# American Institute of Mathematical Sciences

March  2013, 12(2): 851-866. doi: 10.3934/cpaa.2013.12.851

## Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity

 1 School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China, South Korea

Received  September 2011 Revised  February 2012 Published  September 2012

In this paper we investigate three-dimensional compressible magnetohydrodynamic equations with partial viscosity. Local strong solutions to the compressible magnetohydrodynamic equations with large data are established.
Citation: Yu-Zhu Wang, Yin-Xia Wang. Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 851-866. doi: 10.3934/cpaa.2013.12.851
##### References:
 [1] R. A. Admas, "Sobolev Spaces,", Pure and Applied Mathematics, (1975). Google Scholar [2] S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,", Clarendon Press, (1961). Google Scholar [3] G. Chen and D. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data,, J. Differ. Equ., 182 (2002), 344. doi: 10.1006/jdeq.2001.4111. Google Scholar [4] G. 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. Google Scholar [5] Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations,, Nonlinear Analysis, 72 (2010), 4438. doi: 10.1016/j.na.2010.02.019. Google Scholar [6] B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, Comm. Math. Phys., 266 (2006), 595. doi: 10.1007/s00220-006-0052-y. Google Scholar [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 69 (2008), 3637. doi: 10.1016/j.na.2007.10.005. Google Scholar [8] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum,, Nonlinear Analysis: Real world Applications, 10 (2009), 392. doi: 10.1016/j.nonrwa.2007.10.001. Google Scholar [9] 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. Google Scholar [10] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics Math., (2001). Google Scholar [11] 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. Google Scholar [12] X. Hu and D.Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows,, Comm. Math. Phys., 283 (2008), 253. doi: 10.1007/s00220-008-0497-2. Google Scholar [13] X. Hu and D. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations,, J. Diff. Equs., 245 (2008), 2176. doi: 10.1016/j.jde.2008.07.019. Google Scholar [14] X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data,, J. Diff. Equs., 249 (2010), 1179. doi: 10.1016/j.jde.2010.03.027. Google Scholar [15] 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. Google Scholar [16] 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. Google Scholar [17] A. Novotný I. Strašraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford Lecture Ser. Math. Appl., (2004). Google Scholar [18] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., (1980), 67. Google Scholar [19] 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 [20] W. Rudin, "Functional Analysis,", McGraw-Hill, (1991). Google Scholar [21] R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty$,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40 (1993), 17. Google Scholar [22] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics,, SIAM J. Appl. Math, 63 (2003), 1424. doi: 10.1137/S0036139902409284. Google Scholar [23] Y.-Z. Wang, H. J. Zhao and Y.-X. Wang, A logarithmally improved blow up criterion of smooth solutions for the three-dimensional MHD equations,, International Journal of Mathematics, 23 (2012). doi: 10.1142/S0129167X12500279. Google Scholar [24] Y.-Z. Wang, S. Wang and Y.-X. Wang, Regularity criteria for weak solution to the 3D magnetohydrodynamic equations,, Acta Math. Scientia, 32 (2012), 1063. doi: 10.1016/S0252-9602(12)60079-4. Google Scholar [25] Y.-Z. Wang, L. Hu and Y.-X. Wang, A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity,, Boundary Value Problems, (2011). doi: 10.1155/2011/128614. Google Scholar [26] Y.-Z. Wang and Y.-X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity,, Mathematical Methods in the Applied Sciences, 34 (2011), 2125. doi: 10.1002/mma.1510. Google Scholar

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##### References:
 [1] R. A. Admas, "Sobolev Spaces,", Pure and Applied Mathematics, (1975). Google Scholar [2] S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,", Clarendon Press, (1961). Google Scholar [3] G. Chen and D. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data,, J. Differ. Equ., 182 (2002), 344. doi: 10.1006/jdeq.2001.4111. Google Scholar [4] G. 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. Google Scholar [5] Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations,, Nonlinear Analysis, 72 (2010), 4438. doi: 10.1016/j.na.2010.02.019. Google Scholar [6] B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, Comm. Math. Phys., 266 (2006), 595. doi: 10.1007/s00220-006-0052-y. Google Scholar [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 69 (2008), 3637. doi: 10.1016/j.na.2007.10.005. Google Scholar [8] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum,, Nonlinear Analysis: Real world Applications, 10 (2009), 392. doi: 10.1016/j.nonrwa.2007.10.001. Google Scholar [9] 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. Google Scholar [10] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics Math., (2001). Google Scholar [11] 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. Google Scholar [12] X. Hu and D.Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows,, Comm. Math. Phys., 283 (2008), 253. doi: 10.1007/s00220-008-0497-2. Google Scholar [13] X. Hu and D. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations,, J. Diff. Equs., 245 (2008), 2176. doi: 10.1016/j.jde.2008.07.019. Google Scholar [14] X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data,, J. Diff. Equs., 249 (2010), 1179. doi: 10.1016/j.jde.2010.03.027. Google Scholar [15] 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. Google Scholar [16] 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. Google Scholar [17] A. Novotný I. Strašraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford Lecture Ser. Math. Appl., (2004). Google Scholar [18] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., (1980), 67. Google Scholar [19] 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 [20] W. Rudin, "Functional Analysis,", McGraw-Hill, (1991). Google Scholar [21] R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty$,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40 (1993), 17. Google Scholar [22] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics,, SIAM J. Appl. Math, 63 (2003), 1424. doi: 10.1137/S0036139902409284. Google Scholar [23] Y.-Z. Wang, H. J. Zhao and Y.-X. Wang, A logarithmally improved blow up criterion of smooth solutions for the three-dimensional MHD equations,, International Journal of Mathematics, 23 (2012). doi: 10.1142/S0129167X12500279. Google Scholar [24] Y.-Z. Wang, S. Wang and Y.-X. Wang, Regularity criteria for weak solution to the 3D magnetohydrodynamic equations,, Acta Math. Scientia, 32 (2012), 1063. doi: 10.1016/S0252-9602(12)60079-4. Google Scholar [25] Y.-Z. Wang, L. Hu and Y.-X. Wang, A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity,, Boundary Value Problems, (2011). doi: 10.1155/2011/128614. Google Scholar [26] Y.-Z. Wang and Y.-X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity,, Mathematical Methods in the Applied Sciences, 34 (2011), 2125. doi: 10.1002/mma.1510. Google Scholar
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