Mathematical Effects of Linear Visco-elasticity in Quasi-static Biot Models

TL;DR

This paper analyzes the mathematical properties of linear poro-visco-elastic systems, demonstrating their regularizing and dissipative effects within quasi-static Biot models using a coupled pressure-displacement framework.

Contribution

It provides a detailed mathematical analysis of linear poro-visco-elastic systems, including well-posedness, a priori estimates, and characterization of initial conditions in the context of quasi-static Biot equations.

Findings

Visco-elasticity introduces regularizing effects in the system.

The system exhibits dissipative behavior due to visco-elastic effects.

Well-posedness and admissible initial conditions are rigorously established.

Abstract

We investigate and clarify the mathematical properties of linear poro-elastic systems in the presence of classical (linear, Kelvin-Voigt) visco-elasticity. In particular, we quantify the time-regularizing and dissipative effects of visco-elasticity in the context of the quasi-static Biot equations. The full, coupled pressure-displacement presentation of the system is utilized, as well as the framework of implicit, degenerate evolution equations, to demonstrate such effects and characterize linear poro-visco-elastic systems. We consider a simple presentation of the dynamics (with convenient boundary conditions, etc.) for clarity in exposition across several relevant parameter ranges. Clear well-posedness results are provided, with associated a priori estimates on the solutions. In addition, precise statements of admissible initial conditions in each scenario are given.