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Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces
1. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
References:
[1] |
H. Bahouri, J.-Y.Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften 343, 343 (2011).
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-Y. Chemin, Perfect Incompressible Fluids,, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications 14, 14 (1995).
|
[3] |
R. Danchin, Zero Mach number limit in critial spaces for compressible Navier-Stokes equations,, Ann. Sci. Éc. Norm. Supér. (4), 35 (2002), 27.
doi: 10.1016/S0012-9593(01)01085-0. |
[4] |
R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions,, Amer. J. Math., 124 (2002), 1153.
doi: 10.1353/ajm.2002.0036. |
[5] |
B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271.
doi: 10.1098/rspa.1999.0403. |
[6] |
B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions,, J. Math. Pures Appl., 78 (1999), 461.
doi: 10.1016/S0021-7824(99)00032-X. |
[7] |
I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations,, J. Math. Kyoto Univ., 40 (2000), 525.
|
[8] |
J.-F. Gerbeau, C. Le Bris and L. Claude, Mathematical Methods For The Magnetohydrodynamics of Liquid Metals,, Numerical Mathematics and Scientific Computation. Oxford University Press, (2006).
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[9] |
C. C. Hao, Well-posedness to the compressible viscous magnetohydrddynamic system,, Nonlinear Anal. Real World Appl., 12 (2011), 2962.
doi: 10.1016/j.nonrwa.2011.04.017. |
[10] |
L. Hsiao, Q-C. Ju and F.-C. Li, The incompressible limits of compressible Navier-Stokes equations in the whole space with general initial data,, Chin. Ann. Math. Ser. B, 30 (2009), 17.
doi: 10.1007/s11401-008-0039-4. |
[11] |
X-P. Hu and D.-H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows,, SIAM J. Math. Anal., 41 (2009), 1272.
doi: 10.1137/080723983. |
[12] |
S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients,, SIAM J. Math. Anal., 42 (2010), 2539.
doi: 10.1137/100785168. |
[13] |
S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with periodic boundary conditions,, Comm. Math. Phys., 297 (2010), 371.
doi: 10.1007/s00220-010-0992-0. |
[14] |
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media,, 2nd ed., (1984).
|
[15] |
Y. P. Li, Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations,, J. Differential Equations, 252 (2012), 2725.
doi: 10.1016/j.jde.2011.10.002. |
[16] |
N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199.
doi: 10.1016/S0294-1449(00)00123-2. |
[17] |
C. X. Miao and B. Q. Yuan, On the well-posesness of the Cauchy problem for an MHD system in Besov spaces,, Math.Meth.Appl.Sci., 32 (2009), 53.
doi: 10.1002/mma.1026. |
[18] |
H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, 78 (1983).
doi: 10.1007/978-3-0346-0416-1. |
show all references
References:
[1] |
H. Bahouri, J.-Y.Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften 343, 343 (2011).
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-Y. Chemin, Perfect Incompressible Fluids,, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications 14, 14 (1995).
|
[3] |
R. Danchin, Zero Mach number limit in critial spaces for compressible Navier-Stokes equations,, Ann. Sci. Éc. Norm. Supér. (4), 35 (2002), 27.
doi: 10.1016/S0012-9593(01)01085-0. |
[4] |
R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions,, Amer. J. Math., 124 (2002), 1153.
doi: 10.1353/ajm.2002.0036. |
[5] |
B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271.
doi: 10.1098/rspa.1999.0403. |
[6] |
B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions,, J. Math. Pures Appl., 78 (1999), 461.
doi: 10.1016/S0021-7824(99)00032-X. |
[7] |
I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations,, J. Math. Kyoto Univ., 40 (2000), 525.
|
[8] |
J.-F. Gerbeau, C. Le Bris and L. Claude, Mathematical Methods For The Magnetohydrodynamics of Liquid Metals,, Numerical Mathematics and Scientific Computation. Oxford University Press, (2006).
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[9] |
C. C. Hao, Well-posedness to the compressible viscous magnetohydrddynamic system,, Nonlinear Anal. Real World Appl., 12 (2011), 2962.
doi: 10.1016/j.nonrwa.2011.04.017. |
[10] |
L. Hsiao, Q-C. Ju and F.-C. Li, The incompressible limits of compressible Navier-Stokes equations in the whole space with general initial data,, Chin. Ann. Math. Ser. B, 30 (2009), 17.
doi: 10.1007/s11401-008-0039-4. |
[11] |
X-P. Hu and D.-H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows,, SIAM J. Math. Anal., 41 (2009), 1272.
doi: 10.1137/080723983. |
[12] |
S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients,, SIAM J. Math. Anal., 42 (2010), 2539.
doi: 10.1137/100785168. |
[13] |
S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with periodic boundary conditions,, Comm. Math. Phys., 297 (2010), 371.
doi: 10.1007/s00220-010-0992-0. |
[14] |
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media,, 2nd ed., (1984).
|
[15] |
Y. P. Li, Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations,, J. Differential Equations, 252 (2012), 2725.
doi: 10.1016/j.jde.2011.10.002. |
[16] |
N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199.
doi: 10.1016/S0294-1449(00)00123-2. |
[17] |
C. X. Miao and B. Q. Yuan, On the well-posesness of the Cauchy problem for an MHD system in Besov spaces,, Math.Meth.Appl.Sci., 32 (2009), 53.
doi: 10.1002/mma.1026. |
[18] |
H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, 78 (1983).
doi: 10.1007/978-3-0346-0416-1. |
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