Correctness of Biot's model of in situ leaching for incompressible liquid and compressible solid components

[2604.22645] Correctness of Biot's model of in situ leaching for incompressible liquid and compressible solid components Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2604.22645 (math) [Submitted on 24 Apr 2026 ( v1 ), last revised 23 Oct 2026 (this version, v3)] Title: Correctness of Biot's model of in situ leaching for incompressible liquid and compressible solid components Authors: Anvarbek Meirmanov , Akbota Senkebayeva View PDF HTML (experimental) Abstract: We study a mathematical model of in situ leaching of rare metals, in which the joint filtration of two liquids is governed by the microscopic model $\mathbb{A}^{\varepsilon}$. A key difficulty is the unknown (free) boundary $\Gamma(r)$ between solid and liquid components, determined by an additional condition on $\Gamma(r)$; no standard methods exist for this nonlinear problem. To resolve it, we apply the fixed point theorem. For a given function $r(\boldsymbol{x},t)$ from a set $\mathfrak{M}_{(0,T)}$ of sufficiently smooth functions describing the skeleton structure, we consider the auxiliary problem $\mathbb{B}^{\varepsilon}(r)$: an elliptic system for displacements of the liquid and solid components coupled with parabolic equations for the acid concentration. Selecting the weak solution of minimal smoothness, we apply the homogenization method to pass from the microscopic to the macroscopic description. The resulting macroscopic model $\mathbb{H}(r)$ contains a homogenized boundary condition that expresses the normal boundary velocity $V_{N}=\partial r/\partial t$ as a linear function of the acid concentration $c$. Since $c$ depends on $r$ via an operator $\mathbb{F}\colon\mathfrak{M}_{(0,T)}\to\mathfrak{M}_{(0,T)}$, we prove that $\mathbb{F}$ is Lipschitz continuous and, by Banach's theorem, possesses a unique fixed point $r^{*}$, which yields the unique solution $\mathbb{H}=\mathbb{H}(r^{*})$. Comments: v3 Subjects: Analysis of PDEs (math.AP) ; Mathematical Physics (math-ph) Cite as: arXiv:2604.22645 [math.AP] (or arXiv:2604.22645v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2604.22645 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Akbota Senkebayeva [ view email ] [v1] Fri, 24 Apr 2026 15:18:08 UTC (38 KB) [v2] Fri, 28 Aug 2026 10:54:32 UTC (1 KB) (withdrawn) [v3] Fri, 23 Oct 2026 06:26:30 UTC (47 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-04 Change to browse by: math math-ph math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Correctness of Biot's model of in situ leaching for incompressible liquid and compressible solid components#682657
• Long-term reactive transport modelling of stabilized/solidified waste: from dynamic leaching tests to disposal scenarios#436251
• Simultaneous evaporation and imbibition of a droplet on a flooded porous substrate#1109772
• Modelling of dissolved and particulate matter transfer in a fractured medium.#665269
• Etudes et optimisation de réacteurs à sels fondus incinérateurs : du schéma de calcul neutronique aux scénarios de déploiement#172918