Existence analysis of a single-phase flow mixture model with van der Waals pressure

TL;DR

This paper establishes the global existence and long-term behavior of solutions for a single-phase fluid mixture model with van der Waals pressure, combining thermodynamic consistency and mathematical analysis.

Contribution

It introduces a thermodynamically consistent model for fluid mixtures with van der Waals pressure and proves global existence, convergence, and a minimum principle for the system.

Findings

Global-in-time existence of weak solutions is proved.

Solutions converge exponentially fast to equilibrium in 2D.

Numerical examples confirm theoretical convergence results.

Abstract

The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the mass densities. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der Waals equation of state for mixtures. The model consists of parabolic equations with cross diffusion with a hypocoercive diffusion operator. The global-in-time existence of weak solutions in a bounded domain with equilibrium boundary conditions is proved, extending the boundedness-by-entropy method. Based on the free energy inequality, the large-time convergence of the solution to the constant equilibrium mass density is shown. For the two-species model and specific diffusion matrices, an integral inequality is proved, which reveals a minimum principle for the mass fractions. Without mass diffusion, the two-dimensional pressure is shown to converge exponentially fast to a constant. Numerical examples in one space dimension illustrate this convergence.