Finite-time Singularity formation for Strong Solutions to the axi-symmetric $3D$ Euler Equations

Abstract

For all $\epsilon >0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $\{(x,y,z)\in {\mathbb{R}}^3:(1+\epsilon |z|{)}^2\le x^2+y^2\}$ which become singular in finite time. We further show that solutions with 0 swirl are necessarily globally regular. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a $1D$ system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system \cite{EJB} though a number of modifications must be made due to anisotropy and since our domains are not scale-invariant. This seems to be the first construction of singularity formation for finite-energy strong solutions to the actual $3D$ Euler system.

46 pages, 1 figure. arXiv admin note: text overlap with arXiv:1708.09372