DE Seminar: Peter Lavagnino

Abstract: We analyze and discuss several solution techniques and PDE well-posedness results pertaining to the Biot model of poroelaticity. In the existing literature, the model has been analyzed from the point of view of semigroup theory, in which semigroup generation gives strong and mild/generalized solutions. Here we discuss the use of implicit semigroups pertinent to generalizing these techniques. The model is given by a coupled poroelastic system of equations comprising a parabolic fluid pressure and a hyperbolic (inertial) or elliptic (quasi-static) system of elasticity. These equations model the displacements of the porous matrix and the fluid and thus the system is said to be of mixed type describing behavior of a deformable saturated porous medium. It should be noted that the parabolic fluid pressure equation can degenerate under certain conditions, and these will be analyzed in various settings. With this in mind, we present the various regimes in which the model can be studied naturally depending upon the inertial and compressibility parameters. Several cases arise for the dynamics however, no coherent account exists in the literature characterizing them. Furthermore, solutions for the mildly degenerate case are not found in the literature. With this motivation, we seek to provide a full methodology for documenting and preserving solutions in each case. Any existing relevant theory and solution techniques for such cases will be provided. Finally, we establish the existence of weak solutions for the mildly degenerate case so that it may be adopted into the current body of standardized approaches.