On possible isolated blow-up phenomena and regularity criterion of the 3D Navier-Stokes equation along the streamlines
On possible isolated blow-up phenomena and regularity criterion of the 3D Navier-Stokes equation along the streamlines
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3D Navier-Stokes 方程沿流线可能存在的孤立爆炸现象及规律判据
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Tsuyoshi Yoneda
中科院分区：
文献类型：
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作者：
C. Chan;Tsuyoshi Yoneda
摘要：本文的第一个目标是在超临界函数空间条件u ∈ L∞（Lα，λ）（其中3 4（17 1 2 − 1）< α < 3）和div（u）增长率的指数控制下，给出Navier-Stokes方程解u的一种新的正则性准则|u|）沿着u的流线。这个正则性准则大大改进了第一作者的一个结果。然而，我们也指出，全新的思想，包括使用新的超临界函数空间条件是必要的，我们的新的正则性准则在本文中的成功。我们的论文的第二个目标是构建一个无发散矢量场u内的flowinvariant管状区域增加扭曲的流线对一束流线的一端。流线的增加扭曲以这样一种方式控制，即相关量Δ u Δ Lp（2 < p < 3）和Δ div（u| u|）在保持有限能量性质u ∈ L的同时，证明了L 6的爆破.我们还简要地提到这个结构是如何与我们的论文中证明的正则性准则。
Abstract: The first goal of our paper is to give a new type of regularity criterion for solutions u to Navier-Stokes equation in terms of some supercritical function space condition u ∈ L∞(Lα,∗) (with 3 4 (17 1 2 − 1) < α < 3) and some exponential control on the growth rate of div( u |u| ) along the streamlines of u. This regularity criterion greatly improves a previous result of the first author. However, we also point out that totally new idea which involves the use of the new supercritical function space condition is necessary for the success of our new regularity criterion in this paper. The second goal of our paper is to construct a divergence free vector field u within a flowinvariant tubular region with increasing twisting of streamlines towards one end of a bundle of streamlines. The increasing twisting of streamlines is controlled in such a way that the associated quantities ‖u‖Lp (2 < p < 3) and ‖div( u |u| )‖L6 blow up while preserving the finite energy property u ∈ L at the same time. We also briefly mention how this construction is related to the regularity criterion proved in our paper.