Splitting scheme for the stability of strong shock profile

In this paper, we use the Laplace transform and Dirichlet–Neumann map to give a systematical scheme to study the small wave perturbation of general shock profile with general amplitude. Here we use certain non-classical shock waves for a rotationally invariant system of viscous conservation laws to demonstrate this scheme. We derive an explicit solution and show that it converges pointwise to another over-compressive profile exponentially, when the perturbations of the initial data to a given over-compressive shock profile are sufficiently small.