On the incompressible Euler equations and the blow-up problem

Abstract

In this article we review some of the recent results on the Cauchy problem and the issue of finite time blow-up for the 3D incompressible Euler equations. We begin with the refinements Kato's local existence of classical solution and the finite time blow-up criterion by Beale, Kato and Majda, using the tools from harmonic analysis. Then, we present the developments of observation of depletion of nonlinearity of the vortex stretching term, first discovered by Constantin and Fefferman for the case of the Navier-Stokes equations. One consequence of this observation is the regularity control in terms of the direction field of the vorticity, and the other one is the regularity control in terms of the reduced number of vorticity components. We develop those idea both for the generalized Navier-Stokes equations and for the Euler equations. We also present some consequences of the dynamics of the deformation tensor. One of them leads to the spectral dynamics approaches to the Euler equations, which provides us with the local in time enstrophy growth and decay estimates, among others. The other one is characterizations of the set of points leading to the possible finite time singularities. Finally we present studies on the various model problems of the 3D Euler equations. More specifically, we introduce the 2D quasi-geostrophic equation and its one dimensional model equation, the 2D Boussinesq system and a modified Euler system. Finally we present recent result on the nonexistence of the self-similar blow-up.