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Vanishing layer thickness limit of convection in multilayer porous media

Kaijian Sha, Xiaoming Wang

TL;DR

The paper rigorously justifies replacing a vanishing-thickness layer in a multilayer porous medium with a reduced model having one fewer layer. It achieves this via uniform-in-thickness estimates that yield strong $L^{2}$-convergence on finite time intervals and by proving upper semi-continuity of global attractors as the thin layer thickness tends to zero. A finite-time convergence rate ε^{1/4} is established, and attractor convergence is proved using absorbing-set theory. These results provide a solid mathematical justification for thin-layer reductions in nonlinear convection problems and lay groundwork for further extensions such as curved interfaces or simultaneous vanishing of thickness and permeability.

Abstract

Within the Darcy-Boussinesq framework for convection in multilayered porous media, we investigate the singular limit in which the thickness of one layer tends to zero. We establish that the solution of the full system converges to that of the corresponding limiting model with one fewer layer. The convergence is established in two complementary senses: (i) strong $L^{2}$-convergence over arbitrary finite time intervals, and (ii) upper semi-continuity of the global attractors describing the large-time asymptotic behavior.

Vanishing layer thickness limit of convection in multilayer porous media

TL;DR

The paper rigorously justifies replacing a vanishing-thickness layer in a multilayer porous medium with a reduced model having one fewer layer. It achieves this via uniform-in-thickness estimates that yield strong -convergence on finite time intervals and by proving upper semi-continuity of global attractors as the thin layer thickness tends to zero. A finite-time convergence rate ε^{1/4} is established, and attractor convergence is proved using absorbing-set theory. These results provide a solid mathematical justification for thin-layer reductions in nonlinear convection problems and lay groundwork for further extensions such as curved interfaces or simultaneous vanishing of thickness and permeability.

Abstract

Within the Darcy-Boussinesq framework for convection in multilayered porous media, we investigate the singular limit in which the thickness of one layer tends to zero. We establish that the solution of the full system converges to that of the corresponding limiting model with one fewer layer. The convergence is established in two complementary senses: (i) strong -convergence over arbitrary finite time intervals, and (ii) upper semi-continuity of the global attractors describing the large-time asymptotic behavior.

Paper Structure

This paper contains 8 sections, 10 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The mathematical model
  3. Preliminary results
  4. Uniform estimates on the sharp interface model with a thin layer
  5. Convergence
  6. Finite-time convergence
  7. Convergence of attractors
  8. Conclusion

Key Result

Lemma 3.1

If $\psi \in L^r(\Omega)$ for some $r\in (1,\infty)$, we have $\nabla p\in L^r(\Omega)$ and where $C_p = C_p(r, K_i,\Omega)$.

Figures (2)

  • Figure 1: Multilayer porous media
  • Figure 2: The background profile $\phi_b$

Theorems & Definitions (18)

  • Definition 3.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Proposition 3.3
  • Proposition 4.1
  • proof
  • ...and 8 more