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Vanishing permeability limit of convection in multilayer porous media

Kaijian Sha, Xiaoming Wang

Abstract

We analyze the asymptotic behavior of the Boussinesq-Darcy system describing convection in layered porous media in the limit where the permeability of one layer tends to zero. We show that the limiting dynamics are governed by the Boussinesq-Darcy model with an impermeable layer, both in terms of convergence of solutions in L2 on finite time intervals and convergence of the corresponding global attractors. This limit is singular, as the pressure equation becomes degenerate when the permeability vanishes in part of the domain, resulting in a loss of uniqueness of the pressure in the impermeable layer. This difficulty is resolved by combining uniform estimates in the permeable layers with refined control of the pressure equation in the vanishing-permeability layer. The results provide a rigorous description of the zero-permeability limit in layered porous-media convection models.

Vanishing permeability limit of convection in multilayer porous media

Abstract

We analyze the asymptotic behavior of the Boussinesq-Darcy system describing convection in layered porous media in the limit where the permeability of one layer tends to zero. We show that the limiting dynamics are governed by the Boussinesq-Darcy model with an impermeable layer, both in terms of convergence of solutions in L2 on finite time intervals and convergence of the corresponding global attractors. This limit is singular, as the pressure equation becomes degenerate when the permeability vanishes in part of the domain, resulting in a loss of uniqueness of the pressure in the impermeable layer. This difficulty is resolved by combining uniform estimates in the permeable layers with refined control of the pressure equation in the vanishing-permeability layer. The results provide a rigorous description of the zero-permeability limit in layered porous-media convection models.
Paper Structure (10 sections, 10 theorems, 143 equations)

This paper contains 10 sections, 10 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. The mathematic models
  3. Preliminary results
  4. Sharp interface model with degenerate permeability
  5. Uniform estimate for low permeability
  6. Convergence
  7. Finite-time convergence
  8. Convergence of attractor
  9. Conclusion
  10. An elliptic example with explicit convergence rate

Key Result

Lemma 3.3

For any $\psi \in V$, it holds that Furthermore, we have Here the constant $C$ depends only on $\Omega$ and $D_i$.

Theorems & Definitions (19)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3
  • Proposition 5.1
  • ...and 9 more