Weak Solutions and Inertial Limits for Quasi-static Filtrations

TL;DR

This paper develops a mathematical framework for weak solutions in quasi-static filtration systems involving poroelastic solids and free-flow, addressing a complex degenerate limit problem.

Contribution

It introduces a viscoelastic regularization approach to construct weak solutions in the inertial limit, enabling analysis of degenerate filtration models.

Findings

Constructed weak solutions via regularization and limiting processes.

Addresses an open singular limiting problem in filtration models.

Lays groundwork for analyzing nonlinear poroelastic effects.

Abstract

A quasi-static filtration system, comprising a poroelastic solid coupled to an incompressible free-flow, is considered in 3D. Across a flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. The system constitutes a doubly elliptic-parabolic coupling and can be seen as a degenerate case of the inertial Biot-Stokes dynamics of recent interest. These dynamics cannot be easily recovered through a vanishing inertia limit, however, utilizing a viscoelastic regularization of the inertial Biot system allows us to construct weak solutions in the inertial limit; subsequently, we pass to the limit in the regularization parameter to obtain quasi-static weak solutions. This addresses an open singular/degenerate limiting problem in filtrations, and allows for subsequent analysis of uniqueness and regularity. This also provides a foundation for the incorporation of physically-motivated nonlinear poroelastic effects.