Collaborative Research: Numerics and Analysis of Singularities for the Euler Equations

该项目涉及寻找偏微分方程奇异解的方法的发展，并应用于欧拉方程和相关问题。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重要的开放性问题。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。研究者的方法来构建奇异的解决方案是通过互补的分析和数值方法，并将建立在他们以前的结果涉及复杂的数值结构，奇异欧拉解决方案。他们现在提出进一步验证数值结果和一个实奇异解的解析构造，作为复解的扰动。研究人员还将通过将奇点映射为光滑解来展开奇点，目的是对奇点进行严格的分析。这种展开已经在二维Boussinesq流动的相关问题上进行了，并将作为本提案的一部分推广到具有旋流的轴对称流动和三维欧拉流动。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）等流量的无限大值，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。

This project involves the development of methods for finding singular solutions to partial differential equations, with applications to the Euler equations and related problems. Thequestion of singularity formation for the three dimensional Eulerequations of incompressible inviscid fluid flow is an important 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. The investigator's approach to constructing singular solutions is by complementary analytical and numerical methods, and will build on their previous resultsinvolving the numerical construction of complex, singular Euler solutions. They now propose further validation of the numerical results and an analytic construction of a real singular solution, as a perturbation of the complex solution. The investigators will also pursue unfolding of singularities by mapping them to smooth solutions, with the aim of producing a rigorous analysis of singularities. Such unfoldings have been performed for the related problem of 2D Boussinesq flow, and will be generalized to axisymmetric flow with swirl and 3D Euler flow as part of this proposal. 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, i.e., an infinite value in a flow quantity such as the velocity or vorticity (which measures circulation), in afinite time. Due to its implications in turbulence theory, the question of Euler singularities has received intense attention. Successful construction of Euler singularities would solve a majorproblem 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.