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Semigroup Solutions for A Multilayered Filtration System

George Avalos, Galen Richard, Justin T. Webster

TL;DR

This work develops a rigorous operator-theoretic framework for a multilayer FPSI system consisting of Stokes flow above a Biot poroelastic bulk and a mediating 2.5D poroelastic plate. By eliminating the Stokes pressure and formulating a mixed variational resolvent in an auxiliary space, the authors prove that the linear dynamics operator \mathcal{A} generates a \;C_0\;-contraction semigroup on a finite-energy space, yielding strong and mild solutions and enabling nonlinear perturbations via von Kármán plate theory. The results provide a foundation for future stability and spectral analyses, and for extending the model to moving domains and large-deflection plate dynamics, with potential comparisons to Biot-Stokes filtrations without an interface. Overall, the paper advances a robust, verifiable framework for coupled multilayer filtration systems with rigorous well-posedness and pathways to nonlinear and stability investigations.

Abstract

We investigate solutions to a coupled system of partial differential equations that describe a multilayered filtration system. Namely, we study the interaction of a viscous incompressible flow with bulk poroelasticity, via a poroelastic interface. The configuration consists of two 3D toroidal subdomains connected via a plate interface, which permits elastic deformation and perfusive fluid dynamics. The governing dynamics comprise Stokes equations in the bulk fluid region, Biot's equations in the bulk poroelastic region, and the recent poroplate of Mikelić at the interface. Coupling occurs on the top and lower surfaces of the plate, and involves conservation of mass, stress balance, and a certain slip condition for the fluid free-flow. We seek strong (and mild) solutions in the Hilbert space framework via the Lumer-Phillips theorem. The resolvent analysis employs a nonstandard mixed variational formulation which captures the complex, multi-physics coupling at the interface. We explicitly characterize the infinitesimal generator associated to the linear Cauchy problem and establish the generation of a $C_0$-semigroup on a suitably chosen finite-energy space. With the semigroup in hand, we may treat elastic nonlinearities for plate displacements through perturbation theory. These result parallel those for Biot-Stokes filtration systems, and complement the recently established weak solution theory for multilayer filtrations. The agency of the semigroup straightforwardly admits structural (plate) nonlinearity into the dynamics. Future stability and regularity analyses for multilayer filtrations are also made possible by these results, as well as a comparison of spectral and regularity properties between filtration configurations, and the elucidation of the mitigating poroplate dynamics as possibly regularizing and stabilizing.

Semigroup Solutions for A Multilayered Filtration System

TL;DR

This work develops a rigorous operator-theoretic framework for a multilayer FPSI system consisting of Stokes flow above a Biot poroelastic bulk and a mediating 2.5D poroelastic plate. By eliminating the Stokes pressure and formulating a mixed variational resolvent in an auxiliary space, the authors prove that the linear dynamics operator \mathcal{A} generates a \;C_0\;-contraction semigroup on a finite-energy space, yielding strong and mild solutions and enabling nonlinear perturbations via von Kármán plate theory. The results provide a foundation for future stability and spectral analyses, and for extending the model to moving domains and large-deflection plate dynamics, with potential comparisons to Biot-Stokes filtrations without an interface. Overall, the paper advances a robust, verifiable framework for coupled multilayer filtration systems with rigorous well-posedness and pathways to nonlinear and stability investigations.

Abstract

We investigate solutions to a coupled system of partial differential equations that describe a multilayered filtration system. Namely, we study the interaction of a viscous incompressible flow with bulk poroelasticity, via a poroelastic interface. The configuration consists of two 3D toroidal subdomains connected via a plate interface, which permits elastic deformation and perfusive fluid dynamics. The governing dynamics comprise Stokes equations in the bulk fluid region, Biot's equations in the bulk poroelastic region, and the recent poroplate of Mikelić at the interface. Coupling occurs on the top and lower surfaces of the plate, and involves conservation of mass, stress balance, and a certain slip condition for the fluid free-flow. We seek strong (and mild) solutions in the Hilbert space framework via the Lumer-Phillips theorem. The resolvent analysis employs a nonstandard mixed variational formulation which captures the complex, multi-physics coupling at the interface. We explicitly characterize the infinitesimal generator associated to the linear Cauchy problem and establish the generation of a -semigroup on a suitably chosen finite-energy space. With the semigroup in hand, we may treat elastic nonlinearities for plate displacements through perturbation theory. These result parallel those for Biot-Stokes filtration systems, and complement the recently established weak solution theory for multilayer filtrations. The agency of the semigroup straightforwardly admits structural (plate) nonlinearity into the dynamics. Future stability and regularity analyses for multilayer filtrations are also made possible by these results, as well as a comparison of spectral and regularity properties between filtration configurations, and the elucidation of the mitigating poroplate dynamics as possibly regularizing and stabilizing.
Paper Structure (21 sections, 6 theorems, 157 equations, 1 figure)

This paper contains 21 sections, 6 theorems, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Informal Statement of Main Result and Discussion
  3. Novelty of the Work
  4. Comparison with Established Literature
  5. PDE Model and Modeling Background
  6. Spatial Domain and Physical Quantities
  7. Dynamics Model
  8. Energy Identity
  9. The Dynamics Operator
  10. Fluid-Pressure Elimination
  11. Constituent Spaces
  12. Operator and Domain
  13. Precise Statement of Main Results
  14. Dissipativity of the Operator
  15. Maximality
  16. ...and 6 more sections

Key Result

Lemma 4.1

If $q\in H^{0,0,1}(\Omega _{p})$, then $\left. q\right\vert _{\omega _{p}^{+}}$ is well-defined in $H^{-\frac{1}{2}}(\omega _{p}^{+})$. (Mutatis mutandis, for $\omega_p^{-}$.)

Figures (1)

  • Figure 1: Left: 3D domain $\Omega$, including plate pressure $\Omega_p$. Right: 2D vertical cross-section through domain $\Omega$. (Original image from bcmw.)

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Remark 3.1
  • Remark 3.2
  • Definition 3
  • Lemma 4.1
  • proof
  • Definition 4
  • Remark 4.1: Regularity of Traces
  • Theorem 5.1
  • ...and 6 more