Nonlinear stability of large perturbation around the viscous shock wave for the 2D scalar viscous conservation law

Abstract

In this paper, we are interested in nonlinear stability of large perturbation around the planar shock wave for scalar viscous conservation law in two dimensions. We prove that the problem always admits a global classical solution if the initial perturbation is in L1 ∩ H4(ℝ2). The proof is carried out by classical energy estimates based on the maximal principle and the contraction property. More important, for large perturbation problems of certain types of the 2D scalar viscous conservation law, we obtain nonlinear stability of the shock profile for weak shock, and establish the L2 decay rate t-1/4 and L∞ decay rate t-1/2 of solutions toward the planar shock wave. The idea of the proof uses a technique combining the semigroup approach and the energy method to get some smallness estimates, and then to obtain the asymptotic behavior.