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 and 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, vol. 65. Academic Press, New York, 1975.

[2]

S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability," Clarendon Press, Oxford, 1961.

[3]

G. Chen and D. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differ. Equ., 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111.

[4]

G. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632. doi: 10.1007/s00033-003-1017-z.

[5]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Analysis, 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019.

[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), 595-629. doi: 10.1007/s00220-006-0052-y.

[7]

J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. doi: 10.1016/j.na.2007.10.005.

[8]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Analysis: Real world Applications, 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001.

[9]

H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128. doi: 10.1137/S0036141093247366.

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics Math., Springer-Verlag, Berlin, 2001.

[11]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804. doi: 10.1007/s00033-005-4057-8.

[12]

X. Hu and D.Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 253-284. doi: 10.1007/s00220-008-0497-2.

[13]

X. Hu and D. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Diff. Equs., 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.

[14]

X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Diff. Equs., 249 (2010), 1179-1198. doi: 10.1016/j.jde.2010.03.027.

[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), 203-238. doi: 10.1007/s00205-010-0295-9.

[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), 384-387. doi: 10.3792/pjaa.58.384.

[17]

A. Novotný I. Strašraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Ser. Math. Appl., vol. 27, Oxford University Press, Oxford, 2004.

[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., 20 (1980), 67-104.

[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), 445-464. doi: 10.1007/BF01214738.

[20]

W. Rudin, "Functional Analysis," McGraw-Hill, 1991.

[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), 17-51.

[22]

D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math, 63 (2003), 1424-1441. doi: 10.1137/S0036139902409284.

[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), 1250027 (12 pages). doi: 10.1142/S0129167X12500279.

[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), 1063-1072. doi: 10.1016/S0252-9602(12)60079-4.

[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, Volume 2011, Article ID 128614, 14 pages. doi: 10.1155/2011/128614.

[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), 2125-2135. doi: 10.1002/mma.1510.

show all references

References:
[1]

R. A. Admas, "Sobolev Spaces," Pure and Applied Mathematics, vol. 65. Academic Press, New York, 1975.

[2]

S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability," Clarendon Press, Oxford, 1961.

[3]

G. Chen and D. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differ. Equ., 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111.

[4]

G. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632. doi: 10.1007/s00033-003-1017-z.

[5]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Analysis, 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019.

[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), 595-629. doi: 10.1007/s00220-006-0052-y.

[7]

J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. doi: 10.1016/j.na.2007.10.005.

[8]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Analysis: Real world Applications, 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001.

[9]

H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128. doi: 10.1137/S0036141093247366.

[10]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics Math., Springer-Verlag, Berlin, 2001.

[11]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804. doi: 10.1007/s00033-005-4057-8.

[12]

X. Hu and D.Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 253-284. doi: 10.1007/s00220-008-0497-2.

[13]

X. Hu and D. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Diff. Equs., 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.

[14]

X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Diff. Equs., 249 (2010), 1179-1198. doi: 10.1016/j.jde.2010.03.027.

[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), 203-238. doi: 10.1007/s00205-010-0295-9.

[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), 384-387. doi: 10.3792/pjaa.58.384.

[17]

A. Novotný I. Strašraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Ser. Math. Appl., vol. 27, Oxford University Press, Oxford, 2004.

[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., 20 (1980), 67-104.

[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), 445-464. doi: 10.1007/BF01214738.

[20]

W. Rudin, "Functional Analysis," McGraw-Hill, 1991.

[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), 17-51.

[22]

D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math, 63 (2003), 1424-1441. doi: 10.1137/S0036139902409284.

[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), 1250027 (12 pages). doi: 10.1142/S0129167X12500279.

[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), 1063-1072. doi: 10.1016/S0252-9602(12)60079-4.

[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, Volume 2011, Article ID 128614, 14 pages. doi: 10.1155/2011/128614.

[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), 2125-2135. doi: 10.1002/mma.1510.

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