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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 |
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,, , ().
|
[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. |
show all references
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,, , ().
|
[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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