Time decay and exponential stability of solutions to the periodic 3D Navier–Stokes equation in critical spaces

Abstract

Using analysis in frequency space and Fourier methods, we establish that the global solution to the three‐dimensional incompressible periodic Navier–Stokes equation for initial data in the critical Sobolev space decays exponentially fast to zero, and it is exponentially stable as time goes to infinity as soon as the initial data (hence the solution) are mean free; otherwise, the difference to the average does so. Furthermore, we prove that any global nonmean‐free solution vanishes as time goes to infinity, and it is globally exponentially stable. The main tools are the energy methods, the Friedrich's approximating schema, and a crucial change of function that depends explicitly on time. Copyright © 2013 John Wiley & Sons, Ltd.