Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $\mathbf{\epsilon }\in [1,2]$

Abstract

For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\ge 3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\Vert u(t)\Vert }_{{\dot{B}}_{p,q}^{s_p+\epsilon }({\mathbb{R}}^n)}{\gtrsim }_{\phi ,\epsilon ,(p\vee q\vee 2)}{(T-t)}^{-\epsilon /2}$ for all $\epsilon \in [1,2)$ and $p,q\in [1,\frac{n}{2-\epsilon })$, where $s_p:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $\phi$ is the cutoff function used to define the Littlewood-Paley projections. For $\epsilon =2$, we prove the same type of estimate but only for $q=1$: ${\Vert u(t)\Vert }_{{\dot{B}}_{p,1}^{s_p+2}({\mathbb{R}}^n)}{\gtrsim }_{\phi ,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty )$. Under the additional restriction that $p,q\in [1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $\epsilon \in (1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-López and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(\epsilon ,q)=(1,2)$ and $(\epsilon ,q)=(2,1)$.

10 pages