# Preprint 2007-008

# Stability of Viscous Shocks on Finite Intervalls

## Gunilla Kreiss, Heinz-Otto Kreiss and Jens Lorenz

**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/ν 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
which reflect the slow change of the end states.
We show that, for the isolated fast process,
the perturbations decay exponentially in time.