论文标题
Navier-Stokes爆炸率在某些BESOV空间中，其规律性超过了临界值，而不是$ \ boldsymbol {ε\ in [1,2]} $
Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $\boldsymbol{ε\in [1,2]}$
论文作者
论文摘要
对于解决方案的解决方案$ u $，在空间维度$ n \ geq3 $中的Navier-Stokes方程，该方程在有限的时间$ t> 0 $，我们证明了爆炸估算值$ {\ | u（t）\ |} _ {\ dot {b} _ { [1,2）$和$ p，q \ in [1，\ frac {n} {2-ε}）$中的所有$ε\，其中$ s_ {p}：= - 1+ \ 1+ \ frac {n} {n} {p} $是callication-Callitical-Political-Political-Political-Political-Partial-Partical-Parducaritions and $φ$是$φ$是cutoff function define define the Little function definion the Little woody projictions procictions procictions procictionsproc证剂。对于$ε= 2 $，我们证明了相同类型的估算类型，但仅适用于$ q = 1 $：$ {\ | u（t）\ |} _ {\ dot {\ dot {b} _ {p，1}^{s_ {p} 2）} {（t-t）}^{ - 1} $ for [1，\ infty）$中的所有$ p \。在额外的限制下，$ p，q \在[1,2] $和$ n = 3 $中，这些爆炸估计值是罗宾逊，萨德沃斯基和席尔瓦（J.Math。phys。，2012）首先证明的，$ p = q = q = q = 2 $，case $ε\ in（1,2）$，以及mccormick，olson，robins，robhim and robhors，robhor and robhor and robhor，vid vid vid vid （Siam J.Math。Anal。，2016）对于$ P = 2 $，在情况下$（ε，q）=（1,2）$和$（ε，Q）=（2,1）$。
For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+ε}(\mathbb{R}^n)}\gtrsim_{φ,ε,(p\vee q\vee 2)}{(T-t)}^{-ε/2}$ for all $ε\in[1,2)$ and $p,q\in[1,\frac{n}{2-ε})$, where $s_{p}:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $φ$ is the cutoff function used to define the Littlewood-Paley projections. For $ε=2$, we prove the same type of estimate but only for $q=1$: ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{φ,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty)$. Under the additional restriction that $p,q\in[1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $ε\in(1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-López and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(ε,q)=(1,2)$ and $(ε,q)=(2,1)$.
