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

If $u$ is a smooth solution of the Navier–Stokes equations on ${\mathbb R}^3$withfirstblowuptime$T$,weprovelowerboundsfor$u$ in the Sobolev spaces ${\dot{H}}^{3\mathrm{/}2}$, ${\dot{H}}^{5\mathrm{/}2}$, and the Besov space $\dot B^{5/2}{2,1}$,withoptimalratesofblowup:weprovethestronglowerbounds$|u(t)|{\dot H^{3/2}}\ge c(T-t)^{-1/2}$and$|u(t)|{\dot B^{5/2}{2,1}}\ge c(T-t)^{-1}$,butin$\dot H^{5/2}$ we only obtain the weaker result ${\mathrm{lim\hspace{0.17em}sup}}_{t\to T^{-}}(T-t)\mathrm{\parallel }u(t){\mathrm{\parallel }}_{{\dot{H}}^{5\mathrm{/}2}}\ge c$. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of $u$.