$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation

Abstract

We consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front. However, for the transversal directions, we do not have any restrictions on flux function. We first show the contraction property for any large perturbations in $L^2$ of the planar viscous shock. If the initial $L^2$-perturbation is also in $L^1$, the large perturbation converges to zero in $L^2$ as time goes to infinity with $t^{-1/4}$ decay rate. The contraction and decay estimates hold up to dynamical shift. For the results, we do not impose any smallness conditions on the initial value. This result extends the 1D case \cite{Kang-V-1} by the first author and Vasseur to the multi-dimensional case.