Singularity Formation in Fluid Mechanics and Related Equations

The Euler and the Navier-Stokes equations are the most established models in fluid dynamics. Scientists and engineers apply these equations to model various phenomena, including weather patterns, ocean currents, flows around vehicles, aircraft, and ships, as well as blood flow. Mathematicians and physicists believe that understanding the solutions to these equations can lead to an explanation for turbulence. Despite their wide range of applications, there is no theoretical guarantee that smooth solutions to these equations can exist for all time. Mathematically, proving whether smooth solutions to these equations without external forces exist for all time or can break down in finite time has been a longstanding open problem. The potential breakdown mechanism also remains elusive for several related equations with a wide range of applications. The goal of this research is to investigate the potential breakdown mechanism for various equations and develop analytic and numeric tools that provide a theoretical understanding of these mechanisms. This award will also provide opportunities for students to be involved in the latest developments from this research through topics courses and research projects. The project aims to understand whether the incompressible 3D Euler equations and related equations could develop a finite time singularity from a smooth initial condition with finite energy. Our approach builds on PI's recent works on singularity formation in incompressible fluids and the self-similar method for finite time blowup, which consists of the following three steps. Firstly, we construct an approximate blowup profile, which can be obtained either analytically or numerically. Secondly, we impose suitable normalization conditions and prove the nonlinear stability of the approximate blowup profile. Thirdly, we choose a suitable perturbation to construct initial data with desired property and obtain finite time blowup using a rescaling relation. Additionally, the project seeks to develop a novel approach for the stability analysis in the self-similar variables that is robust enough to be applied to study a larger class of nonlinear partial differential equations. 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.