Weak solutions for some compressible multicomponent fluid models

TL;DR

This paper proves the existence of weak solutions for a viscous bi-fluid model, extending mathematical techniques from compressible Navier-Stokes equations to complex multifluid systems with multiple densities.

Contribution

It introduces a novel approach to establish weak solutions for a realistic multifluid model with two densities, using adapted Lions–Feireisl and DiPerna–Lions methods.

Findings

Existence of weak solutions for large initial data.

Transformation of the multifluid model to an academic system similar to Navier–Stokes.

Potential generalization to models with more than two species.

Abstract

The principle purpose of this work is to investigate a "viscous" version of a "simple" but still realistic bi-fluid model described in [Bresch, Desjardin, Ghidaglia, Grenier, Hillairet] whose "non-viscous" version is derived from physical considerations in \cite[Ishii, Hibiki]{ISHI} as a particular sample of a multifluid model with algebraic closure. The goal is to show existence of weak solutions for large initial data on an arbitrarily large time interval. We achieve this goal by transforming the model to an academic system which resembles to the compressible Navier-Stokes equations, with however two continuity equations and a momentum equation endowed with pressure of complicated structure dependent on two variable densities. The new "academic system" is then solved by an adaptation of the Lions--Feireisl approach for solving compressible Navier--Stokes equation, completed with several observations related to the DiPerna--Lions transport theory inspired by [Maltese, Michalek, Mucha, Novotny, Pokorny, Zatorska] and [Vasseur, Wen, Yu]. We also explain how these techniques can be generalized to a model of mixtures with more then two species. This is the first result on the existence of weak solutions for any realistic multifluid system.