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.
Lavagnino et al. (Thu,) studied this question.