Homogenization in algebras with mean value

TL;DR

This paper develops a comprehensive theory for homogenization in algebras with mean value, including nonergodic cases, and applies it to stochastic fluid flow models, expanding the scope of homogenization techniques.

Contribution

It introduces a detailed homogenization framework for nonergodic algebras with mean value, answering a key open question and extending homogenization to stochastic fluid models.

Findings

Homogenization theory is extended to nonergodic algebras with mean value.

A compactness result for Young measures in these algebras is established.

Homogenization of a stochastic Ladyzhenskaya model is successfully achieved.

Abstract

In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.