Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

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

Pages: 1775 – 1813



Anne-Laure Dalibard (Faculté des Sciences et Ingénierie, Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, Paris, France)

Charlotte Perrin (Aix Marseille Université, CNRS, Institut de Mathématiques de Marseille, France)


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.


compressible Navier–Stokes equations, singular limit, free boundary problem, viscous shock waves, nonlinear stability

2010 Mathematics Subject Classification

35L67, 35Q35

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program Grant agreement No. 637653, project BLOC “Mathematical Study of Boundary Layers in Oceanic Motion”. C. P. was partially supported by a CNRS PEPS JCJC grant. This work was supported by the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).

Received 7 February 2019

Accepted 1 April 2020

Published 11 December 2020