Weak solutions for the Stokes system for compressible fluids with general pressure

TL;DR

This paper establishes the existence and uniqueness of global weak solutions for the compressible Stokes system with general pressure, using Lagrangian formulation and advanced flow analysis techniques.

Contribution

It introduces a novel approach to prove uniqueness of solutions for compressible Stokes systems with non-monotone pressure using Lagrangian transformations and BMO space properties.

Findings

Proved existence of global weak solutions for the system.

Established uniqueness of solutions under non-monotone pressure.

Developed a method based on regular Lagrangian flows and weighted flow analysis.

Abstract

We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the transformation to the original Eulerian coordinates. For nonnegative and bounded initial density the solution is also nonnegative for all $t$ and belongs to $L^\infty([0,\infty)\times\mathbb{T}^d)$. A key point of our considerations is the uniqueness of such transformation. Since the velocity might not be Lipschitz continuous, we develop a method which relies on the results of Crippa \& De Lellis, concerning regular Lagriangian flows. The uniqueness is obtained thanks to the application of a certain weighted flow and detail analysis based on the properties of the $BMO$ space.