Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations

Abstract

Abstract. The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution timeasymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain RˆTn ́1pn ě 2q, where Tn ́1 is the n ́ 1-dimensional torus.