# American Institute of Mathematical Sciences

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

## Local strong solutions to the compressible viscous magnetohydrodynamic equations

 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

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.
Citation: Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & 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. doi: 10.1016/j.aim.2010.08.017. Google Scholar [2] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788. Google Scholar [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. doi: 10.1007/s00209-007-0120-9. Google Scholar [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. doi: 10.1016/j.nonrwa.2007.10.001. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2011.06.019. Google Scholar [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. doi: 10.1016/j.jde.2008.07.019. Google Scholar [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. doi: 10.1002/cpa.21382. Google Scholar [9] S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559. doi: 10.1007/PL00005543. Google Scholar [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. doi: 10.4310/MAA.2005.v12.n3.a2. Google Scholar [11] T. Kato, Quasi-linear Equations Of Evolution with Applications to Partial Differential Equations,, Springer, (1974). Google Scholar [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. doi: 10.1007/s00205-012-0538-z. Google Scholar [13] M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations,, Adv.Math.Sci.Appl., 22 (2012), 319. Google Scholar [14] M. Kotschote and R. Zacher, Strong solutions in the dynamical theory of compressible two phase fluids,, , (). Google Scholar [15] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Society for Industrial and Applied Mathematics, (1973). Google Scholar [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. doi: 10.1137/120893355. Google Scholar [17] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models,, Oxford University Press, (1998). Google Scholar [18] A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, (1984). doi: 10.1007/978-1-4612-1116-7. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF01214738. Google Scholar [21] J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces,, Math. Bohem., 127 (2002), 311. Google Scholar [22] J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces,, Math. Z., 203 (1990), 429. doi: 10.1007/BF02570748. Google Scholar [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. doi: 10.1007/s00033-012-0245-5. Google Scholar [24] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar [25] J. H. Wu, Regularity criteria for the generalized MHD equations,, Comm. Partial Differential Equations, 33 (2008), 285. doi: 10.1080/03605300701382530. Google Scholar [26] J. H. Wu, Global regularity for a class of generalized magnetohydrodynamic equations,, J. Math. Fluid Mech., 13 (2011), 295. doi: 10.1007/s00021-009-0017-y. Google Scholar

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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. doi: 10.1016/j.aim.2010.08.017. Google Scholar [2] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788. Google Scholar [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. doi: 10.1007/s00209-007-0120-9. Google Scholar [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. doi: 10.1016/j.nonrwa.2007.10.001. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2011.06.019. Google Scholar [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. doi: 10.1016/j.jde.2008.07.019. Google Scholar [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. doi: 10.1002/cpa.21382. Google Scholar [9] S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559. doi: 10.1007/PL00005543. Google Scholar [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. doi: 10.4310/MAA.2005.v12.n3.a2. Google Scholar [11] T. Kato, Quasi-linear Equations Of Evolution with Applications to Partial Differential Equations,, Springer, (1974). Google Scholar [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. doi: 10.1007/s00205-012-0538-z. Google Scholar [13] M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations,, Adv.Math.Sci.Appl., 22 (2012), 319. Google Scholar [14] M. Kotschote and R. Zacher, Strong solutions in the dynamical theory of compressible two phase fluids,, , (). Google Scholar [15] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Society for Industrial and Applied Mathematics, (1973). Google Scholar [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. doi: 10.1137/120893355. Google Scholar [17] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models,, Oxford University Press, (1998). Google Scholar [18] A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, (1984). doi: 10.1007/978-1-4612-1116-7. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF01214738. Google Scholar [21] J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces,, Math. Bohem., 127 (2002), 311. Google Scholar [22] J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces,, Math. Z., 203 (1990), 429. doi: 10.1007/BF02570748. Google Scholar [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. doi: 10.1007/s00033-012-0245-5. Google Scholar [24] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar [25] J. H. Wu, Regularity criteria for the generalized MHD equations,, Comm. Partial Differential Equations, 33 (2008), 285. doi: 10.1080/03605300701382530. Google Scholar [26] J. H. Wu, Global regularity for a class of generalized magnetohydrodynamic equations,, J. Math. Fluid Mech., 13 (2011), 295. doi: 10.1007/s00021-009-0017-y. Google Scholar
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