Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

We consider the 2D Euler equations on (Formula presented) (Formula presented) in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index (Formula presented) (Formula presented). We show weak existence for every (Formula presented) (Formula presented) initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional (Formula presented) (Formula presented) regularity. For every (Formula presented) (Formula presented) and for certain regularity indices (Formula presented) (Formula presented) of the Kraichnan noise, we show also pathwise uniqueness for every Lp initial vorticity. This result is not known without noise.