Weak solutions to Allen-Cahn-like equations modelling consolidation of porous media

TL;DR

This paper establishes the existence of weak solutions for a system of Allen-Cahn-like equations modeling the consolidation process in saturated porous media, using energy estimates, fixed point theory, and numerical schemes.

Contribution

It introduces a novel approach combining energy estimates, fixed point methods, and finite difference schemes to analyze weak solutions of coupled Allen-Cahn-like equations in porous media.

Findings

Existence of weak solutions via energy estimates and fixed point theory

Local existence of strong solutions for a regularized system

Numerical demonstration of solution negativity

Abstract

We study the weak solvability of a system of coupled Allen-Cahn-like equations resembling cross-diffusion which is arising as a model for the consolidation of saturated porous media. Besides using energy like estimates, we cast the special structure of the system in the framework of the Leray-Schauder fixed point principle and ensure this way the local existence of strong solutions to a regularised version of our system. Furthermore, weak convergence techniques ensure the existence of weak solutions to the original consolidation problem. The uniqueness of global-in-time solutions is guaranteed in a particular case. Moreover, we use a finite difference scheme to show the negativity of the vector of solutions.