Nonlinear stability of semidiscrete shock waves
by
Sylvie Benzoni-Gavage
Maply, Lyon I University
Coauthors: Frédéric Rousset

Semidiscrete shocks are traveling wave solutions of conservative Lattice Dynamical Systems -- obtained through space discretization of conservation laws. The existence of semidiscrete shocks -- of small strength -- can be shown by a center manifold argument for mixed type functional differential equations. The talk concerns the orbital stability of these semidiscrete shock waves. The now well-known result of Chow, Mallet-Paret and Shen (JDE 1998) does not apply to those waves, by lack of a spectral gap. However, their approach is connected to an alternative, PDE-type, approach. Taking advantage of the interplay between the two points of view, it is possible to derive a Green's function for the linearized problem about a reference wave. As in the works of Zumbrun et al. concerning continuous shock waves -- viscous or relaxation shocks--, our Green's function can be decomposed and estimated in a way that leads to nonlinear stability. Here the main difficulty is to take care of the infinite dimensions in estimating the Green's function. This is why the analysis is for the moment limited to the upwind scheme, for which we do have an Evans function.

Date received: May 21, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-58.