Abstract

We resolve the issue of uniqueness of weak solutions for linear, inertial fluid-poroelastic-structure interactive dynamics. The model we study comprises a three-dimensional Biot poroelastic system coupled to a three-dimensional incompressible Stokes flow via a two-dimensional interface, where kinematic, stress-matching, and tangential-slip conditions are prescribed. Previous work provided a construction of weak solutions, these satisfying an associated 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 (Lamé) 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 nondegenerate cases separately, two different approaches are utilized. In the former, energy estimates are obtained for arbitrary weak solutions through systematic decoupling of the constituent dynamics, and well-posedness of weak solutions is inferred. In the latter case, an abstract semigroup approach due to Ball is utilized which relies on 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 for weak solutions in hyperbolic-parabolic systems.

Keywords

Mathematical Subject Classification

Milestones

Authors