Seminar and Oral Examination
Mathematics/Psychology : 401
Date & Time
December 6, 2021, 11:00 am – 12:00 pm
The mathematical theory of poroelasticity was developed for geoscience applications (e.g., petroleum engineering). More recently, it has been incorporated into biological models, owing to the poroelastic nature of biological tissues. Poroelastic systems, which relate porous structural deformations to saturated fluid pressure, are typically described by Biot's equations. For the parameters of physical interest, a quasi-static approximation leads to dynamics which can be represented as an implicit evolution. Moreover, compressibility in Biot's equations is a significant consideration, and, in the incompressible limit, Biot's model degenerates.
We present a biologically-motivated multilayered system, describing the coupled dynamics of a 3D poroelastic structure, a poroelastic plate, and a free Stokes flow. We will largely focus on related two sub-problems, to gain a better understanding this extremely complex system. First, we give a complete well-posedness analysis of the poroelastic plate utilizing variational tools; secondly, we give a semigroup analysis of a coupled Biot-Stokes problem. The most interesting cases are singular limits, leading us to abstract implicit, degenerate evolutions, of which we provide a brief overview.