The Existence, uniqueness, and regularity of weak solutions for a thermodynamically consistent two-phase flow model in porous media

TL;DR

This paper establishes the existence, uniqueness, and regularity of weak solutions for a thermodynamically consistent two-phase flow model in porous media, using advanced mathematical techniques and PDE theory.

Contribution

It provides a rigorous mathematical proof of solution properties for a recent two-phase flow model, including new approximation methods and regularity results.

Findings

Existence of weak solutions for the model.

Uniqueness of solutions under certain smoothness conditions.

Regularity of solutions using elliptic PDE theory.

Abstract

Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model proposed in [25,35]. To this end, firstly, we introduce a fully implicit time semi-discrete approximation and a fully discrete approximation for an appropriate weak formulation of the thermodynamically consistent model. Next, by using the zeros of a vector field theorem, we prove the existence of the weak solution for the fully discrete approximation. Then the existence of weak solutions for the fully implicit time semi-discrete approximation and the weak formulation of the model are derived by the weak convergence technique and the energy stability estimate. Subsequently, by the Gr{\" o}nwall inequality, we prove the uniqueness result under the smoothness assumption on the chemical potential. Finally, combined with the regularity theory of elliptic partial differential equations (PDE), the regularity of the weak solution for the model with complete Neumann boundary conditions is established.