Existence and stability of partially congested propagation fronts in a one-dimensional Navier-Stokes model

TL;DR

This paper studies the existence and stability of partially congested traveling wave solutions in a one-dimensional Navier-Stokes model with a singular pressure law, showing convergence to free-congested fronts and nonlinear stability under small perturbations.

Contribution

It introduces a detailed analysis of shock profiles with congestion effects, demonstrating convergence to free-congested solutions and establishing their nonlinear stability.

Findings

Profiles converge to free-congested traveling fronts as pressure singularity vanishes.

Profiles are asymptotically nonlinearly stable under small perturbations.

Quantifies perturbation size relative to pressure singularity.

Abstract

In this paper, we analyze the behavior of viscous shock profiles of one-dimensional compressible Navier-Stokes equations with a singular pressure law which encodes the effects of congestion. As the intensity of the singular pressure tends to 0, we show the convergence of these profiles towards free-congested traveling front solutions of a two-phase compressible-incompressible Navier-Stokes system and we provide a refined description of the profiles in the vicinity of the transition between the free domain and the congested domain. In the second part of the paper, we prove that the profiles are asymptotically nonlinearly stable under small perturbations with zero integral, and we quantify the size of the admissible perturbations in terms of the intensity of the singular pressure.