Homogenization of elliptic systems with stratified structure revisited

Homogenization of Trajectory Statistical Solutions for the 3D Incompressible Micropolar Fluids with Rapidly Oscillating Terms

“…Proceeding as the proofs of [58] (Lemma 2.1), we can obtain the following lemma about the existence and estimate of weak solutions to problem (15) and ( 16). The difference is that we replace the the positive constant R = 2λ…”

“…Suppose that (H) holds. Then, to every w 0 = (u 0 , ω 0 ) ∈ H and each ∈ (0, 1], there corresponds at least one weak solution w (t) of problems (15) and (16). Moreover, there exists a time t…”

“…In terms of Lemma 1, we give the definition of the trajectory space and kernel of Equation (15), respectively, as T + = {w(t) : w(t) is a weak solution of (15) and w(t) ∈ X for all t ∈ [0, ∞)}, K = {w(t) : w(t) is a weak solution of (15) and w(t) ∈ X for all t ∈ (−∞, +∞)}.…”

“…The homogenization theory of partial differential equations is very fruitful; see e.g., [12][13][14][15] and the references therein. The homogenization of attractors for evolution equations with rapidly oscillating terms has been widely studied.…”

Compactness and stable regularity in multiscale homogenization

“…In the special case of onescale homogenization, Dong, Li and Wang in [15,14] obtained the uniform interior Lipschitz estimate by the compactness method. In [32], Xu and Niu established the optimal convergence rate and uniform regularity for more general one-scale laminate structure A ε = A(x, ρ(x)/ε), where ρ : Ω → R n satisfies proper nondegenerate condition. Our result in Theorem 1.2 provides a multiscale counterpart of the previous results, and also demonstrates that the Lipschitz estimate is stable, namely, it holds without separation of scales (1.2).…”