2D Navier–Stokes equation in Besov spaces of negative order

The Navier–Stokes equation in the bidimensional torus is considered, with initial velocity in the Besov spaces Bpr−s+2−2r and forcing term in Lr(0,T;Bpq−s) for suitable indices s,r,p,qs,r,p,q. Results of local existence and uniqueness are proven in the case −1<−s+2−2r<0 and of global existence in the case −12<−s+2−2r<0.