On putative self-similarity for incompressible 3D Euler

TL;DR

This paper investigates the conditions under which hypothetical self-similar finite-time blowup solutions of the 3D Euler equations can exist, establishing lower bounds on the similarity exponent based on initial data and solution properties.

Contribution

It proves lower bounds on the similarity exponent for self-similar blowup solutions of 3D Euler, including cases with specific profile properties and axisymmetry.

Findings

If initial data has finite energy, then rac{b3}{2}

Existence of smooth self-similar profiles implies rac{b3}{2}

For axisymmetric solutions, rac{b3}{2} in general settings.

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 $\gamma$ which governs the rate of zooming in must be larger than $2/5$. If a smooth globally self-similar blowup profile exists, and this profile satisfies an outgoing property, we prove that $\gamma \geq 1/2$. For axisymmetric solutions, we establish the bound $\gamma\geq 1/2$ in more general settings, including ones in which the outgoing property is not present.