Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones

Kaijian Sha, Xiaoming Wang, Hao Wu

Abstract

We study the asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones in the material properties. As the transition-layer width tends to zero, we prove the upper semi-continuous convergence of the global attractor, invariant measure, and Nusselt number to their counterparts in the limiting sharp-interface model. We also show that the global attractors have finite fractal dimensions, with an explicit upper bound uniform in the transition-layer width. The analysis combines a carefully designed background temperature/contaminant profile together with a novel choice of phase space that ensures global well-posedness of the model and asymptotic compactness of the solution semigroup, and a new interpolation inequality. The phase space is associated with fractional powers of the principal elliptic operator with discontinuous coefficients. These results provide a rigorous long-time validation of the sharp-interface Darcy--Boussinesq model and extend our earlier finite-time convergence theory (H. Dong and X. Wang, SIAM J. Appl. Math. 85 (2025), 1621--1642) to the long-time regime.

Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones

Abstract

We study the asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones in the material properties. As the transition-layer width tends to zero, we prove the upper semi-continuous convergence of the global attractor, invariant measure, and Nusselt number to their counterparts in the limiting sharp-interface model. We also show that the global attractors have finite fractal dimensions, with an explicit upper bound uniform in the transition-layer width. The analysis combines a carefully designed background temperature/contaminant profile together with a novel choice of phase space that ensures global well-posedness of the model and asymptotic compactness of the solution semigroup, and a new interpolation inequality. The phase space is associated with fractional powers of the principal elliptic operator with discontinuous coefficients. These results provide a rigorous long-time validation of the sharp-interface Darcy--Boussinesq model and extend our earlier finite-time convergence theory (H. Dong and X. Wang, SIAM J. Appl. Math. 85 (2025), 1621--1642) to the long-time regime.
Paper Structure (19 sections, 32 theorems, 231 equations)

This paper contains 19 sections, 32 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Model Description
  3. The sharp interface model
  4. The diffuse interface model
  5. Homogenization of boundary conditions and the reduced systems
  6. Well-posedness
  7. Preliminaries
  8. Well-posedness in $\mathcal{H}^s$
  9. Existence of Global Attractors
  10. Sharp interface model
  11. Diffuse interface model
  12. Asymptotics in the Long-time Regime
  13. Upper semi-continuity of the attractors
  14. Upper semi-continuity of invariant measures
  15. Upper semi-continuity of the Nusselt number
  16. ...and 4 more sections

Key Result

Proposition 3.2

For every $\psi_0\in \mathcal{H}$, problem Sharps--bc admits a global weak solution $({\boldsymbol{u}},p,\psi)$ on $[0,T]$ in the sense of Definition def-Sharp-weak. In addition, if $\psi_0\in \mathcal{V}$, then the solution is unique and satisfies

Theorems & Definitions (59)

  • Remark 2.1
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.1
  • Remark 3.1
  • Theorem 3.3
  • Corollary 3.4
  • Lemma 3.2: $L^r$-estimate for $\psi$
  • proof
  • Lemma 3.3: $\mathcal{H}^s$-estimate for $\psi$
  • ...and 49 more