On a Bound for Amplitudes of Navier–Stokes Flow with almost Periodic Initial Data

Abstract

For any bounded (real) initial data it is known that there is a unique global solution to the two-dimensional Navier–Stokes equations. This paper is concerned with a bound for the sum of the modulus of amplitudes when initial velocity is spatially almost periodic in 2D. In the case of general dimension, it is bounded on local time of existence shown by Giga et al. (Methods Appl Anal 12:381–393,2005). A class of initial data is given such that the sum of the modulus of amplitudes of a solution is bounded on any finite time interval. It is shown by an explicit example that such a bound may diverge to infinity as the time goes to infinity at least for complex initial data.