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Shu Gu

Publications and source records attributed to Shu Gu.

6 recordsLinked to original sources
arXiv2020

Large-scale Regularity of Nearly Incompressible Elasticity in Stochastic Homogenization

In this paper, we systematically study the regularity theory of the linear system of nearly incompressible elasticity. In the setting of stochastic homogenization, we develop new techniques to establish the large-scale estimates of displacement and pressure, which are uniform in both the scale parameter and the incompressibility parameter. In particular, we obtain the boundary estimates in a new class of Lipschitz domains whose boundaries are smooth at large scales and bumpy at small scales.

math.AP↗
arXiv2017

Periodic homogenization of Green's functions for Stokes systems

This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green's functions $(G_\varepsilon,\Pi_\varepsilon)$ (and fundamental solutions $(\Gamma_\varepsilon, Q_\varepsilon)$) 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 $\Pi_\varepsilon$. Also, for the first time we prove the \textit{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 $G_\varepsilon,\nabla G_\varepsilon, \Pi_\varepsilon$ 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.

math.AP↗
arXiv2016

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

The paper concerns the sharp boundary regularity estimates in homogenization of Dirichlet problem for Stokes systems. We obtain the Lipschitz estimates for velocity term and $L^\infty$ estimate for pressure term, under some reasonable smoothness assumption on rapidly oscillating periodic coefficients. The approach is based on convergence rates, originally investigated by S. Armstrong and Z. Shen in \cite{SZ,SZW12}, however the argument developed here does not rely on the Rellich estimates. In this sense, we find a new way to obtain the sharp uniform boundary estimates without imposing the symmetry assumption on coefficients. Additionally, we emphasize that $L^\infty$ estimate for the pressure term does require the $O(\varepsilon^{1/2})$ convergence rate, locally at least, compared to $O(\varepsilon^\lambda)$ for the velocity term, where $\lambda\in(0,1/2)$.

math.AP↗
arXiv2015

Homogenization of Stokes Systems and Uniform Regularity Estimates

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and $L^\infty$ estimates for the pressure as well as a Liouville property for solutions in $\mathbb{R}^d$. We also obtain the boundary $W^{1,p}$ estimates in a bounded $C^1$ domain for any $1<p<\infty$.

math.AP↗