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

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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.