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We consider the 2D Euler equations on $\mathbb{R}^2$ in vorticity form, withunbounded initial vorticity, perturbed by a suitable non-smooth Kraichnantransport noise, with regularity index $\alpha\in (0,1)$.  We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks tothe noise, the solutions that we construct are limits in law of a regularizedstochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$regularity.  For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ ofthe Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initialvorticity. This result is not known without noise.

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