Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations

We prove the time-asymptotic stability of composite waves consisting superposition a viscous shock and rarefaction for one-dimensional compressible barotropic Navier-Stokes equations. Our result solves long-standing problem first mentioned in 1986 by Matsumura Nishihara [28]. The same authors introduced it officially as an open 1992 [29] was again described very challenging 2018 survey paper [26]. main difficulty is due to incompatibility standard anti-derivative method, used study shocks, energy method rarefactions. Instead our proof uses a-contraction with shifts theory recently developed two authors. This based, can seamlessly handle different kinds.