This paper studies a Cauchy problem for a scalar multidimensional (multi-d) viscous convex conservation law, in which the initial data is a planar viscous shock with a multi-d periodic perturbation. We show that if the wave-strength and perturbation are both small, then the viscous shock is stable in the \(L^\infty ({\mathbb R}^n)\) space with an exponential decay rate. One contribution of the paper is to establish a new framework to study the stability of viscous shocks in multiple dimensions, where the elementary energy method with the antiderivative technique can be used. The idea is to decompose the multi-d perturbation into a one-dimensional function and a multi-d remainder, where the former can well define its antiderivative and the latter satisfies the Poincaré inequality over a periodic domain. 

　　Publication: 

　　SIAM Journal on Mathematical Analysis, Volume 55, Issue 3 (2023) 

　　http://dx.doi.org/10.1137/21M1462453 

　　Author: 

　　Qian Yuan 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. 

　　Email: qyuan@amss.ac.cn