In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces $H^s,s>\frac{1}{2}$. This complements the seminal work of Foias and Temam (1989) [26] on $H^1$ based Gevrey classes, thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust of the paper consists in showing that the existence times in the much stronger Gevrey norms (i.e. the norms defining the analytic Gevrey classes which quantify the radius of real-analyticity of solutions) match the best known persistence times in Sobolev classes. Additionally, as in the case of persistence times in the corresponding Sobolev classes, our existence times in Gevrey norms are optimal for $\frac{1}{2}<s<\frac{5}{2}$.

本文研究基于Sobolev空间的时变解析Gevrey类中三维Navier-Stokes方程强解的存在时间 $H^s，s>\frac{\mathrm{1个}}{2}$。这是对Foias和Temam（1989）[26]的开创性工作的补充。$H^{\mathrm{1个}}$基于Gevrey类的模型，因此使我们能够改进某些类初始数据的解的分析半径的估计。本文的主要目的在于表明，在更强的Gevrey规范（即定义解析Gevrey类的规范，这些标准量化解决方案的实际解析度的半径）中的存在时间与Sobolev类中最著名的持续时间相匹配。此外，就像在相应的Sobolev类中的持续时间一样，我们在Gevrey规范中的存在时间对于$\frac{\mathrm{1个}}{2}<s<\frac{5}{2}$。