Well-posedness for the Navier–Stokes equations with datum in Sobolev–Fourier–Lorentz spaces

In this note, for s∈Rs∈R and 1≤p,r≤∞1≤p,r≤∞, we introduce and study Sobolev–Fourier–Lorentz spaces H˙Lp,rs(Rd). In the family spaces H˙Lp,rs(Rd), the critical invariant spaces for the Navier–Stokes equations correspond to the value s=dp−1. When the initial datum belongs to the critical spaces H˙Lp,rdp−1(Rd) with d≥2d≥2, 1≤p<∞1≤p<∞, and 1≤r<∞1≤r<∞, we establish the existence of local mild solutions to the Cauchy problem for the Navier–Stokes equations in spaces L∞([0,T];H˙Lp,rdp−1(Rd)) with arbitrary initial value, and existence of global mild solutions in spaces L∞([0,∞);H˙Lp,rdp−1(Rd)) when the norm of the initial value in the Besov spaces B˙Lp˜,∞dp˜−1,∞(Rd) is small enough, where p˜ may take some suitable values.