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.
Categories: math.AP