论文标题
在I型奇点条件下，就$ \ Mathbb r^3 $的Euler方程的压力而言
On a Type I singularity condition in terms of the pressure for the Euler equations in $\mathbb R^3$
论文作者
论文摘要
我们证明了平滑溶液的压力$ u \ in c（[0，t）的爆炸标准; w^{2，q}（\ Mathbb r^3））$，$ q> 3 $不可压缩的Euler方程。我们表明，$ t = t $炸毁仅发生时才发生，仅当$$ \ int_0 ^t \ int_0 ^t \ left \ weft \ {\ int_0 ^s \ | d ^2 p（τ）\ | | _ {l ^\ in {l ^\ infty}dτ\ exp_ （τ）\ | _ {l^\ infty}dτd\ s \ right）\ right \} dsdt \，= +\ infty。 {c} {（t-t）^2} $，$ c <1 $ as $ t \ nearrow t $。在$ \ int_0 ^t \ | u（t）\ | _ {l ^\ infty（b（x_0，ρ））} dt <+\ infty $的其他假设下，我们获得了这些结果的本地化版本。
We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions $u\in C([0, T); W^{2,q} (\mathbb R^3))$, $q>3$ of the incompressible Euler equations. We show that a blow up at $t=T$ happens only if $$\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (τ)\|_{L^\infty} dτ\exp \left( \int_{s} ^t \int_0 ^{\s} \|D^2 p (τ)\|_{L^\infty} dτd\s \right) \right\}dsdt \, = +\infty.$$ As consequences of this criterion we show that there is no blow up at $t=T$ if $ \|D^2 p(t)\|_{L^\infty} \le \frac {c}{(T-t)^2}$ with $c<1$ as $t\nearrow T$. Under the additional assumption of $\int_0 ^T \|u(t)\|_{L^\infty (B(x_0, ρ))} dt <+\infty$, we obtain localized versions of these results.
