标题： 三维纳维-斯托克斯方程解的正则性与奇异性 显示英文标题

标题： Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations

摘要： 三维不可压缩纳维-斯托克斯方程在域$[0, L]^{3}_{per}$上解的正则性问题通过涡度场$\Omega_{m}(t)$的高阶矩，以$L^{2m}$范数（$1 \leq m < \infty$）的形式进行研究。发现量$$ D_{m}(t) = \Omega_{m}^{\alpha_{m}} ,\qquad\qquad\alpha_{m} = \frac{2m}{4m-3}, $$的集合在问题中提供了自然的缩放，从而在有限的时间区间$[0, T]$上导致了一组有界的平均时间$<D_{m}>_{T}$。 $D_{m+1}/D_{m}$的行为在被称为“好”和“坏”的区间$[0, T]$上被研究，这些区间被连接点（中性）$\tau_{i}$所隔开。 对于$m$的较大但有限的值，初始数据\big （$\Omega_{m}(0) \leq \varpi_{0}O(\Gr^{4})$\big ）较大时，发现存在一个上界$$ \Omega_{m} \leq c_{av}^{2}\varpi_{0}\Gr^{4} ,\qquad\varpi_{0} = \nu L^{-2}, $$，该上界通过良好/不良区间之间的垂直壁中的无限小间隙或窗口被穿透，解可以通过这些间隙或窗口逃逸。 虽然这个结果与Leray \cite{Leray}和Scheffer \cite{Scheff76}的结果一致，但对 $\Omega_{m}$的估计对应于远低于纳维-斯托克斯方程有效性的长度尺度。 显示英文摘要

摘要： Higher moments of the vorticity field $\Omega_{m}(t)$ in the form of $L^{2m}$-norms ($1 \leq m < \infty$) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier-Stokes equations on the domain $[0, L]^{3}_{per}$. It is found that the set of quantities $$ D_{m}(t) = \Omega_{m}^{\alpha_{m}} ,\qquad\qquad\alpha_{m} = \frac{2m}{4m-3}, $$ provide a natural scaling in the problem resulting in a bounded set of time averages $<D_{m}>_{T}$ on a finite interval of time $[0, T]$. The behaviour of $D_{m+1}/D_{m}$ is studied on what are called `good' and `bad' intervals of $[0, T]$ which are interspersed with junction points (neutral) $\tau_{i}$. For large but finite values of $m$ with large initial data \big($\Omega_{m}(0) \leq \varpi_{0}O(\Gr^{4})$\big), it is found that there is an upper bound $$ \Omega_{m} \leq c_{av}^{2}\varpi_{0}\Gr^{4} ,\qquad\varpi_{0} = \nu L^{-2}, $$ which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray \cite{Leray} and Scheffer \cite{Scheff76}, this estimate for $\Omega_{m}$ corresponds to a length scale well below the validity of the Navier-Stokes equations.