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arXiv math/0112128math.AP
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Navier-斯托克斯方程的近恒等变换

Near identity transformations for the Navier-Stokes equations

Peter Constantin

摘要:

Navier-斯托克斯方程及其各种近似可以用近恒等映射来描述,近恒等映射是物理空间的扩散粒子路径变换。主动速度由扩散路径变换和使用韦伯公式的虚速度得到。主动涡量是由扩散路径变换和利用柯西公式得到的虚涡量。在粘性为零的情况下,虚速度和虚涡量服从扩散方程,形式上可归结为被动平流。除了与粘度成比例外,这些扩散方程的系数涉及近恒等变换的二阶导数,并且与Christoffel系数有关。如果近恒等变换过分偏离恒等变换,则重新计算。最大拟能的最小时间之间的两个连续的复位的下界。

英文摘要:

The Navier-Stokes equations and their various approximations can be described in terms of near identity maps, that are diffusive particle path transformations of physical space. The active velocity is obtained from the diffusive path transformation and a virtual velocity using the Weber formula. The active vorticity is obtained from the diffusive path transformation and a virtual vorticity using a Cauchy formula. The virtual velocity and the virtual vorticity obey diffusive equations, which reduce to passive advection formally, if the viscosity is zero. Apart from being proportional to the viscosity, the coefficients of these diffusion equations involve second derivatives of the near identity transformation and are related to the Christoffel coefficients. If and when the near-identity transformation departs excessively from the identity, one resets the calculation. Lower bounds on the minimum time between two successive resettings are given in terms of the maximum enstrophy.

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