Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes

TL;DR

This paper investigates the existence duration of strong solutions to the 3D Navier-Stokes equations within Sobolev-Gevrey classes, improving estimates of analyticity radius and demonstrating optimal persistence times for certain regularity levels.

Contribution

It extends prior work by analyzing solutions in stronger Gevrey norms, showing that their existence times match the best known Sobolev persistence times, and improves analyticity radius estimates.

Findings

Existence times in Gevrey norms match Sobolev class persistence times.

Analyticity radius estimates are improved for certain initial data.

Existence times are proven to be optimal for 1/2 < s < 5/2.

Abstract

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