A catalogue of singularities

TL;DR

This paper classifies finite-time singularities in PDEs, especially in free-surface flows, by analyzing the self-similar behavior near singular points through dynamical systems theory.

Contribution

It introduces a novel framework using infinite-dimensional dynamical systems to classify singularities based on fixed points and their stability.

Findings

Classification of singularities via fixed points and stability analysis

Identification of self-similar solutions as fixed points in dynamical systems

Use of center-manifold and chaos theory to understand singularity structures

Abstract

This paper is an attempt to classify finite-time singularities of PDEs. Most of the problems considered describe free-surface flows, which are easily observed experimentally. We consider problems where the singularity occurs at a point, and where typical scales of the solution shrink to zero as the singularity is approached. Upon a similarity transformation, exact self-similar behaviour is mapped to the fixed point of a {\it infinite dimensional dynamical system} representing the original dynamics. We show that the dynamics close to the fixed point is a useful way classifying the structure of the singularity. Specifically, we consider various types of stable and unstable fixed points, centre-manifold dynamics, limit cycles, and chaotic dynamics.