Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas

This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states at x=±∞, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant γ satisfies 1≦γ≦2, then the solution is proved to tend to the rarefaction wave as t→∞ under no smallness conditions of both the difference of asymptotic values at x=±∞ and the initial data. The proof is given by an elementary L²-energy method.