FRG: Collaborative Research: Singularity Formation for the Three-Dimensional Euler Equations and Related Problems

不可压缩无粘流体流动的三维欧拉方程的奇点形成问题是数学和物理学中一个著名的开放问题。欧拉奇点的存在可能对物理流体动力学有重大影响，特别是在湍流的开始和结构中起作用。本研究将采用数值与解析相结合的方法来研究欧拉奇点的形成，并考察这些奇点对流体动力湍流的意义。该项目的核心是一种构造奇异欧拉解的新方法，从使用复时空和复平面上的奇点的半解析方法开始。结果应符合严格的分析和直接的数值验证。对奇点结构进行完整的数值和解析验证是本方案的主要组成部分。还将开展涉及界面内波奇点形成和不可压缩非耗散磁流体力学的几个相关项目。不可压缩欧拉方程是一个描述无粘流体流动的偏微分方程组。尽管这些方程已经被发现了近250年，但关于解的本质的基本数学问题仍然是开放的。特别是，三维欧拉方程的解是否能在有限时间内形成奇点，即速度或涡量（测量循环）等流量的无穷大值，目前尚不清楚。由于其在湍流理论中的意义，欧拉奇点问题在过去的20年里受到了广泛的关注。欧拉奇点的成功构造将解决一个重大的数学问题，并为解决奇点的形成建立一种新的方法。对这些奇点的流体动力学理解可能会导致对湍流结构的重要见解，这是经典物理学的主要开放问题之一。这反过来又可能导致理解和模拟湍流的重要新方法，这在广泛的工程应用中是必不可少的，包括飞机和船只的设计。

The question of singularity formation for the three-dimensional Euler equations of incompressible inviscid fluid flow is a celebrated open problem in mathematics and physics. The existence of Euler singularities is likely to have substantial implications for physical fluid dynamics, in particular a role in the onset and structure of turbulence. This research will utilize a combination of numerical and analytical methods to study Euler singularity formation, as well as examine the significance of these singularities for fluid dynamic turbulence. A centerpiece of the project is a new method for constructing singular Euler solutions, starting from a semi-analytic approach using complex space-time and singularities in the complex plane. The results should be amenable to rigorous analysis and direct numerical validation. A full numerical and analytic validation of the singularity construction forms a main component of this proposal. Several related projects involving singularity formation in interfacial internal waves, and in incompressible nondissipative magnetohydrodynamics will also be undertaken. The incompressible Euler equations are a system of partial differential equations that describe the flow of inviscid fluids. Although these equations have been known for nearly 250 years, basic mathematical questions concerning the nature of solutions are still open. In particular, it is still not known whether solutions of the three dimensional Euler equations can form a singularity, that is, an infinite value in a flow quantity such as the velocity or vorticity (which measures circulation) in finite time. Due to its implications in turbulence theory, the question of Euler singularities has received intense attention over the last 20 years. Successful construction of Euler singularities would solve a major problem of mathematics and would establish a new method for addressing singularity formation. A fluid dynamic understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open problems of classical physics. This in turn could lead to important new methods for understanding and simulating turbulent flows, essential in a wide range of engineering applications, including the design of aircraft and watercraft.