We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible barotropic Navier-Stokes equations. Our result solves a long-standing problem first mentioned in 1986 by Matsumura and Nishihara in Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. The same authors introduced it officially as an open problem in 1992 in Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas and it was again described as very challenging open problem in 2018 in the survey paper Waves in compressible fluids: viscous shock, rarefaction, and contact waves. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscous shocks, and the energy method used for the stability of rarefactions. Instead of the anti-derivative method, our proof uses the a-contraction with shifts theory recently developed by two of the authors. This method is energy based, and can seamlessly handle the superposition of waves of different kinds. 

　　Publication: 

　　Advances in Mathematics, Volume 419, 15 April 2023, 108963 

　　Author: 

　　Moon-Jin Kang 

　　Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea 

　　Alexis F. Vasseur 

　　Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA 

　　Yi Wang 

　　Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, PR China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China 

　　Email: wangyi@amss.ac.cn