Homogenization of some degenerate pseudoparabolic variational inequalities
Mariya Ptashnyk

TL;DR
This paper develops a multiscale analysis for a degenerate pseudoparabolic variational inequality modeling two-phase flow with dynamical capillary pressure in perforated domains, using regularization, penalty methods, and two-scale convergence.
Contribution
It introduces a homogenization approach for a complex nonlinear degenerate variational inequality in perforated domains, establishing existence and deriving macroscopic limits.
Findings
Existence of solutions for the microscopic problem.
Uniform a priori estimates independent of regularization and microstructure scale.
Derivation of a macroscopic obstacle problem via two-scale convergence.
Abstract
Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.
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