We consider the Navier–Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier’s slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global existence of a strong solution for large data. We also show several estimates for the strong solution with constants explicitly depending on the thickness of the thin domain. The proofs of these results are based on a standard energy method and a good product estimate for the convection and viscous terms following from a detailed study of average operators in the thin direction. We use the average operators to decompose a three-dimensional vector field on the thin domain into the almost two-dimensional average part and the residual part, and derive good estimates for them which play an important role in the proof of the product estimate.

我们在Navier的滑移边界条件下，围绕给定的封闭曲面，在三维弯曲薄域中考虑Navier-Stokes方程。当薄域的厚度足够小时，我们就建立了针对大数据的强大解决方案的全局存在。我们还根据常数根据薄域的厚度显示了强解的几种估计。这些结果的证明是基于标准的能量方法以及对流和粘性项的良好乘积估计，这是通过对平均算子在细方向上的详细研究得出的。我们使用平均算子将稀疏域上的三维矢量场分解为几乎二维的平均部分和残差部分，