On putative self-similarity for incompressible 3D Euler

Abstract

We consider hypothetical solutions of 3D Euler which blow up in finite time in a self-similar fashion. We prove that if the initial data has finite kinetic energy, then the similarity exponent ${\hat{I}}^3$ which governs the rate of zooming in must be at least $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that ${\hat{I}}^3\ge 1/2$. For axisymmetric solutions, we establish the bound ${\hat{I}}^3\ge 1/2$ under the sole assumption that the velocity profile is $C^2$ smooth.