Properties of Navier-Stokes mild solutions in sub-critical Besov spaces whose regularity exceeds the critical value by $\boldsymbol{\epsilon\in(0,1)}$

Abstract

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.