Weak Solutions in Nonlinear Poroelasticity with Incompressible Constituents

TL;DR

This paper establishes the existence and uniqueness of weak solutions for a nonlinear, degenerate poroelastic system with incompressible constituents, relevant to biomechanics and tissue perfusion, using a novel fixed point approach.

Contribution

It introduces a fixed point strategy and a new uniqueness proof for weak solutions in nonlinear poroelasticity with incompressible constituents.

Findings

Proved existence of weak solutions for the system.

Established uniqueness of weak solutions via novel energy estimates.

Laid groundwork for analyzing strong solutions and further uniqueness results.

Abstract

We consider quasi-static poroelastic systems with incompressible constituents. The nonlinear permeability is taken to be dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure are considered. Such dynamics are motivated by applications in biomechanics and, in particular, tissue perfusion. This system represents a nonlinear, implicit, degenerate evolution problem. We provide a direct fixed point strategy for proving the existence of weak solutions, which is made possible by a novel result on the uniqueness of weak solution to the associated linear system (the permeability a given function of space and time). The linear uniqueness proof is based on novel energy estimates for arbitrary weak solutions, rather than just for constructed solutions (as limits of approximants). The results of this work provide a foundation for addressing strong solutions, as well uniqueness of weak solutions for the nonlinear porous media system.