Mathematical analysis and homogenization of a free boundary problem for in situ leaching of rare earths with a special periodic structure

Díaz J.I. Meirmanov A. Senkebayeva A.
2026 World Scientific

Bulletin of Mathematical Sciences
2026

In this paper, we consider an initial-boundary value problem modeling in situ leaching of rare earths with a special periodic structure by an acid solution, improving some previous studies by the second author and collaborators. At the microscopic scale, fluid motion in the pore space is described by the Stokes equations for a slightly compressible fluid coupled with the deformation of the elastic skeleton, governed by the Lamé system and the diffusion equation for the acid solution. Due to rock dissolution, the interface between liquid and solid phases is unknown (it is a free boundary) and must be determined as part of the solution. To overcome this difficulty, we introduce a family of approximate microscopic models with prescribed pore geometry and establish their well-posedness in a weak formulation. By using a priori estimates and Galerkin’s method, we obtain existence results and apply the method of two-scale convergence for periodic structures to derive the corresponding homogenized macroscopic model. Finally, a fixed-point argument yields existence and uniqueness for the resulting macroscopic system.

diffusion , Free boundary problem , homogenization , structures with special periodicity

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Departamento de Análisis Matemático-Matemática Aplicada, Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Madrid, Spain
L. N. Gumilyov Eurasian National University, Physics and Technical Faculty, Astana, Kazakhstan
Kazakh-British Technical University, Almaty, Kazakhstan

Departamento de Análisis Matemático-Matemática Aplicada
L. N. Gumilyov Eurasian National University
Kazakh-British Technical University

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