| ABSTRACT |
For all ?>0, we prove the existence of finite-energy strong solutions to the axi-symmetric 3D Euler equations on the domains {(x,y,z)∈R3:(1+?|z|)2≤x2+y2} 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. |
