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Uniqueness of Weak Solutions for Biot-Stokes Interactions

George Avalos, Justin T. Webster

TL;DR

This work resolves the longstanding question of uniqueness for weak solutions to the fully dynamic Biot-Stokes FPSI system across a 2D interface. It develops two complementary proofs: a degenerate-case argument at $c_0=0$ using hyperbolic-regularization and Temam-type regularity to decouple components, and a nondegenerate-case argument at $c_0>0$ based on Ball's semigroup framework by explicitly characterizing the adjoint $ ext{A}^*$. Together with prior weak-existence results and an energy inequality, the paper establishes uniqueness and, as a corollary, continuous dependence on data for all $c_0\, ext{with}\, c_0\, obreak o 0$. The analysis provides a robust framework for weak solutions of hyperbolic-parabolic coupled systems (and potential extensions to other poroelastic couplings) and clarifies the role of fluid content $oldsymbol{ abla}oldsymbol{ abla} eq 0$ via the Biot content $oldsymbol{ abla}oldsymbol{u}$. The results yield rigorous well-posedness that supports future investigations into nonlinear elastodynamics and more complex interface couplings in FPSI models.

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 (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 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.

Uniqueness of Weak Solutions for Biot-Stokes Interactions

TL;DR

This work resolves the longstanding question of uniqueness for weak solutions to the fully dynamic Biot-Stokes FPSI system across a 2D interface. It develops two complementary proofs: a degenerate-case argument at using hyperbolic-regularization and Temam-type regularity to decouple components, and a nondegenerate-case argument at based on Ball's semigroup framework by explicitly characterizing the adjoint . Together with prior weak-existence results and an energy inequality, the paper establishes uniqueness and, as a corollary, continuous dependence on data for all . The analysis provides a robust framework for weak solutions of hyperbolic-parabolic coupled systems (and potential extensions to other poroelastic couplings) and clarifies the role of fluid content via the Biot content . The results yield rigorous well-posedness that supports future investigations into nonlinear elastodynamics and more complex interface couplings in FPSI models.

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 (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 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.

Paper Structure

This paper contains 12 sections, 7 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Approaches to Uniqueness
  3. Notation
  4. Biot-Stokes Model and Spatial Domain
  5. Relevant Definitions and Past Result
  6. Operators and Spaces
  7. Definition of Weak Solutions and Discussion
  8. Definition of the Semigroup Generator $\mathcal{A}$: $c_0>0$
  9. Statement of Previous Results oldpaper
  10. Main Result and Proof
  11. Proof for $c_0=0$ Case: Hyperbolic Regularization
  12. Proof for $c_0>0$ Case: Ball's Method

Key Result

Theorem 2.1

The operator $\mathcal{A}$ on $X$,defined by diffaction, with domain $\mathcal{D}(\mathcal{A})$ given in Definition diffdomain, is the generator of a strongly continuous semigroup $\{e^{{\mathcal{A}} t}: t\geq 0\}$ of contractions on $X$. Thus, for $\mathbf y_0 \in \mathcal{D}(\mathcal{A})$, we have

Theorems & Definitions (17)

  • Definition 1
  • Remark 2.1
  • Definition 2: Domain of ${\mathcal{A}}$
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 3.1
  • Corollary 3.2
  • proof : Proof of Theorem \ref{['th:main']} with $c_0=0$
  • Proposition 3.3
  • ...and 7 more