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

Abstract We consider the 2D Euler equations on R 2 in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index α ∈ ( 0 , 1 ) . We show weak existence for every H ˙ − 1 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 L 2 ( [ 0 , T ] ; H − α ) regularity. For every p > 3 / 2 and for certain regularity indices α ∈ ( 0 , 1 / 2 ) of the Kraichnan noise, we show also pathwise uniqueness for every L p initial vorticity. This result is not known without noise.