ON A BOUND FOR AMPLITUDES OF NAVIER-STOKES FLOW WITH ALMOST PERIODIC INITIAL DATA

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, Inui, Mahalov and Matsui in 2005. A class of initial data is given such that the sum of the modulus of amplitudes of a solution is bounded on any nite time interval. It is shown by an explicit example that such a bound may diverge to in nity as the time goes to in nity at least for complex initial data