Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity
We consider the 2D Euler equations on the full space in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index in (0,1). We show weak existence for every $\dot{H}^{-1}$ 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 $L^2([0,T];H^{-\alpha})$ regularity. For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we also show pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise.
Area: CS5 - Stochastic PDEs for physical models (Margherita Zanella)
Keywords: 2D Euler equation, Kraichnan noise, regularization by noise.
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