L2 decay for large perturbations of viscous shocks for multi-D Burgers equation

In this paper, 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 [Formula: see text] of the planar viscous shock. If the initial [Formula: see text]-perturbation is also in [Formula: see text], the large perturbation converges to zero in [Formula: see text] as time goes to infinity with [Formula: see text] 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 [M.-J. Kang and A. F. Vasseur, [Formula: see text]-contraction for shock waves of scalar viscous conservation laws, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017) 139–156] by the first author and Vasseur to the multi-dimensional case.