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

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.Weak solutions; cross-diffusion system; energy method; Leray-Schauder fixed point theorem; finite differences; consolidation of porous media

On degenerate coupled transport processes in porous media with memory phenomena

“…In [], the authors studied a model of specific structure of a heat and mass transfer arising from textile industry and proved the global existence for one‐dimensional problems in [] and three‐dimensional problems in []. In [], the authors proved the existence of the weak solutions to systems modeling the consolidation of saturated porous media. In [], the author proved the local existence of weak solutions to degenerate quasilinear problems, where the coefficient function in front of the time derivative may vanish at a set of zero measure.…”

“…In [28,29,30], the authors studied a model of specific structure of a heat and mass transfer arising from textile industry and proved the global existence for one-dimensional problems in [28,29] and three-dimensional problems in [30]. In [21], the authors proved the existence of the weak solutions to systems modeling the consolidation of saturated porous media.…”