Uniform estimates for Stokes equations in a domain with a small hole and applications in homogenization problems
Uniform estimates for Stokes equations in a domain with a small hole and applications in homogenization problems
复制标题
小孔域斯托克斯方程的均匀估计及其在均质化问题中的应用
DOI：
10.1007/s00526-021-02104-4
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发表时间：
2015
影响因子：
2.1
通讯作者：
Yong Lu
中科院分区：
文献类型：
--
作者：
Yong Lu
本文研究了一类具有收缩洞的区域上Stokes方程的Dirichlet问题。一个典型的观察是，当洞的大小为零时，域的Lipschitz范数为无穷大。因此，如果，经典的结果表明，当洞的大小趋于零时，解的估计可能趋于无穷大。在固定区域上存在收缩洞的情况下，我们给出了所有方程解的一致估计的完整描述。我们证明了一致估计成立当且仅当（当）。然后，我们给出了两个应用在流体力学中的均匀化问题的研究：一个推广的限制算子和Bogovskii型算子的构造在穿孔域与定量估计的运营商范数。
We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in. 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, the classical results indicate that theestimate 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 uniformestimates of the solution for all. We show that the uniformestimate holds if and only if(when). 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.