H-Principle and Fluid Dynamics

流体动力学是物理学的一个分支，它研究流体的运动，即液体和气体，它使用复杂的数学来描述运动，做出有用的预测，并找到有效的方法来解决科学和工程中的实际问题。虽然这些描述背后的许多理论都植根于19世纪的物理学，但由于其中涉及的数学对象的复杂性，许多方面仍然知之甚少。本研究项目主要关注两个问题。(1)理论上，当作用于流体的力及其初始状态已知时，数学模型应能确定流体的未来演变；然而，对于最简单和最流行的模型来说，这种期望的严格数学证明是缺失的。用外行的话来说，原因是我们不理解表面上“平稳和平静”的流体运动“如何以及为什么”演变成“混乱和动荡”的运动。(2)即使我们期望模型具有完全的可预测性，然而，计算往往是如此复杂，以至于实际上是不可能的。由于这个原因，科学家们已经开发出了运动的统计描述，可以在不需要大量计算的情况下推断出它们的典型行为。从描述运动的原始数学模型中严格证明这些统计描述仍然是现代科学的一大挑战。本项目涉及流体动力学数学中最常用的偏微分方程组，欧拉方程和纳维-斯托克斯方程。该项目的三个主要主题如下。(A)流体粘度趋于零时，Navier-Stokes方程解收敛于Euler方程解。一个主要的问题是湍流理论的一个原则，即对于固定宏观尺度的流体运动，能量耗散率与粘度无关。最近，Lars Onsager关于低正则解的能量耗散的一个著名猜想的解决，为证明这种反常耗散的存在铺平了道路。(B)解的规律性和奇异性。千禧年奖的一个著名问题是，在有限时间内，Navier-Stokes方程的光滑解是否会出现奇点。Navier-Stokes方程可以“嵌入”在一个单参数的模型家族中，其中参数描述粘性项的正则化强度。虽然关于Navier-Stokes方程的一些已知结果已经推广到后者，但仍有相当多的结果缺乏合适的对应。(C)黎曼流形等距嵌入的刚性和柔性。(A)点提到的Onsager猜想的最新解决方案是基于项目负责人Laszlo Szekelyhidi Jr.的发现，该发现与约翰·纳什（John Nash）在50年代解决的黎曼几何经典问题有意想不到的联系。然而，几何框架中的Onsager猜想的对应物尚未完全解决。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Fluid dynamics is the branch of physics that studies the motion of fluids, that are liquids and gases, and it uses sophisticated mathematics to describe the motions, make useful predictions and find efficient ways to solve practical problems in science and engineering. While much of the theory behind such descriptions are rooted in the physics of the nineteenth century, due to the complexity of the mathematical objects involved in them many aspects are still poorly understood. Two problems are of primary interest in this research project. (1) In theory the mathematical models should determine the future evolution of the fluid when the forces acting on it and its initial status are known; nonetheless a rigorous mathematical proof of this expectation is missing for the simplest and most popular models. In layman terms the reason is that we do not understand "how and why" fluid motions which are apparently "smooth and calm" evolve into "chaotic and turbulent" ones. (2) Even if we expect the complete predictability of the models, nonetheless the computations are often so complex that become practically impossible. For this reason scientists have developed statistical descriptions of the motions, which allow to infer their typical behavior without resorting to heavy computations. The rigorous justification of these statistical descriptions from the primary mathematical models describing the motions is still a big challenge of modern science.This project is concerned with some of the most popular systems of partial differential equations used in the mathematics of fluid dynamics, the Euler and Navier-Stokes equations. The three main themes of the project are the following. (A) The convergence of solutions of the Navier-Stokes equations to the Euler equations when the viscosity of the fluid goes to zero. A primary concern is a tenet of the theory of turbulence that the rate of energy dissipation is, for fluid motions of a fixed macroscopic scale, independent of the viscosity. The recent resolution of a well-known conjecture of Lars Onsager on the energy dissipation for low regularity solutions of Euler could pave the way to a mathematical proof of the existence of such anomalous dissipation. (B) Regularity and singularity of the solutions. One of the celebrated Millennium prize problems asks whether smooth solutions of the Navier-Stokes equations develop singularities in finite time. The Navier-Stokes equations can be "embedded" in a one-parameter family of models where the parameter describes the regularizing strength of the viscous term. While some of the known results concerning the Navier-Stokes equations have been extended to the latter family, there is still quite a few which lack a suitable counterpart. (C) Rigidity and flexibility of the isometric embeddings of Riemannian manifolds. The recent resolution of the Onsager conjecture mentioned at point (A) is based on the discovery of the project leader and Laszlo Szekelyhidi Jr. of unexpected connections with a classical problem in Riemannian geometry, solved by John Nash in the fifties. The counterpart of the Onsager conjecture in the geometric framework is however not yet fully solved.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.