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

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.

75 page (to appear in Handbook on Evolutionary Differential Equations)

References in corpus (8)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation
• Spectral Dynamics of the Velocity Gradient Field in Restricted Flows
• Nonexistence of self-similar singularities for the 3D incompressible Euler equations
• On the Collapse of Tubes Carried by 3D Incompressible Flows
• Computational Euler History
• Ortho-normal quaternion frames, Lagrangian evolution equations and the three-dimensional Euler equations
• Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

Cited by in corpus (10)

• Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice
• Development of high vorticity structures in incompressible 3D Euler equations
• Renormalization and universality of blowup in hydrodynamic flows
• Development of high vorticity structures and geometrical properties of the vortex line representation
• The Navier-Stokes equations for Incompressible Flows: solution properties at potential blow-up times
• On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations
• Localized non blow-up criterion of the Beale-Kato-Majda type for the 3D Euler equations
• Global regularity for the 3D Navier-Stokes and the 3D Euler equations
• Nonexistence of self-similar singularities in the ideal magnetohydrodynamics
• On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations