Homogenization of an elliptic transmission system modeling the flux of oxygen from blood vessels to tissues

University essay from

Author: Filippo Di Tillio; [2021]

Keywords: ;

Abstract: Motivated by the study of the hypoxia problem in cancerous tissues, we propose a system of coupled partial differential equations defined on a heterogeneous, periodically perforated  domain describing the flux of oxygen from blood vessels towards the tissue and the corresponding oxygen diffusion within the tissue. Using heuristics based on dimensional analysis, we rephrase the initially parabolic problem as a semi-linear elliptic transmission problem. Focusing on the elliptic case, we are able to define a microscopic $\varepsilon$-dependent problem that is the starting point of our mathematical analysis; here $\varepsilon$ is linked to the scale of heterogeneity.   We study the well-posedness of the microscopic problem as well as the passage to the periodic homogenization limit. Additionally, we derive the strong formulation of the two-scale macroscopic limit problem. Finally, we prove a corrector estimate. This specific ingredient allows us to estimate, in an {\em a priori} way,  the discrepancy between solutions to the microscopic and, respectively, macroscopic problem. Our working techniques include energy-type estimates, fixed-point type iterations, monotonicity arguments, as well as the two-scale convergence tool.

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