三维NS流中涡量的振荡和可积性
Oscillations and integrability of the vorticity in the 3D NS flows
摘要：
在Navier-Stokes（NS）正则性问题的研究中，越来越清楚的是，一个更现实的改进先验界的途径是试图摆脱爆破型参数领域中能级估计的尺度（所考虑的解在可能的爆破时间内是正则/光滑的），而不是试图改进任意Leray弱解的正则性。本文章是在这个方向上的贡献，更确切地说，它是显示-在任意程度的代数/多项式型爆破配置文件的背景下-一个非常弱的条件对涡度方向场（成员资格在一个本地$bmo$空间加权任意多个矩阵）-足以打破能级标度的涡度的界限。同时，所得到的边界将由宏观尺度长涡丝所描述的三维NS临界情景转化为无奇异性情景。
英文摘要：
In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the blow-up-type arguments (the solution in view is regular/smooth up to the possible blow-up time) rather than to try to improve regularity of arbitrary Leray's weak solutions. The present article is a contribution in this direction; more precisely, it is shown--in the context of an algebraic/polynomial-type blow-up profile of arbitrary degree--that a very weak condition on the vorticity direction field (membership in a local $bmo$ space weighted with arbitrary many logarithms)--suffices to break the energy-level scaling in the bounds on the vorticity. At the same time, the obtained bounds transform a 3D NS criticality scenario depicted by the macro-scale long vortex filaments into a no-singularity scenario.
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