Exactly self-similar blow-up of the generalized De Gregorio equation

Abstract

We study exactly self-similar blow-up profiles for the generalized De Gregorio model for the three-dimensional Euler equation: w t + a u w x = u x w , u x = H w . We show that for any α ∈ ( 0 , 1 ) such that | a α | is sufficiently small, there is an exactly self-similar C α solution that blows up in finite time. This simultaneously improves on the result in Elgindi and Jeong (2020 Arch. Ration. Mech. Anal. 235 1763–817) by removing the restriction 1 / α ∈ Z and Chen et al (2021 Commun. Pure Appl. Math. 74 1282–350), which only deals with asymptotically self-similar blow-ups.