ON NONLINEAR STABILITY OF GENERAL UNDERCOMPRESSIVE VISCOUS SHOCK-WAVES

We study the nonlinear stability of general undercompressive viscous shock waves. Previously, the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation, using new pointwise methods. By examining time invariants associated with the linearized equations, we can now overcome a new difficulty in the general case, namely, nonlinear movement of the shock. We introduce a coordinate transformation suitable to treat this new aspect, and demonstrate our method by analyzing a model system of generic type. We obtain sharp pointwise bounds and L(p) behavior of the solution for all p, 1 less than or equal to p less than or equal to infinity.