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Shock Waves in Conservation Laws with Physical Viscosity
Tai-Ping Liu, Institute of Mathematics, Academia Sinica, Taipei, Taiwan, and Stanford University, CA, and Yanni Zeng, University of Alabama at Birmingham, AL
Memoirs of the American Mathematical Society
2014; 168 pp; softcover
Volume: 234
ISBN-10: 1-4704-1016-8
ISBN-13: 978-1-4704-1016-2
List Price: US$89
Individual Members: US$53.40
Institutional Members: US$71.20
Order Code: MEMO/234/1105
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Expected publication date is March 3, 2015.

The authors study the perturbation of a shock wave in conservation laws with physical viscosity. They obtain the detailed pointwise estimates of the solutions. In particular, they show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small but independent. The authors' assumptions on the viscosity matrix are general so that their results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. The authors' analysis depends on accurate construction of an approximate Green's function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that the author can close the nonlinear term through Duhamel's principle.

Table of Contents

  • Introduction
  • Preliminaries
  • Green's functions for systems with constant coefficients
  • Green's function for systems linearized along shock profiles
  • Estimates on Green's function
  • Estimates on crossing of initial layer
  • Estimates on truncation error
  • Energy type estimates
  • Wave interaction
  • Stability analysis
  • Application to magnetohydrodynamics
  • Bibliography
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