数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (2): 447-498.doi: 10.1016/S0252-9602(10)60058-6

• 论文 • 上一篇    下一篇

EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

 Blake Barker, Olivier Lafitte, Kevin Zumbrun   

  1. 1.Department of Mathematics, Indiana University, Bloomington, IN 47402, USA;
    2.LAGA, Institut Galilee, Universite Paris 13, 93 430 Villetaneuse and CEA Saclay, DM2S/DIR, 91 191 Gif sur Yvette Cedex, France
  • 收稿日期:2009-12-13 出版日期:2010-03-20 发布日期:2010-03-20
  • 基金资助:

    This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.

EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

 Blake Barker, Olivier Lafitte, Kevin Zumbrun   

  1. 1.Department of Mathematics, Indiana University, Bloomington, IN 47402, USA;
    2.LAGA, Institut Galilee, Universite Paris 13, 93 430 Villetaneuse and CEA Saclay, DM2S/DIR, 91 191 Gif sur Yvette Cedex, France
  • Received:2009-12-13 Online:2010-03-20 Published:2010-03-20
  • Supported by:

    This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.

摘要:

For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with γ-law gas equation of state, γ≥1, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes.
In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. Considered as three-dimensional solutions,
undercompressive shocks are Lax-type (Alfven) waves. For the monatomic and diatomic cases γ=5/3 and γ= 7/5,
with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well, both with respect to two-dimensional and (in the neutral sense of convergence to nearby Riemann solutions) three-dimensional perturbations.

关键词: stability of viscous shock waves, Evans function, magnetohydrodynamics

Abstract:

For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with γ-law gas equation of state, γ≥1, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes.
In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. Considered as three-dimensional solutions,
undercompressive shocks are Lax-type (Alfven) waves. For the monatomic and diatomic cases γ=5/3 and γ= 7/5,
with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well, both with respect to two-dimensional and (in the neutral sense of convergence to nearby Riemann solutions) three-dimensional perturbations.

Key words: stability of viscous shock waves, Evans function, magnetohydrodynamics

中图分类号: 

  • 35B35