Poro-elastic dynamics, in the case of 3D bulk elasticity as well as 2D plate models, have received recent interest in their application to biological models of tissues. The intrinsic nature of most biological tissues is both porous and elastic, whose dynamics suggests a filtration system---that is, a 3D incompressible Stokes flow running adjacent to a saturated poro-elastic structure (governed by inertial Biot equations). The two dynamics are coupled at a 2D interface by the Beavers-Joseph-Saffman (slip-type) conditions. We demonstrate semigroup well-posedness of this filtration system via a Lumer-Philips approach in a laterally periodic box. Notably, we utilize an approach of Avalos and Triggiani for the elimination of Stokes pressure, which includes a delicate contribution to the definition of the generator. In particular, demonstrating maximality brings about a mixed, variational problem, which is addressed by the LBB conditions. Time-permitting, we will discuss the construction of weak solutions, including in the biologically-motivated and degenerate case of quasi-static dynamics with incompressible constituents.
Semigroup Generation for a Coupled Biot-Stokes Filtration System
Elena Gurvich, UMBC
Authors: Elena Gurvich, Justin Webster
2023 AWM Research Symposium
Early Career Researchers in Mathematical Biology and Differential Equations [Organized by Rayanne Luke, Sarah Strikwerda, and Prajakta Bedekar]