Weak solutions to Allen-Cahn-like equations modelling consolidation of porous media
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.}{Weak solutions; cross--diffusion system; energy method; Leray--Schauder fixed point theorem; finite differences; consolidation of porous media