Homogenization of Brinkman flows in heterogeneous dynamic media

Abstract

In this paper, we study Brinkman's equations with microscale properties that are highly heterogeneous in space and time. The time variations are controlled by a stochastic particle dynamics described by an SDE. The particle dynamics can be thought as particle deposition that often occurs in filter problems. Our main results include the derivation of macroscale equations and showing that the macroscale equations are deterministic. The latter is important for our (also many other) applications as it greatly simplifies the macroscale equations. We use the asymptotic properties of the SDE and the periodicity of the Brinkman's coefficient in the space variable to prove the convergence result. The SDE has a unique invariant measure that is ergodic and strongly mixing. The macro scale equations are derived through an averaging principle of the slow motion (fluid velocity) with respect to the fast motion (particle dynamics) and also by averaging the Brinkman's coefficient with respect to the space variable. Our results can be extended to more general nonlinear diffusion equations with heterogeneous coefficients.

Stochastic homogenization for a diffusion-reaction model

“…Isolating the term (1.2), a uniform estimate in H 2 (D) 3 was needed to be able to pass to the limit. Unfortunately, while the random variable u ε in [2] was uniformly bounded in the Sobolev space H 2 (D) 3 , in the current paper it is only uniformly bounded in the Sobolev space H 1 (D) 3 . Instead, the limit in the term (1.2) will be performed by using a Khasminskii type argument, following an idea already introduced in [4] where as mentioned earlier, our term α(·, v ε )u ε does not satisfy the same assumptions.…”

“…These particles have a faster motion than the motion of the fluid flow and are driven by a stochastic perturbation of Brownian type. A simpler version of this model has been studied in [2], where the diffusion A was considered to be constant and equal to 1. The heterogeneous diffusion brings an additional difficulty and makes this problem more realistic since one deals with heterogeneous permeability fields in most porous media problems.…”

“…There is not much in the literature dealing with averaging systems for porous media when spatial heterogeneities are present. We refer to our previous paper [2], where to our knowledge, it was the first paper where time and spatial scales have been considered for porous media in a stochastic setting. In [5], the authors prove an averaging principle for a very general class of stochastic PDEs.…”

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

“…Research in this direction is still at its infancy, despite the importance of such problems in both applied and fundamental sciences. Some relevant interesting work have recently been undertaken, mainly for parabolic SPDEs; see for instance [3,8,10,11,21,43,44]. We also note the closely related work [3,25,15,16] dealing with stochastic homogenization for SPDEs with small parameters.…”

Stochastic homogenization of multicontinuum heterogeneous flows

“…The techniques used to pass to the limit in the mathematical model (1.1) are a generalization of the techniques used in our recent paper [8], where a simpler reaction diffusion model was considered. Let us mention that, like in [8] and [9] the random coefficient α does depend on the spatial variable x/ε and the process (v ε 1 , v ε 2 ) which is ergodic for frozen (u ε 1 , u ε 2 ). Hence, the passage to the limit involves a combination of two kind of convergences: averaging (in time) and homogenization (in space).…”