Uniform estimates for Stokes equations in a domain with a small hole and applications in homogenization problems

Abstract

We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in ${\mathbb {R}}^d, \ d\ge 2$. A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if $p\ne 2$, the classical results indicate that the $W^{1,p}$ estimate of the solution may go to infinity as the size of the hole tends to zero. With the presence of the shrinking hole in a fixed domain, we give a complete description for the uniform $W^{1,p}$ estimates of the solution for all $1<p<\infty$. We show that the uniform $W^{1,p}$ estimate holds if and only if $d'<p<d$ ($p=2$ when $d=2$). We then give two applications in the study of homogenization problems in fluid mechanics: a generalization of the restriction operator and a construction of Bogovskii type operator in perforated domains with a quantitative estimate of the operator norm.