Asymptotic stability of composite waves of two viscous shocks for relaxed compressible Navier-Stokes equations

This paper investigates the time asymptotic stability of composite waves formed by two shock waves within the context of one-dimensional relaxed compressible Navier-Stokes equations. We demonstrate that the composite waves consisting of two viscous shocks achieve asymptotic nonlinear stability under the condition of having two small, independent wave strengths and the presence of minor initial perturbations. Furthermore, the solutions of the relaxed system are observed to globally converge over time to those of the classical system as the relaxation parameter approaches zero. The methodologies are grounded in relative entropy, the $a$-contraction with shifts theory and fundamental energy estimates.