Lower bounds on blowing-up solutions of the three-dimensional navier-stokes equations in H^3/2, H^5/2, and B^5/2_2,1

If u is a smooth solution of the Navier-Stokes equations on with first blowup time T, we prove lower bounds for u in the Sobolev spaces and the Besov space with optimal rates of blowup: we prove the strong lower bounds and we obtain lim sup c, a weaker result. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of u.