# Convergence rates and $W^{1,p}$ estimates in homogenization theory of Stokes systems in Lipschitz domains

@article{Xu2016ConvergenceRA, title={Convergence rates and \$W^\{1,p\}\$ estimates in homogenization theory of Stokes systems in Lipschitz domains}, author={Qiang Xu}, journal={arXiv: Analysis of PDEs}, year={2016} }

Concerned with the Stokes systems with rapidly oscillating periodic coefficients, we mainly extend the recent works in \cite{SGZWS,G} to those in term of Lipschitz domains.
The arguments employed here are quite different from theirs, and the basic idea comes from \cite{QX2}, originally motivated by \cite{SZW2,SZW12,TS}. We obtain an almost-sharp $O(\varepsilon\ln(r_0/\varepsilon))$ convergence rate in $L^2$ space, and a sharp $O(\varepsilon)$ error estimate in $L^{\frac{2d}{d-1}}$ space by a…

#### 7 Citations

Uniform boundary estimates in homogenization of higher-order elliptic systems

- Mathematics
- 2017

AbstractThis paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $$\mathcal {L}_\varepsilon $$Lε, with rapidly oscillating periodic…

Quantitative Estimates in Homogenization of Parabolic Systems of Elasticity in Lipschitz Cylinders

- Mathematics
- 2017

In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with…

Periodic homogenization of Green’s functions for Stokes systems

- Mathematics, PhysicsCalculus of Variations and Partial Differential Equations
- 2019

This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green’s functions $$(G_\varepsilon ,\Pi _\varepsilon )$$(Gε,Πε) (and fundamental solutions $$(\Gamma…

Optimal Boundary Estimates for Stokes Systems in Homogenization Theory

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2017

A new way is found to obtain the sharp uniform boundary estimates without imposing the symmetry assumption on coefficients of the Dirichlet problem for Stokes systems.

Error Estimates of Reiterated Stokes Systems via Fourier Transform Methods.

- Mathematics
- 2019

In this paper, we are interested in the error estimates of the reiterated Stokes systems in a bounded $C^{1,1}$ domain with Dirichlet boundary conditions. And we have obtained the $O(\varepsilon)$…

Homogenization of some degenerate pseudoparabolic variational inequalities

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

Abstract Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this…

Large-scale Regularity of Nearly Incompressible Elasticity in Stochastic Homogenization

- Mathematics
- 2020

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…

#### References

SHOWING 1-10 OF 43 REFERENCES

Convergence rates for general elliptic homogenization problems in a bounded Lipschitz domain

- Mathematics
- 2015

The paper extends the results obtained by C. Kenig, F. Lin and Z. Shen in \cite{SZW2} to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no…

Boundary Estimates in Elliptic Homogenization

- Mathematics
- 2015

For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the…

Uniform Regularity Estimates in Homogenization Theory of Elliptic Systems with Lower Order Terms on the Neumann Boundary Problem

- Mathematics
- 2015

In this paper, we mainly employed the idea of the previous paper to study the sharp uniform $W^{1,p}$ estimates with $1<p\leq \infty$ for more general elliptic systems with the Neumann boundary…

Convergence Rates for General Elliptic Homogenization Problems in Lipschitz Domains

- Computer Science, MathematicsSIAM J. Math. Anal.
- 2016

The paper finds the new weighted-type estimates for the smoothing operator at scale $\varepsilon$ and obtains sharp convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Shen for elasticity systems in [preprint, arXiv:1505.00694v1, 2015].

Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2013

A sharp order estimate for the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$ is obtained.

Uniform W^{1,p} Estimates for Systems of Linear Elasticity in a Periodic Medium

- Mathematics
- 2011

Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for…

Convergence Rates in L2 for Elliptic Homogenization Problems

- Mathematics
- 2011

We study rates of convergence of solutions in L2 and H1/2 for a family of elliptic systems $${\{\mathcal{L}_\varepsilon\}}$$ with rapidly oscillating coefficients in Lipschitz domains with Dirichlet…

Hölder continuity of solutions of elliptic systems

- Mathematics
- 1971

AbstractThe purpose of this note is to observe that a variant of the method of Morrey, as exposed in [4] and [5], can be used to show that weak solutions of a certain class of elliptic systems of…

Error estimate and unfolding for periodic homogenization

- Physics, Mathematics
- 2004

This paper deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding. We give the upper bound of the distance between the unfolded…

Homogenization of Stokes Systems and Uniform Regularity Estimates

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2015

Uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients are established and a Liouville property for solutions in $\mathbb{R}^d$ is obtained.