Incompressible Euler Equations: the blow-up problem and related results

TL;DR

This survey reviews recent mathematical advances on the 3D incompressible Euler equations, focusing on singularity formation, regularity criteria, simplified models, and the ongoing challenge of understanding finite-time blow-up.

Contribution

It summarizes recent developments in criteria for blow-up, regularity conditions, and simplified models, providing a comprehensive overview of the current state of research.

Findings

Review of Kato's local well-posedness result

Discussion of Beale-Kato-Majda blow-up criterion

Analysis of scenarios excluding finite-time singularities

Abstract

The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent approaches to the problem. We first review Kato's classical local well-posedness result in the Sobolev space and derive the celebrated Beale-Kato-Majda criterion for finite time blow-up. Then, we discuss recent refinements of the criterion as well as geometric type of theorems on the sufficiency condition for the regularity of solutions. After that we review results excluding some of the scenarios leading to finite time singularities. We also survey studies of various simplified model problems. A dichotomy type of result between the finite time blow-up and the global in time regular dynamics is presented, and a spectral dynamics approach to study local in time behaviors of the enstrophy is also reviewed. Finally, progresses on the problem of optimal regularity for solutions to have conserved quantities are presented.