Asymptotic stability of planar viscous shock wave to three-dimensional relaxed compressible Navier-Stokes equations

Abstract

This paper establishes the nonlinear time-asymptotic stability of shifted planar viscous shock waves for the three-dimensional relaxed compressible Navier-Stokes equations, in which a modified Maxwell-type model replaces the classical Newtonian constitutive relation. Under the assumptions of sufficiently small shock strength and initial perturbations, we prove that planar viscous shock waves are nonlinearly stable. The main steps of our analysis are as follows. First, using the relative entropy method together with the framework of $a$-contraction with shifts, we derive energy estimates for the weighted relative entropy of perturbations. We then successively obtain high-order and dissipation estimates via direct energy arguments, which provide the required a priori bounds. Combining these estimates with a local existence result, we establish the global asymptotic stability of the shifted planar viscous shock wave. Finally, we show that as the relaxation parameter tends to zero, solutions of the relaxed system converge globally in time to those of the classical Navier-Stokes system.