Memoirs of the American Mathematical Society 2014; 168 pp; softcover Volume: 234 ISBN10: 1470410168 ISBN13: 9781470410162 List Price: US$89 Individual Members: US$53.40 Institutional Members: US$71.20 Order Code: MEMO/234/1105
Not yet published. 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 NavierStokes 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
