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Weak and Strong Solutions for A Fluid-Poroelastic-Structure Interaction via A Semigroup Approach

George Avalos, Elena Gurvich, Justin T. Webster

Abstract

A filtration system, comprising a Biot poroelastic solid coupled to an incompressible Stokes free-flow, is considered in 3D. Across the flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. In the inertial, linear, and non-degenerate case, the hyperbolic-parabolic coupled problem is posed through a dynamics operator on an appropriate energy space, adapted from Stokes-Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. $C_0$-semigroup generation for the dynamics operator is obtained with a non-standard maximality argument. The latter employs a mixed-variational formulation in order to invoke the Babuška-Brezzi theorem. The Lumer-Philips theorem yields semigroup generation, and thereby, strong and generalized solutions are obtained. As the dynamics are linear, a standard argument by density obtains weak solutions; we extend this argument to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial Biot-Stokes filtration, we provide a clear elucidation of strong and weak solutions, as well as their regularity through associated estimates.

Weak and Strong Solutions for A Fluid-Poroelastic-Structure Interaction via A Semigroup Approach

Abstract

A filtration system, comprising a Biot poroelastic solid coupled to an incompressible Stokes free-flow, is considered in 3D. Across the flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. In the inertial, linear, and non-degenerate case, the hyperbolic-parabolic coupled problem is posed through a dynamics operator on an appropriate energy space, adapted from Stokes-Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. -semigroup generation for the dynamics operator is obtained with a non-standard maximality argument. The latter employs a mixed-variational formulation in order to invoke the Babuška-Brezzi theorem. The Lumer-Philips theorem yields semigroup generation, and thereby, strong and generalized solutions are obtained. As the dynamics are linear, a standard argument by density obtains weak solutions; we extend this argument to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial Biot-Stokes filtration, we provide a clear elucidation of strong and weak solutions, as well as their regularity through associated estimates.
Paper Structure (24 sections, 4 theorems, 153 equations)

This paper contains 24 sections, 4 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Literature Review and Further Discussion
  3. Notation
  4. Biot Dynamics
  5. Physical Configuration and Coupled Model
  6. Energy Balance
  7. Relevant Definitions and Main Results
  8. Operators and Spaces for the Inertial Biot-Stokes System
  9. Definition of Weak Solutions
  10. Statement of Main Results
  11. Dynamics Operator
  12. Elimination of Fluid-Pressure and Associated Green's Maps
  13. Domain and Properties of ${\mathcal{A}}$
  14. Semigroup Generation: Theorem \ref{['th:main1']}
  15. Dissipativity
  16. ...and 9 more sections

Key Result

Theorem 2.1

The operator $\mathcal{A}$ on $X$ (defined by A, 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 $e^{\ma

Theorems & Definitions (20)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1
  • Remark 2.1
  • Remark 2.2: Regularity and Weak Solutions I
  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 10 more