	J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001.
[5]	J. S. Fan, H. J. Gao and G. Nakamura, Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations, Taiwanese J. Math., 15 (2011), 1059-1073.
[6]	J. S. Fan, S. Jiang and G. Nakamura, Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data, J. Differential Equations, 251 (2011), 2025-2036. doi: 10.1016/j.jde.2011.06.019.
[7]	X. P. Hu and D. H. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.
[8]	X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.
[9]	S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543.
[10]	S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.
[11]	T. Kato, Quasi-linear Equations Of Evolution with Applications to Partial Differential Equations, Springer, 1974.
[12]	M. Kotschote, Strong solutions to the Navier-Stokes equations for a compressible fluid of Allen-Cahn type, Arch. Rational Mech. Anal., 206 (2012), 489-514. doi: 10.1007/s00205-012-0538-z.
[13]	M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations, Adv.Math.Sci.Appl., 22 (2012), 319-347.
[14]	M. Kotschote and R. Zacher, Strong solutions in the dynamical theory of compressible two phase fluids, arXiv:1306.2565v1.
[15]	P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, 1973.
[16]	H. L. Li, X. Y. Xu and J. W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387. doi: 10.1137/120893355.
[17]	P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models, Oxford University Press, New York, 1998.
[18]	A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.
[19]	A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
[20]	A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.
[21]	J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces, Math. Bohem., 127 (2002), 311-327.
[22]	J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.
[23]	X. K. Pu and B. L. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538. doi: 10.1007/s00033-012-0245-5.
[24]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.
[25]	J. H. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.
[26]	J. H. 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.

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References:

[1]	C. S. Cao and J. H. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.
[2]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.
[3]	R. Denk, M. Hieber and J. Prüss, Optimal $L_p-L_q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.
[4]	J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001.
[5]	J. S. Fan, H. J. Gao and G. Nakamura, Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations, Taiwanese J. Math., 15 (2011), 1059-1073.
[6]	J. S. Fan, S. Jiang and G. Nakamura, Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data, J. Differential Equations, 251 (2011), 2025-2036. doi: 10.1016/j.jde.2011.06.019.
[7]	X. P. Hu and D. H. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.
[8]	X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.
[9]	S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543.
[10]	S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.
[11]	T. Kato, Quasi-linear Equations Of Evolution with Applications to Partial Differential Equations, Springer, 1974.
[12]	M. Kotschote, Strong solutions to the Navier-Stokes equations for a compressible fluid of Allen-Cahn type, Arch. Rational Mech. Anal., 206 (2012), 489-514. doi: 10.1007/s00205-012-0538-z.
[13]	M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations, Adv.Math.Sci.Appl., 22 (2012), 319-347.
[14]	M. Kotschote and R. Zacher, Strong solutions in the dynamical theory of compressible two phase fluids, arXiv:1306.2565v1.
[15]	P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, 1973.
[16]	H. L. Li, X. Y. Xu and J. W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387. doi: 10.1137/120893355.
[17]	P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models, Oxford University Press, New York, 1998.
[18]	A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.
[19]	A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
[20]	A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.
[21]	J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces, Math. Bohem., 127 (2002), 311-327.
[22]	J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.
[23]	X. K. Pu and B. L. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538. doi: 10.1007/s00033-012-0245-5.
[24]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.
[25]	J. H. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.
[26]	J. H. 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.

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