Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient

TL;DR

This paper investigates a complex cross-diffusion system modeling fluid mixtures driven by pressure gradients, establishing global existence of solutions using advanced mathematical techniques.

Contribution

It introduces a novel analysis of a coupled parabolic-hyperbolic system for fluid mixtures, proving global existence of solutions under realistic boundary conditions.

Findings

Proved global-in-time existence of classical solutions.

Decomposed the system into porous-medium and transport equations.

Applied parabolic regularity and renormalized solutions theory.

Abstract

The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy's law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.