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

Abstract

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 ${\dot{B}}_{2,1}^{5/2}$, with optimal rates of blowup: we prove the strong lower bounds $\Vert u(t){\Vert }_{{\dot{H}}^{3/2}}\ge c(T-t{)}^{-1/2}$ and $\Vert u(t){\Vert }_{{\dot{B}}_{2,1}^{5/2}}\ge c(T-t{)}^{-1}$, but in ${\dot{H}}^{5/2}$ we only obtain the weaker result ${\mathrm{limsup}}_{t\to T^{-}}(T-t)\Vert u(t){\Vert }_{{\dot{H}}^{5/2}}\ge c$. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of $u$.