  We consider mild solutions to the Navier-Stokes initial-value problem which
belong to certain ranges
$Z_{p,q}^{s}(T,n):=\widetilde{L}^{1}(0,T;\dot{B}_{p,q}^{s+2}(\mathbb{R}^{n}))\cap\widetilde{L}^{\infty}(0,T;\dot{B}_{p,q}^{s}(\mathbb{R}^{n}))$
of Chemin-Lerner spaces. For $n=3$, $\epsilon\in(0,1)$ and
$f\in\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^{3})$, Chemin and
Gallagher (Tunis. J. Math., 2019) construct a local solution
$u\in\cap_{T'\in(0,T_{f,\epsilon}^{*})}Z_{\infty,\infty}^{-1+\epsilon}(T',3)$
with maximal existence time
${T_{f,\epsilon}^{*}\gtrsim_{\varphi,\epsilon}{\|f\|}_{\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^{3})}^{-2/\epsilon}}$,
where $\varphi$ is the cutoff function used to define the Littlewood-Paley
projections. We improve on this result as follows: for $n\geq 1$,
$\epsilon\in(0,1)$, $s\in(-1,\infty)$, $p,q\in[1,\infty]$, and initial data
$f\in\dot{B}_{p,q}^{s}(\mathbb{R}^{n})\cap\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^{n})$,
we prove that there exists a unique local solution
$u\in\cap_{T'\in(0,T^*_f)}\left(Z_{p,q}^{s}(T',n)\cap
Z_{\infty,\infty}^{-1+\epsilon}(T',n)\right)$ which, along with its maximal
existence time $T_{f}^{*}\in(0,\infty]$, is independent of $\epsilon,s,p,q$. If
$T_{f}^{*}$ is finite, then we have the blow-up estimate (with explicit
dependence on $\epsilon$)
${\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^{n})}\gtrsim_{\varphi}\epsilon(1-\epsilon){(T_{f}^{*}-t)}^{-\epsilon/2}$
for all $t\in(0,T_{f}^{*})$. The solution is unique among all solutions in the
larger class
$\cap_{T'\in(0,T_{f}^{*})}\cup_{\alpha\in(2,\infty)}L^{\alpha}(0,T';L^{\infty}(\mathbb{R}^{n}))$,
and if $T_{f}^{*}<\infty$ then $u\notin
L^{2}(0,T_{f}^{*};L^{\infty}(\mathbb{R}^{n}))$. We also establish additional
properties of the solution, depending on the Besov spaces to which the initial
data belongs.
