On the doubly non-local Hele-Shaw-Cahn-Hilliard system: derivation and \(2D\) well-posedness (Q6536756)

This paper is concerning the flow of two incompressible immiscible fluids in a thin highly heterogeneous Hele-Shaw cell. The basic model is the nonlocal Cahn-Hilliard-Navier-Stokes system (at the microscale). The sigma-convergence theory is used, previously obtained as a generalization of two-scale convergence for thin periodic structures -- see [\textit{W. Jäger} and \textit{J. L. Woukeng}, ``Sigma-convergence for thin heterogeneous domains and application to the upscaling of Darcy-Lapwood-Brinkman flow'', Preprint, \url{arXiv:2309.09004}; \textit{M. Neuss-Radu} and \textit{W. Jäger}, SIAM J. Math. Anal. 39, No. 3, 687--720 (2007; Zbl 1145.35017)]. The new element of this paper is the obtained upscaled model. Existence results for the considered system and uniform estimates are given in Section 2, by using the Wiener amalgam and some local Sobolev spaces. Fundamentals of algebras with mean value are used to give very interesting and important elements concerning sigma-convergence for thin heterogeneous domains, in Section 3. Useful results for the homogenization process and the homogenized system are given in Section 4. The obtained ``homogenized system'' (4.35) is a Hele-Shaw equation with memory, which can be also used for transient flow modelling the tumours growth. To the best of the authors knowledge ``this is the first time that such a system is obtained in the literature''. The analysis of the homogenized system (4.35) is given in Section 5, with important assumptions concerning the regularity of the coefficients appearing in (1.2). Some results of \textit{F. Della Porta} and \textit{M. Grasselli} [Commun. Pure Appl. Anal. 15, No. 2, 299--317 (2016; Zbl 1334.35226)] are also used. Important tools are the Gagliardo-Nirenberg inequality, the Laplace transform in some particular spaces and the Moser-type iteration. Very interesting examples are given in the last section. Periodic, almost periodic and perturbed periodic microstructures are analyzed.