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Fluid-Structure Interaction with Porous Media: The Beaver-Joseph condition in the strong sense

Tim Binz, Matthias Hieber, Arnab Roy

Abstract

This article considers fluid structure interaction describing the motion of a fluid contained in a porous medium. The fluid is modelled by Navier-Stokes equations and the coupling between fluid and the porous medium is described by the classical Beaver-Joseph or the Beaver-Joseph-Saffman interface condition. In contrast to previous work these conditions are investigated for the first time in the strong sense and it is shown that the coupled system admits a unique, global strong solution in critical spaces provided the data are small enough. Furthermore, a Serrin-type blow-up criterium is developed and higher regularity estimates at the interface are established, which say that the solution is even analytic provided the forces are so.

Fluid-Structure Interaction with Porous Media: The Beaver-Joseph condition in the strong sense

Abstract

This article considers fluid structure interaction describing the motion of a fluid contained in a porous medium. The fluid is modelled by Navier-Stokes equations and the coupling between fluid and the porous medium is described by the classical Beaver-Joseph or the Beaver-Joseph-Saffman interface condition. In contrast to previous work these conditions are investigated for the first time in the strong sense and it is shown that the coupled system admits a unique, global strong solution in critical spaces provided the data are small enough. Furthermore, a Serrin-type blow-up criterium is developed and higher regularity estimates at the interface are established, which say that the solution is even analytic provided the forces are so.
Paper Structure (15 sections, 21 theorems, 114 equations, 1 figure)

This paper contains 15 sections, 21 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the model: Beaver-Joseph and Beaver-Joseph-Saffman conditions
  3. Preliminaries: semilinear evolution equations, function spaces, boundary perturbations, $\mathcal{R}$-sectorial operators
  4. Semilinear evolution equations
  5. Function spaces
  6. Dirichlet operators and perturbation theory for boundary conditions
  7. $\mathcal{R}$-sectorial operators
  8. Main results for Beaver-Joseph-Saffman interface conditions
  9. Main results for Beaver-Joseph interface conditions
  10. The linearized system
  11. Laplacian and Stokes operator with Neumann boundary conditions
  12. The linearized system subject to Beaver-Joseph-Saffman boundary conditions
  13. The linearized equation subject to Beaver-Joseph boundary conditions
  14. Strong well-posedness
  15. Higher regularity

Key Result

Proposition 3.1

Let $u_0 \in X_{\gamma,\mu}$, $f \in \mathbb{E}_{0,\mu}(0,T)$. Suppose Assumption (A) and let $\mu \in (\mu_c,1]$. a) Then there exists $T' = T'(u_0) \in (0,T)$ such that eq:acp-abstract admits a unique solution $u \in \mathbb{E}_{1,\mu}(0,T')$. Moreover, the solution depends continuously on the dat

Figures (1)

  • Figure 1: Fluid-porous media interaction

Theorems & Definitions (41)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 4.1: Strong well-posedness in critical spaces
  • Remark 4.2
  • Corollary 4.3: Finite-in-time-Blow up
  • Corollary 4.4: Higher interior regularity
  • Corollary 4.5: Higher regularity at the interface
  • Remark 4.6
  • ...and 31 more