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
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