Stability of viscous scalar shock fronts in several dimensions

We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation. In this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions. These multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts (Lu) and in shock waves with phase changes. The proof here is a rigorous version of arguments that are used to derive effective equations (such as the Kuramoto-Sivashinsky equation) to describe the behavior of fronts. The one dimensional stability for scalar conservation laws was proven by II ' in and Oleinik (IO) using the "integrated equation" (see below) and a max- imum principle. Another proof, based on weighted norms and spectral theory for the linearized problem, was given by Sattinger (S). The multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by Kawashima and Mat- sumura (KM) and Goodman (Go) and completed by Liu (Li). These proofs use L2 energy estimates for the integrated equation. We consider equations of the form (!) ut + f(u)x + g(u)y=:uxx + uyy, where f(u) is a strictly convex function of u :