Finite-energy solutions for compressible two-fluid Stokes system

TL;DR

This paper establishes the existence of global weak solutions for a complex compressible two-fluid Stokes system with a single velocity field, using innovative compactness and approximation techniques outside classical methods.

Contribution

It introduces a novel approach to prove weak solution existence for a two-fluid system with algebraic pressure law, extending beyond traditional Lions-Feireisl frameworks.

Findings

Proved weak sequential stability of solutions.

Constructed weak solutions via Lagrangian and truncation methods.

Demonstrated solutions exist globally in time.

Abstract

We prove existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions-Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier-Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via unconventional approximation using the Lagrangian formulation, truncations and stability result of trajectories for rough velocity fields.