Semigroup Methods for Poroelastic Multi-Physics Systems Describing Biological Tissues

Abstract

This thesis presents novel work in the mathematical theory of poroelasticity, which was first phenomenologically developed by Biot and Zenisek during the mid-last century. The theory relates saturated porous structural deformations to fluid pressure changes within, and blossomed through many applications in the geosciences (e.g., seismic and petroleum engineering). At the outset of the 21st century, poroelasticity proved to be a revolutionary incorporation to the biological fields (e.g. biomedical engineering, arterial stents, scaffolding), owing to the poroelastic nature of biological tissues. For the parameters of physical interest, a quasi-static approximation induces dynamics, which can be represented as an implicit evolution. Moreover, compressibility in Biot's equations is a significant consideration. In the incompressible limit, Biot’s model degenerates This dissertation will present a biologically motivated multilayered system, composed of the coupled dynamics of a 3D poroelastic structure, a poroelastic plate, and an incompressible free Stokes flow. We propose two constituent sub-problems, to gain a better understanding of this extremely complex system. First, a complete well-posedness analysis of the poroelastic plate is shown utilizing variational tools. Secondly, Biot-Stokes filtration is proposed with Beavers-Joseph-Saffman coupling conditions on a fixed 2D interface. A semigroup approach is used to bypass the issues with mismatched trace regularities on the interface; thus guaranteeing strong and generalized solutions. Then the existence of weak solutions, including the degenerate case, is provided by argument by density. The most interesting cases are singular limits, which lead to the use of the theory of abstract implicit, degenerate evolutions, of which the appendix supplies a brief overview. Thus, this thesis provides a clear elucidation of strong solutions and the construction of weak solutions for inertial linear Biot-Stokes filtration systems and uniquely for a poroelastic plate, as well as their regularity through associated estimates.