Compressible fluids and active potentials

TL;DR

This paper analyzes one-dimensional compressible fluid models with degenerate diffusion, proving solution smoothness and existence results, especially when the active potential influences the momentum equation.

Contribution

It introduces a unified approach using Bresch-Desjardins entropy to establish solution regularity and existence for models with active potentials.

Findings

Solutions remain smooth when diffusion coefficients are non-vanishing.

Global existence results are established for the models.

The active potential plays a key role in the analysis.

Abstract

We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.