Monge-Ampere 几何与涡旋
Monge-Ampere Geometry and Vortices
AI总结：
本文通过更高辛几何技术研究Monge-Ampere几何，探讨其在湍流中涡旋形成中的应用，并展示如何通过拓扑信息和对称性分析理解流体动力学中的关键关系。
摘要：
我们介绍了一种基于更高辛几何技术的新方法来研究Monge-Ampere几何。我们的工作受到将Monge-Ampere几何应用于不可压缩纳维-斯托克斯流体中压力泊松方程的应用启发。尽管该方程对于压力来说是一个椭圆问题，但它也可以被视为一个非线性偏微分方程，将压力、涡度和应变率联系起来。因此，它是理解湍流中涡旋形成的关键诊断关系。我们通过与流体配置空间的余切丛相关的（更高）拉格朗日子流形来研究该方程。使用我们对（更高）Monge-Ampere结构的定义，我们研究了余切丛上的相关度量及其在（更高）拉格朗日子流形上的拉回。这些度量的符号由涡度与应变率之间的关系决定，其标量曲率可以以物理语境解释为涡度、应变及其梯度的累积。我们证明了在二维流中，如何从拉格朗日子流形的Monge-Ampere几何中推导出拓扑信息。我们还展示了某些不可压缩纳维-斯托克斯方程的解，如Hill的球形涡旋和Arnol'd-Beltrami-Childress流的可积情况，具有使这些解从（更高）辛约简视角出发的对称性。
英文摘要：
We introduce a new approach to Monge-Ampere geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampere geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampere structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampere geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.
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