Nonexistence of self-similar singularities for the 3D incompressible Euler equations

Abstract

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in ${\mathbb{R}}^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.

This version refines the previous one by relaxing the condition of compact support for the vorticity