Variational Approach to Homogenization of Doubly-Nonlinear Flow in a Periodic Structure

TL;DR

This paper develops a variational framework for homogenizing doubly-nonlinear flow systems in periodic structures, deriving a two-scale formulation and a homogenized problem using null-minimization principles and two-scale convergence.

Contribution

It introduces a variational approach to homogenize doubly-nonlinear flow systems with periodic coefficients, utilizing Fitzpatrick's null-minimization and two-scale convergence methods.

Findings

Derived a two-scale formulation for the nonlinear flow system.

Established a homogenized problem as the limit when ε approaches zero.

Applied null-minimization principles to formulate the inclusions.

Abstract

This work deals with the homogenization of an initial- and boundary-value problem for the doubly-nonlinear system $$ D_t w -\nabla\cdot \vec z = \nabla\cdot \vec h(x,t,x/\varepsilon), \qquad w\in \alpha(u,x/\varepsilon), \qquad \vec z\in \vec\gamma(\nabla u,x/\varepsilon). $$ Here $\varepsilon$ is a positive parameter, and the prescribed mappings $\alpha$ and $\vec\gamma$ are maximal monotone with respect to the first variable and periodic with respect to the second one. The two inclusions are here formulated as null-minimization principles, via the theory of Fitzpatrick [MR 1009594]. As $\varepsilon\to 0$, a two-scale formulation is derived via Nguetseng's notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved.