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

TL;DR

This paper establishes optimal lower bounds on the blowup rates of smooth solutions to the 3D Navier--Stokes equations in specific Sobolev and Besov spaces, using new inequalities derived from dyadic decompositions.

Contribution

It introduces new inequalities for the nonlinear term in Sobolev and Besov spaces and derives optimal blowup rate bounds for solutions in these spaces.

Findings

Lower bound in  ext{H}^{3/2} is (T-t)^{-1/2}

Lower bound in  ext{B}^{5/2}_{2,1} is (T-t)^{-1}

Weaker bound in  ext{H}^{5/2} involving limsup

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