On putative self-similarity for incompressible 3D Euler

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 at least $2/5$.

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 $\gamma\geq 1/2$ under the sole assumption that the velocity profile is $C^2$ smooth.