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 R^ 3 with first blowup time T, we prove lower bounds for u in the Sobolev spaces H?^(3/2) , H?^( 5/2) , and the Besov space B?^(5/2)_( 2,1 ), with optimal rates of blowup: we prove the strong lower bounds ||u(t)||_(H?^(3/2))= c(T - t) ^(-1/2) and ||u(t)||_(B?^(5/2)_( 2,1))= c(T - t) -1 , but in H?^(5/2) we only obtain the weaker result lim supt?T - (T -t)||u(t)||_(H?^(5/2)) = c. 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.