Y spaces and global smooth solution of fractional Navier-Stokes equations with initial value in the critical oscillation spaces

For fractional Navier-Stokes equations and critical initial spaces X, one used to establish the well-posedness in the solution space which is contained in C(R+,X). In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces Ym,β where Ym,β is not contained in C(R+,B˙∞1−2β,∞). Consequently, for 12<β<1, we establish the global well-posedness of fractional Navier-Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov-Morrey spaces (B˙p,qγ1,γ2(Rn))n or any Triebel-Lizorkin-Morrey spaces (F˙p,qγ1,γ2(Rn))n where 1≤p,q≤∞,0≤γ2≤np,γ1−γ2=1−2β. These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel-Lizorkin spaces etc.