Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation
Feimin Huang1 and Akitaka Matsumura2
||Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing, China|
||Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan|
Received: 21 February 2008 Accepted: 20 April 2009 Published online: 2 June 2009
Communicated by P. Constantin
In this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible Navier-Stokes equation. By introducing a new linear diffusion wave special to this case, we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small, the unique global solution in time to the full compressible Navier-Stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts (in space) of two viscous shock waves are uniquely determined by the initial perturbations. We then apply the idea to study a half space problem for the full compressible Navier-Stokes equation and obtain a similar result.
Research is supported in part by NSFC Grant No. 10471138, NSFC-NSAF Grant No. 10676037 and 973 project of China, Grant No. 2006CB805902, in part by Japan Society for the Promotion of Science, the Invitation Fellowship for Research in Japan (Short-Term).
Research is supported in part by Grant-in-Aid for Scientific Research (B) 19340037, Japan.