Asymptotics toward one-dimensional rarefaction wave for the solution of two-dimensional compressible Euler equation with an artificial viscosity

Abstract

This paper is concerned with the asymptotic behavior toward one-dimensional rarefaction wave for the solution of two-dimensional compressible Euler equation with an artificial viscosity. It is shown that if the initial data are suitably close to a constant state and their asymptotic values at x=±∞ are chosen so that the Riemann problem for the corresponding one-dimensional hyperbolic system admits the weak rarefaction wave, then the solution is proved to tend toward the one-dimensional rarefaction wave as t→+∞. The proof is given by using stability results of one-dimensional rarefaction wave and an elementary L² energy method.