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arXiv 0705.2446math.AP
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三维Navier-斯托克斯方程的速度方向正则性判据

Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity

Alexis Vasseur

摘要:

在这篇短文中,我们给出了三维斯托克斯方程解$u$的正则性与速度方向的行为之间的联系$u/|u|$。结果表明,控制$\Div (u/|u|)$在一个合适的$L_t^p(L_x^q)$规范是足够的,以确保全球的正则性。这个结果让人想起了由Constantin和Frederman首先提出的涡度方向的判据。但在这种情况下,条件不是涡量,而是速度本身。这个证明基于非常标准的方法,依赖于速度方向的发散和能量沿流线的增长之间的直接关系。

英文摘要:

In this short note, we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation, and the behavior of the direction of the velocity $u/|u|$. It is shown that the control of $\Div (u/|u|)$ in a suitable $L_t^p(L_x^q)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. But in this case the condition is not on the vorticity, but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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