A catalogue of singularities
Jens Eggers, Marco A. Fontelos

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.
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Taxonomy
TopicsQuantum chaos and dynamical systems · Fluid dynamics and aerodynamics studies · Fluid Dynamics and Thin Films
