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.