Singularity Formation and Propagation in Incompressible Fluids

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This project revolves around the mathematical analysis of certain types of coherent structures in fluids. Roughly speaking, a coherent structure is a well-defined feature in fluid flow that can be easily distinguished and mathematically described in a simple manner. One example is a vortex, where all fluid particles rotate about a single point. Another example is a sharp front, which may arise when a fluid consists of warm and cold regions separated by a thin moving transition region. Many more examples of coherent structures exist in the natural world and we encounter them daily. While it seems to be an extremely formidable problem to give an efficient way to describe general fluid flows, describing the evolution of coherent structures seems to be amenable to mathematical analysis. Moreover, since coherent structures are often observed to be dominant in physical and numerical experiments, describing their evolution is of great importance. A simple question that one could ask in this regard is: What happens to a "strong" vortex under small perturbations? Does the vortex persist or does it quickly disintegrate? Another question of interest is whether nice fluid flows can develop coherent structures that possess a singularity, that is, infinite velocity or velocity gradient. Such questions lie at the core of this project. Mathematically, the research focuses primarily on the dynamics of both weak and strong solutions to the incompressible Euler equations and related models. The project investigates a novel approach to the classical problem of finite-time singularity formation in fluids with zero viscosity -- that is, the emergence of certain singular structures in a fluid without external influence. Previous work provided an example of finite-time singularity formation for strong solutions to the incompressible Euler equations in certain settings. Part of the project involves extending and strengthening these results as well as investigating the applicability of the methods to other questions including the dynamics of vortex patches in the 2D Euler equation. Another part of the project studies the stability of certain singular weak solutions to the incompressible Euler equation and related models, specifically, the stability of singular vortices with respect to smoother perturbations.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.