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 (1)

• Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

Cited by in corpus (4)

• On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations
• On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations
• Global regularity for the 3D Navier-Stokes and the 3D Euler equations
• Nonexistence of self-similar singularities in the ideal magnetohydrodynamics