  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) 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 $\frac{1}{2} < s < \frac{5}{2}$.
