Uniqueness of Weak Solutions for Biot-Stokes Interactions

TL;DR

This paper establishes the uniqueness of weak solutions for a complex fluid-poroelastic-structure interaction model involving a 3D Biot system coupled with Stokes flow, addressing challenges posed by low regularity and dynamic coupling.

Contribution

It introduces two novel approaches to prove uniqueness for weak solutions in hyperbolic-parabolic coupled systems, including energy estimates and an abstract semigroup method.

Findings

Proved uniqueness of weak solutions for the coupled Biot-Stokes system.

Developed energy estimates through decoupling of dynamics.

Applied semigroup theory for the hyperbolic component.

Abstract

We resolve the issue of uniqueness of weak solutions for linear, inertial fluid-poroelastic-structure coupled dynamics. The model comprises a 3D Biot poroelastic system coupled to a 3D incompressible Stokes flow via a 2D interface, where kinematic, stress-matching, and tangential-slip conditions are prescribed. Our previous work provided a construction of weak solutions, these satisfying an associated finite energy inequality. However, several well-established issues related to the dynamic coupling, hinder a direct approach to obtaining uniqueness and continuous dependence. In particular, low regularity of the hyperbolic (Lam\'e) component of the model precludes the use of the solution as a test function, which would yield the necessary a priori estimate. In considering degenerate and non-degenerate cases separately, we utilize two different approaches. In the former, energy estimates are obtained for arbitrary weak solutions through a systematic decoupling of the constituent dynamics, and well-posedness of weak solutions is inferred. In the latter case, an abstract semigroup approach is utilized to obtain uniqueness via a precise characterization of the adjoint of the dynamics operator. The results here can be adapted to other systems of poroelasticity, as well as to the general theory of weak solutions for hyperbolic-parabolic coupled systems.