Remarks on the decay of Fourier coefficients to solutions of Navier-Stokes system

Abstract

We first investigate the large time decay of the Fourier coefficients to the global mild solution of Navier-Stokes system with periodic boundary condition and with small enough initial data. In particular, this result ensures the fast decay of Fourier coefficients to any global solution u ∈ C ( [ 0 , ∞ [ ; H ˙ 1 2 ( T 3 ) ) u\in C([0,\infty [;\dot H^{\frac 12}({\mathbb T}^3)) of Navier-Stokes system. Then we extend this result to Navier-Stokes system with an additional low frequency damping term in the whole space. Finally, we prove that the radius of analyticity R ( t ) R(t) to this modified system with small initial data in H ˙ 1 2 ( R 3 ) \dot {H}^{\frac 12}(\mathbb {R}^3) is bounded from below by a positive constant times t , t, which makes sharp contrast with the corresponding result for the classical Navier-Stokes system that R ( t ) t − 1 2 R(t)t^{-\frac 12} is bounded from below by a positive constant.