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If $u$ is a smooth solution of the Navier--Stokes equations on ${\mathbbR}^3$ with first blowup time $T$, we prove lower bounds for $u$ in the Sobolevspaces $\dot H^{3/2}$, $\dot H^{5/2}$, and the Besov space $\dotB^{5/2}_{2,1}$, with optimal rates of blowup: we prove the strong lower bounds$\|u(t)\|_{\dot H^{3/2}}\ge c(T-t)^{-1/2}$ and $\|u(t)\|_{\dotB^{5/2}_{2,1}}\ge c(T-t)^{-1}$, but in $\dot H^{5/2}$ we only obtain the weakerresult $\limsup_{t\to T^-}(T-t)\|u(t)\|_{\dot H^{5/2}}\ge c$. The proofsinvolve new inequalities for the nonlinear term in Sobolev and Besov spaces,both of which are obtained using a dyadic decomposition of $u$.

Lower bounds on blowing-up solutions of the 3D Navier--Stokes equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$