Lower Bounds on Blowing-Up Solutions of the Three-Dimensional Navier-Stokes Equations in đot H3/2, • H5/2, and đot B5/22, 1

If $u$ is a smooth solution of the Navier--Stokes equations on ${\mathbb R}^3$ with first blowup time $T$, we prove lower bounds for $u$ in the Sobolev spaces $\dot H^{3/2}$, $\dot H^{5/2}$, and the Besov space $\dot B^{5/2}_{2,1}$, with optimal rates of blowup: we prove the strong lower bounds $\|u(t)\|_{\dot H^{3/2}}\ge c(T-t)^{-1/2}$ and $\|u(t)\|_{\dot B^{5/2}_{2,1}}\ge c(T-t)^{-1}$; in $\dot H^{5/2}$ we obtain $\limsup_{t\to T^-}(T-t)\|u(t)\|_{\dot H^{5/2}}\ge c$, 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 $u$.