Abstrakt
The principle purpose of this work is to investigate a "viscous" version of a "simple" but still realistic bi-fluid model described in BRESCH et al. (in: GIGA, NOVOTNY (eds) Handbook of mathematical analysis in mechanics of viscous fluids, 2018) whose "non-viscous" version is derived from physical considerations in Ishii and HIBIKI (Thermo-fluid dynamics of two-phase flow, Springer, Berlin, 2006) as a particular sample of a multifluid model with algebraic closure. The goal is to show the existence of weak solutions for large initial data on an arbitrarily large time interval.
We achieve this goal by transforming the model to a transformed two-densities system which resembles the compressible Navier-Stokes equations, with, however, two continuity equations and amomentum equation endowed with the pressure of a complicated structure dependent on two variable densities. The new "transformed two-densities 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 et al. (J Differ Equ 261:4448-4485, 2016) and Vasseur et al. (JMath Pures Appl 125:247-282, 2019).
We also explain how these techniques can be generalized to a model of mixtures with more than two species. This is the first result on the existence of weak solutions for any realistic multifluid system.