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December  2017, 10(4): 1235-1253. doi: 10.3934/krm.2017047

The stability of contact discontinuity for compressible planar magnetohydrodynamics

Haiyan Yin ,

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

* Corresponding author: Haiyan Yin

Received  October 2016 Revised  January 2017 Published  March 2017

This paper is concerned with the planar magnetohydrodynamicswith initial data whose behaviors at far fields $x\rightarrow \pm\infty$ are different. Motivated by the relationship between planar magnetohydrodynamics and Navier-Stokes, we can prove that the solutions to the planar magnetohydrodynamics tend time-asymptotically to a viscous contact wave which is constructed from a contact discontinuity solution of the Riemann problemon Euler system. This result is proved by the method of elementary energy estimates.

Keywords: Planar magnetohydrodynamics, Riemann problem, contact discontinuity, large time behavior, stability.
Mathematics Subject Classification: Primary: 76W05, 76D33; Secondary: 35B35.
Citation: Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic & Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047
References:
[1]

F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal., 54 (1974), 373-392.  doi: 10.1007/BF00249197.  Google Scholar

[2]

G. Q. Chen and D. H. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[3]

G. Q. Chen and D. H. 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.  Google Scholar

[4]

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

[5]

B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

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J. S. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Commun. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1.  Google Scholar

[7]

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.  Google Scholar

[8]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344.  doi: 10.1007/BF00276840.  Google Scholar

[9]

X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.  Google Scholar

[10]

F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0.  Google Scholar

[11]

F. M. Huang, A. Matsumura and X. D. Shi, On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math., 41 (2004), 193-210.   Google Scholar

[12]

F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7.  Google Scholar

[13]

F. M. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: Ⅰ. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728.  doi: 10.3934/krm.2010.3.685.  Google Scholar

[14]

F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.  Google Scholar

[15]

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.  Google Scholar

[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.  Google Scholar

[17]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.  doi: 10.1007/BF01466726.  Google Scholar

[18]

B. Q. Lv and B. Huang, On strong solutions to the Cauchy problem of the two-dimensional compressible magnetohydrodynamic equations with vacuum, Nonlinearity, 28 (2015), 509-530.  doi: 10.1088/0951-7715/28/2/509.  Google Scholar

[19]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.   Google Scholar

[20]

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.  Google Scholar

[21]

X. H. Qin, T. Wang and Y. Wang, Global stability of wave patterns for compressible Navier-Stokes system with free boundary, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1192-1214.  doi: 10.1016/S0252-9602(16)30062-5.  Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[23]

A. Vasseur and Y. Wang, The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method, SIAM J. Math. Anal., 47 (2015), 4350-4359.  doi: 10.1137/15M1023439.  Google Scholar

[24]

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

[25]

X. Ye and J. W. Zhang, On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit, J. Differential Equations, 260 (2016), 3927-3961.  doi: 10.1016/j.jde.2015.10.049.  Google Scholar

show all references

References:
[1]

F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal., 54 (1974), 373-392.  doi: 10.1007/BF00249197.  Google Scholar

[2]

G. Q. Chen and D. H. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[3]

G. Q. Chen and D. H. 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.  Google Scholar

[4]

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

[5]

B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[6]

J. S. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Commun. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1.  Google Scholar

[7]

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.  Google Scholar

[8]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344.  doi: 10.1007/BF00276840.  Google Scholar

[9]

X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.  Google Scholar

[10]

F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0.  Google Scholar

[11]

F. M. Huang, A. Matsumura and X. D. Shi, On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math., 41 (2004), 193-210.   Google Scholar

[12]

F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7.  Google Scholar

[13]

F. M. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: Ⅰ. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728.  doi: 10.3934/krm.2010.3.685.  Google Scholar

[14]

F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.  Google Scholar

[15]

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.  Google Scholar

[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.  Google Scholar

[17]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.  doi: 10.1007/BF01466726.  Google Scholar

[18]

B. Q. Lv and B. Huang, On strong solutions to the Cauchy problem of the two-dimensional compressible magnetohydrodynamic equations with vacuum, Nonlinearity, 28 (2015), 509-530.  doi: 10.1088/0951-7715/28/2/509.  Google Scholar

[19]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.   Google Scholar

[20]

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.  Google Scholar

[21]

X. H. Qin, T. Wang and Y. Wang, Global stability of wave patterns for compressible Navier-Stokes system with free boundary, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1192-1214.  doi: 10.1016/S0252-9602(16)30062-5.  Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[23]

A. Vasseur and Y. Wang, The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method, SIAM J. Math. Anal., 47 (2015), 4350-4359.  doi: 10.1137/15M1023439.  Google Scholar

[24]

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

[25]

X. Ye and J. W. Zhang, On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit, J. Differential Equations, 260 (2016), 3927-3961.  doi: 10.1016/j.jde.2015.10.049.  Google Scholar

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