Properties of Navier-Stokes mild solutions in sub-critical Besov spaces whose regularity exceeds the critical value by

We consider mild solutions to the Navier-Stokes initial-value problem which belong to certain ranges $Z_{p,q}^s(T,n):={\tilde{L}}^1(0,T;{\dot{B}}_{p,q}^{s+2}({\mathbb{R}}^n))\cap {\tilde{L}}^{\mathrm{\infty }}(0,T;{\dot{B}}_{p,q}^s({\mathbb{R}}^n))$ of Chemin-Lerner spaces. For $n=3$, $\epsilon \in (0,1)$ and $f\in {\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1+\epsilon }({\mathbb{R}}^3)$, Chemin and Gallagher (Tunis. J. Math., 2019) construct a local solution $u\in {\cap }_{T^{\mathrm{\prime }}\in (0,T_{f,\epsilon }^{\ast })}{Z}_{\mathrm{\infty },\mathrm{\infty }}^{-1+\epsilon }(T^{\mathrm{\prime }},3)$ with maximal existence time $T_{f,\epsilon }^{\ast }{\gtrsim }_{\phi ,\epsilon }{\mathrm{\parallel }f\mathrm{\parallel }}_{{\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1+\epsilon }({\mathbb{R}}^3)}^{-2\mathrm{/}\epsilon }$, where $\phi$ is the cutoff function used to define the Littlewood-Paley projections. We improve on this result as follows: for $n\ge 1$, $\epsilon \in (0,1)$, $s\in (-1,\mathrm{\infty })$, $p,q\in [1,\mathrm{\infty }]$, and initial data $f\in {\dot{B}}_{p,q}^s({\mathbb{R}}^n)\cap {\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1+\epsilon }({\mathbb{R}}^n)$, we prove that there exists a unique local solution $u\in {\cap }_{T^{\mathrm{\prime }}\in (0,T_f^{\ast })}(Z_{p,q}^s(T^{\mathrm{\prime }},n)\cap Z_{\mathrm{\infty },\mathrm{\infty }}^{-1+\epsilon }(T^{\mathrm{\prime }},n))$ which, along with its maximal existence time $T_f^{\ast }\in (0,\mathrm{\infty }]$, is independent of $\epsilon ,s,p,q$. If $T_f^{\ast }$ is finite, then we have the blow-up estimate (with explicit dependence on $\epsilon$) ${\mathrm{\parallel }u(t)\mathrm{\parallel }}_{{\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1+\epsilon }({\mathbb{R}}^n)}{\gtrsim }_{\phi }\epsilon (1-\epsilon ){(T_f^{\ast }-t)}^{-\epsilon \mathrm{/}2}$ for all $t\in (0,T_f^{\ast })$. The solution is unique among all solutions in the larger class ${\cap }_{T^{\mathrm{\prime }}\in (0,T_f^{\ast })}{\cup }_{\alpha \in (2,\mathrm{\infty })}{L}^{\alpha }(0,T^{\mathrm{\prime }};L^{\mathrm{\infty }}({\mathbb{R}}^n))$, and if $T_f^{\ast }<\mathrm{\infty }$ then $u\mathrm{\notin }{L}^2(0,T_f^{\ast };L^{\mathrm{\infty }}({\mathbb{R}}^n))$. We also establish additional properties of the solution, depending on the Besov spaces to which the initial data belongs.