  For a solution $u$ to the Navier-Stokes equations in spatial dimension
$n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate
${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^n)}\gtrsim_{\varphi,\epsilon,(p\vee
q\vee 2)}{(T-t)}^{-\epsilon/2}$ 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 $\varphi$ is the cutoff function used to
define the Littlewood-Paley projections. For $\epsilon =2$, we prove the same
type of estimate but only for $q=1$:
${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{\varphi,(p\vee
2)}{(T-t)}^{-1}$ 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-L\'{o}pez and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases
$(\epsilon,q)=(1,2)$ and $(\epsilon,q)=(2,1)$.
