Periodic homogenization of Green’s functions for Stokes systems

Abstract

This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green's functions (Gε, Πε) (and fundamental solutions (Γε, Qε)) for the Stokes system with periodically oscillating coefficients (including a system of linear incompressible elasticity). Particular emphasis will be placed on the new oscillation estimates for the pressure component Πε. Also, for the first time we prove the adjustable uniform estimates (i.e., Lipschitz estimate for velocity and oscillation estimate for pressure) by making full use of the Green's functions. Via these estimates, we establish the asymptotic expansions of Gε, ∇Gε, Πε and more, with a tiny loss on the errors. Some estimates obtained in this paper are new even for Stokes system with constant coefficients, and possess potential applications in homogenization of Stokes or elasticity system.2010 Mathematics Subject Classification. 35B27, 76M50, 35C20.

Regularity for the stationary Navier–Stokes equations over bumpy boundaries and a local wall law

“…Third, we provide explicit quantitative regularity estimates in the non-perturbative regime. Fourth, in the vein of the seminal works [9,10] and of [61,48], we provide pointwise estimates for the large-scale decay of the velocity and pressure parts of the Green function associated to the Stokes system in bumpy John half-spaces; see Subsection 1.3 and Appendix B. These estimates are pivotal to construct the first-order boundary layers in Subsection 4.2.…”

“…Proof. We will carry out a delicate oscillation estimate of the pressure originating from [48]. We first consider the estimate (i), i.e., x 3 > 2 and y 3 > 2.…”

Convergence rates of nonlinear Stokes problems in homogenization

“…In 2015, Gu [7] also proved convergence rates in L 2 and H 1 of Dirichlet problems for linear Stokes systems. In 2017, Gu and Shen [10] got the asymptotic behaviors of the Green functions as well as the convergence rates in L p and L ∞ for solutions. Recently, other authors have also been interested in the regularity estimates for the Stokes problems, see [3,5,11,23] and their references for more results.…”

Estimates for Green functions of Stokes systems in two dimensional domains

“…In [7], the authors constructed Green function for a conormal derivative problem when coefficients are variably partially BMO. We also refer the reader to [16] for Green functions of Stokes systems with oscillating periodic coefficients.…”