论文标题
弯曲的薄域中的Navier-Stokes方程，第一部分：Stokes Operator的均匀估计值
Navier-Stokes equations in a curved thin domain, Part I: uniform estimates for the Stokes operator
论文作者
论文摘要
在本文和即将发表的论文[41,42]中，我们研究了在Navier的滑动边界条件下，在给定闭合表面周围的三维弯曲薄域中的Navier-Stokes方程。我们专注于本文中曲面薄域的Stokes操作员的研究。建立了Stokes操作员的统一规范等效性以及Stokes和Laplace操作员的均匀差异估计值，其中常数与弯曲的薄域的厚度无关。为了证明这些结果，我们表明了基于对矢量场和边界上的矢量场和表面数量的仔细分析，表明了弯曲薄域对向量拉普拉斯操作员的均匀估计。我们还提供了弯曲的薄域和矢量场的例子，均匀的Korn不等式无效，但标准的Korn不等式具有常数，随着薄域的厚度往往为零。
In the series of this paper and the forthcoming papers [41,42] we study the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. We focus on the study of the Stokes operator for the curved thin domain in this paper. The uniform norm equivalence for the Stokes operator and a uniform difference estimate for the Stokes and Laplace operators are established in which constants are independent of the thickness of the curved thin domain. To prove these results we show a uniform Korn inequality and a uniform a priori estimate for the vector Laplace operator on the curved thin domain based on a careful analysis of vector fields and surface quantities on the boundary. We also present examples of curved thin domains and vector fields for which the uniform Korn inequality is not valid but a standard Korn inequality holds with a constant that blows up as the thickness of a thin domain tends to zero.
