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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[7]

I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations, J. Math. Kyoto Univ., 40 (2000), 525-540.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations, J. Math. Kyoto Univ., 40 (2000), 525-540.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

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