Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones
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arXiv
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| Format: | Preprint |
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2026
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| author | Sha, Kaijian Wang, Xiaoming Wu, Hao |
| author_facet | Sha, Kaijian Wang, Xiaoming Wu, Hao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01362 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones Sha, Kaijian Wang, Xiaoming Wu, Hao Analysis of PDEs 35Q35, 35Q86, 76D03, 76S99, 76R99 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. |
| title | Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones |
| topic | Analysis of PDEs 35Q35, 35Q86, 76D03, 76S99, 76R99 |
| url | https://arxiv.org/abs/2603.01362 |