The method of $a$-contraction with shifts used for long-time behavior toward viscous shock

TL;DR

This paper refines the $a$-contraction method with shifts for analyzing the long-time behavior of viscous shocks in barotropic Navier-Stokes flows, avoiding the use of the effective velocity variable, which is beneficial for more complex systems.

Contribution

It introduces a new approach to the $a$-contraction method that does not rely on the effective velocity variable, simplifying analysis for systems with additional effects.

Findings

Enhanced understanding of shock stability in Navier-Stokes flows.

Method applicable to systems with capillary and boundary effects.

Potential for broader application in fluid dynamics stability analysis.

Abstract

We revisit the method of $a$-contraction with shifts used for long-time behavior of barotropic Navier-Stokes flows perturbed from a Riemann shock. For the usage of the method of $a$-contraction with shifts, we do not employ the effective velocity $h$ variable even for higher order estimates. This approach would be important when handling the barotropic Navier-Stokes system with other effects, for example, such as capillary effect and boundary effect.