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Analysis Seminar
Stability of viscous shocks on bounded intervals
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Consider a viscous conservation law 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. 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, which reflect the slow change of the end states. We show that for this problem the perturbation decays exponentially fast in time. |