On the stability of viscous shock fronts for certain conservation laws in two-dimensional space

Abstract

We prove nonlinear stability of planar shock front solutions for certain viscous scalar conservation laws in two space dimensions. Let us admit the planar shock front solution in x-direction, where the flux function is only in the x-direction and its convexity is not necessarily assumed. For either the nondegenerate or degenerate cases, if the initial disturbance is sufficiently small, then the solution approaches to the shifted planar shock front solution as t$ \rightarrow \infty$. Here, the shift function may have different asymptotic states in $y$-direction. The proofs are given by applying an elementary weighted energy method to the ``integrated equation" which is equivalent to the original one.