论文标题
Navier-Stokes的属性在亚临界空间中的温和解决方案，其规律性超过临界值，$ \ boldsymbol {ε\ in（0,1）} $
Properties of Navier-Stokes mild solutions in sub-critical Besov spaces whose regularity exceeds the critical value by $\boldsymbol{ε\in(0,1)}$
论文作者
论文摘要
我们考虑属于某些范围的Navier-Stokes初始值问题的温和解决方案$ Z_ { ^{n}）））\ cap \ widetilde {l}^{\ infty}（0，t; \ dot {b} _ {p，q}^{s}^{s}（\ mathbb {r} Chemin-Lerner空间。 $ n = 3 $，$ε\ in（0,1）$和$ f \ in \ dot {b} _ {\ infty，\ infty，\ infty}^{ - 1+ε}（\ Mathbb {r}^{3}^{3}^{3}^{3}） $ u \ in \ cap_ {t'\ in（0，t_ {f，ε}^{*}）} z _ {\ infty，\ infty，\ infty}^{ - 1+ε}（t'，t'，3）$具有最大存在时间$ {其中$φ$是用于定义Littlewood-Paley投影的截止功能。我们对此结果进行改进如下：对于$ n \ geq 1 $，$ε\ in（0,1）$，$ s \ in（-1，\ infty）$，$ p，q \ in [1，\ infty] $和初始数据$ f \ in \ dot {b} _ { $ u \ in \ cap_ {t'\ in（0，t^*_ f）} \ left（z_ {p，q}^{s}^{s}（t'，t'，n）\ cap z _ {\ infty，\ infty，\ infty，\ infty}^{ - 1+am}（t's） $ t_ {f}^{*} \ in（0，\ infty] $，独立于$ε，s，p，q $。如果$ t_ {f} {f}^{*} $是有限的$ {\ | u（t）\ |} _ {\ dot {b} _ {\ infty，\ infty}^{ - 1+ε}（\ Math bb {r}^{n}）}} \gtrsim_φε（1-ε）{（t_ {f}^{*} - t）}^{ - ε/2} $对于所有$ t \ in（0，t_ {f}^{*}）$。 $ \ cap_ {t'\ in（0，t_ {f}^{*}）} \ cup_ {α\ in（2，\ infty）} l^α（0，t'; l^{\ infty} $ u \ notin l^{2}（0，t_ {f}^{*}; l^{\ infty}（\ mathbb {r}^{n}））$。
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 $n=3$, $ε\in(0,1)$ and $f\in\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{3})$, Chemin and Gallagher (Tunis. J. Math., 2019) construct a local solution $u\in\cap_{T'\in(0,T_{f,ε}^{*})}Z_{\infty,\infty}^{-1+ε}(T',3)$ with maximal existence time ${T_{f,ε}^{*}\gtrsim_{φ,ε}{\|f\|}_{\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{3})}^{-2/ε}}$, where $φ$ is the cutoff function used to define the Littlewood-Paley projections. We improve on this result as follows: for $n\geq 1$, $ε\in(0,1)$, $s\in(-1,\infty)$, $p,q\in[1,\infty]$, and initial data $f\in\dot{B}_{p,q}^{s}(\mathbb{R}^{n})\cap\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{n})$, we prove that there exists a unique local solution $u\in\cap_{T'\in(0,T^*_f)}\left(Z_{p,q}^{s}(T',n)\cap Z_{\infty,\infty}^{-1+ε}(T',n)\right)$ which, along with its maximal existence time $T_{f}^{*}\in(0,\infty]$, is independent of $ε,s,p,q$. If $T_{f}^{*}$ is finite, then we have the blow-up estimate (with explicit dependence on $ε$) ${\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{n})}\gtrsim_φε(1-ε){(T_{f}^{*}-t)}^{-ε/2}$ for all $t\in(0,T_{f}^{*})$. The solution is unique among all solutions in the larger class $\cap_{T'\in(0,T_{f}^{*})}\cup_{α\in(2,\infty)}L^α(0,T';L^{\infty}(\mathbb{R}^{n}))$, 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.
