Local and global well-posedness of one-dimensional free-congested equations

TL;DR

This paper analyzes a one-dimensional congestion model with two phases, establishing local well-posedness for large perturbations and global stability for small ones, including asymptotic stability of traveling waves.

Contribution

It proves local well-posedness for large initial perturbations and global stability for small perturbations in a two-phase congestion model with a free boundary.

Findings

Local well-posedness in weighted Sobolev spaces for large perturbations.

Global stability and asymptotic stability of traveling waves for small perturbations.

Introduction of a new unknown with better estimates for stability analysis.

Abstract

This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics. We investigate the Cauchy problem for initial data which are small perturbations in the non-congested zone of travelling wave profiles. We prove two different results. First, we show that for arbitrarily large perturbations, the Cauchy problem is locally well-posed in weighted Sobolev spaces. The solution we obtain takes the form (vs, us)(t, x -- x(t)), where x < x(t) is the congested zone and x > x(t) is the non-congested zone. The set {x = x(t)} is a free surface, whose evolution is coupled with the one of the solution. Second, we prove that if the initial perturbation is sufficiently small, then the solution is global. This stability result relies on coercivity properties of the linearized operator around a travelling wave, and on the introduction of a new unknown which satisfies better estimates than the original one. In this case, we also prove that travelling waves are asymptotically stable.