Stochastic homogenization of $Λ$-convex gradient flows

Abstract

This paper is dedicated to Alexander Mielke on the occasion of his 60th birthday.

Two-scale homogenization of abstract linear time-dependent PDEs

“…Stochastic homogenization via unfolding. Recently, in [23,14] Heida and the first two authors reconsidered the notion of two-scale convergence in the mean from the perspective of an unfolding operator (see also [38]). This approach is motivated by (and shares many similarities with) the well-established notion of periodic unfolding [8].…”

“…This approach is motivated by (and shares many similarities with) the well-established notion of periodic unfolding [8]. In the following we briefly recall its definition (for more detail, see [14]).…”

“…Note that this is the abstract variant of the classical result [4, Theorem 4.1.1] (see Section 5.2 for the periodic case). We assume throughout that m = 1 and that ( 10)- (14).…”

“…In this work we propose an abstract strategy for homogenization of evolutionary equations. The methods we present are based on a combination of three concepts: The abstract unified theory for evolutionary equations introduced in [26] (see also [19,27]), their homogenization theory developed in [40] (with plenty of applications, see references below), and the stochastic unfolding procedure recently introduced in [23,14].…”

“…Periodic two-scale convergence [24,1,18] and the method of periodic unfolding [8,39,9,20] are well-suited for problems involving periodic coefficients. For stochastic homogenization problems (e.g., for models describing random heterogeneous materials) the notion of stochastic two-scale convergence [4,2,51,13] and the stochastic unfolding method [14,23] are available as convenient tools.…”