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

TL;DR

This paper proves the nonlinear stability of planar viscous shock waves in a 3D relaxed compressible Navier-Stokes system with a modified Maxwell model, and shows convergence to classical Navier-Stokes solutions as relaxation vanishes.

Contribution

It establishes the global nonlinear asymptotic stability of shock waves in a relaxed 3D Navier-Stokes model using energy methods and relative entropy, including relaxation limit analysis.

Findings

Planar viscous shock waves are nonlinearly stable under small perturbations.

Solutions of the relaxed system converge to classical Navier-Stokes solutions as relaxation parameter tends to zero.

Energy estimates and relative entropy methods are effective for stability analysis.

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.