与Navier-Stokes方程相关的Riemann空间：几何方法综述
Riemann spaces associated with the Navier-Stokes equations: a survey of the geometric approach
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中文总结 AI 辅助
本文综述了作者提出的Navier-Stokes方程的十四维Riemann几何方法，其中Ricci平坦性等价于方程解，并展示了Euler与Lagrangian描述的几何统一，以及Cartan不变量在构造三维Taylor-Green涡旋中的应用。
中文摘要 AI 辅助
我们综述了作者在过去十五年间发展的Navier-Stokes方程的几何方法。其核心对象是一个十四维的Riemann空间，该空间的度量恰好当Navier-Stokes系统的解成立时是Ricci平坦的，因此粘性不可压缩流体的行为成为曲率的一种性质。该空间分解为一个平坦的六维部分和两对对偶的坐标四元组，即Euler坐标和Lagrangian坐标，这展示了流体的两种经典描述作为单一几何的两个半部分。我们还回顾了相关的六维度量，其可积条件是流体的不可压缩性；该度量与E. Cartan意义下的二阶常微分方程的射影几何之间的联系；以及利用Cartan不变量构造Taylor-Green涡旋的三维类比。最后指出了将同一构造应用于刚体旋转方程的可能性。
英文摘要
We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely on the solutions of the Navier-Stokes system, so that the behaviour of a viscous incompressible fluid becomes a property of curvature. The space decomposes into a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, which exhibits the two classical descriptions of a fluid as two halves of a single geometry. We also recall the associated six-dimensional metric, whose integrability condition is the incompressibility of the fluid; the link with the projective geometry of second-order ordinary differential equations in the sense of E. Cartan; and the use of Cartan's invariants to construct a three-dimensional analogue of the Taylor-Green vortex. An application of the same construction to the equations of rotation of a rigid body is indicated.
发表机构
- Vladimir Andrunachievici Institute of Mathematics and Computer Science（弗拉基米尔·安德鲁纳奇耶维奇数学与计算机科学研究所）
- Moldova State University（摩尔多瓦国立大学）
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