Weak-strong uniqueness for bi-fluid compressible system with algebraic closure

TL;DR

This paper proves the weak-strong uniqueness for a two-fluid compressible viscous system with algebraic pressure closure, using the relative entropy method to handle complex nonlinear terms.

Contribution

It establishes the weak-strong uniqueness principle for a bi-fluid system with algebraic pressure closure, addressing nonlinear challenges in the relative entropy framework.

Findings

Weak-strong uniqueness is proven for the bi-fluid system.

The relative entropy method is successfully applied to a complex nonlinear system.

The analysis handles additional nonlinear terms in the transport equation.

Abstract

We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotn\'{y} and Pokorn\'{y} [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech., 2022]. In this paper, we establish the weak-strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase $+$ obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.