Stability of Viscous Shocks on Finite Intervals

Abstract

Consider the Cauchy problem for a system of viscous conservation laws with a solution consisting of a thin, viscous shock layer connecting smooth regions. We expect the time-dependent behavior of such a solution to involve two processes. One process consists of the large-scale evolution of the solution. This process is well modeled by the corresponding inviscid equations. The other process is the adjustment in shape and position of the shock layer to the large-scale solution. The time scale of the second process is much faster than the first, 1/v compared to 1. The second process can be divided into two parts, adjustment of the shape and of the position. During this adjustment the end states are essentially constant. In order to answer the question of stability we have developed a technique where the two processes can be separated. To isolate the fast process, we consider the region in the vicinity of the shock layer. The equations are augmented with special boundary conditions that reflect the slow change of the end states. We show that, for the isolated fast process, the perturbations decay exponentially in time.