Existence time for the 3D Navier–Stokes equations in a generalized Gevrey class

Abstract

Gevrey class technique is a widely used tool for studying higher regularity properties of solutions to dissipative equations. Maximal radius in a Gevrey class determines a small length scale associated to the decay of the Fourier power spectrum and turbulence. In this paper, we consider existence theory for the three dimensional, incompressible Navier–Stokes equations, in a certain generalized Gevrey class of functions, which contains all the analytic and sub-analytic Gevrey classes, and is in turn contained in the space of all C∞ functions. This class has been in focus recently, pertaining to the study of the attractor for the 2D Navier–Stokes equations, particularly in relation to a question posed by Peter Constantin as to whether or not zero is in the attractor of the 2D Navier–Stokes equations. We show that in the 3D case, the differential inequality that one obtains in this class is almost linear, and the corresponding existence time is better than the reciprocal of any algebraic power of the norm of the initial data (in this class). Subsequently, we compare the existence times with well-known ones in Sobolev classes for certain types of initial data. We also obtain a lower bound for the maximal radius in the generalized Gevrey class under consideration here. The first two authors are delighted to dedicate this paper to Professor Edriss Titi on the occasion of his 60th birthday. They admire Professor Titi’s enormous research contributions to the field of fluid dynamics, as well as his mathematical acumen, novel ideas, and infectious enthusiasm for the subject, as evidenced by his numerous lucid, and enthralling, lectures. This research was completed after the passing of Professor Basil Nicolaenko. The first two authors are confident that he too would have shared their sentiments concerning Professor Titi.