Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces

Abstract

We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spaces $F{\dot{B}}_{p,q}^{1-2\beta +3/p^{\prime }}$. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical case $\beta =1/2$. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.