Weak Solutions for a Poro-elastic Plate System

TL;DR

This paper develops a mathematical framework for weak solutions to a simplified poro-elastic plate model derived from the Biot system, addressing non-autonomous and degenerate cases with potential applications in related models.

Contribution

It introduces a novel weak solution theory for a coupled poro-elastic plate system with time-dependent permeability, extending existing methods to non-autonomous, degenerate problems.

Findings

Existence of weak solutions established.

Uniqueness under regularity conditions.

Analysis of inertial case via semigroup theory.

Abstract

We consider a recent plate model obtained as a scaled limit of the three dimensional Biot system of poro-elasticity. The result is a "2.5" dimensional linear system that couples traditional Euler-Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We alow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions to implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate, and exposits a variety of interesting related models and associated analytical investigations.