Homogenization of some degenerate pseudoparabolic variational inequalities

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.