Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions

Abstract

This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave r ( x t ) r(\frac {x}{t}) connecting u + u_{+} and u − u_{-} for the scalar viscous conservation law in two space dimensions. We assume that the initial data u 0 ( x , y ) u_{0}(x,y) tends to constant states u ± u_{\pm } as x → ± ∞ x \rightarrow \pm \infty , respectively. Then, the convergence rate to r ( x t ) r(\frac {x}{t}) of the solution u ( t , x , y ) u(t,x,y) is investigated without the smallness conditions of | u + − u − | |u_{+}-u_{-}| and the initial disturbance. The proof is given by elementary L 2 L^{2} -energy method.