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December  2014, 7(4): 739-753. doi: 10.3934/krm.2014.7.739

Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces

Yanmin Mu 1,

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received  April 2014 Revised  July 2014 Published  November 2014

We study the convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible model with ill-prepared initial data in critical Besov spaces. Under the condition that the initial data is small in some norm, we show that the convergence holds globally as the Mach number goes to zero. Moreover, we also obtain the convergence rate.
Keywords: Compressible isentropic magnetohydrodynamic equations, critical Besov spaces., low Mach number limit, incompressible magnetohydrodynamic equations.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35B4.
Citation: Yanmin Mu. Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces. Kinetic and Related Models, 2014, 7 (4) : 739-753. doi: 10.3934/krm.2014.7.739
References:
[1]

H. Bahouri, J.-Y.Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften 343, Springer, Heidelberg, 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, The Clarendon Press, Oxford University Press, New York, 1998.

[3]

R. Danchin, Zero Mach number limit in critial spaces for compressible Navier-Stokes equations, Ann. Sci. Éc. Norm. Supér. (4), 35 (2002), 27-75. 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), 1153-1219. 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), 2271-2279. 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), 461-471. 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), 525-540.

[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, Oxford, 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), 2962-2972. 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), 17-26. 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), 1272-1294. 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), 2539-2553. 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), 371-400. doi: 10.1007/s00220-010-0992-0.

[14]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., New York, Pergamon, 1984.

[15]

Y. P. Li, Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations, J. Differential Equations, 252 (2012), 2725-2738. 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), 199-224. 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), 53-76. doi: 10.1002/mma.1026.

[18]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 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, Springer, Heidelberg, 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, The Clarendon Press, Oxford University Press, New York, 1998.

[3]

R. Danchin, Zero Mach number limit in critial spaces for compressible Navier-Stokes equations, Ann. Sci. Éc. Norm. Supér. (4), 35 (2002), 27-75. 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), 1153-1219. 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), 2271-2279. 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), 461-471. 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), 525-540.

[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, Oxford, 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), 2962-2972. 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), 17-26. 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), 1272-1294. 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), 2539-2553. 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), 371-400. doi: 10.1007/s00220-010-0992-0.

[14]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., New York, Pergamon, 1984.

[15]

Y. P. Li, Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations, J. Differential Equations, 252 (2012), 2725-2738. 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), 199-224. 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), 53-76. doi: 10.1002/mma.1026.

[18]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

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