10 Ottobre, 2023 14:30 in punto
Sezione di Probabilità e Statistica Matematica
Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity
Mario Maurelli, Università di Pisa
Aula Seminari - III piano - polimi-it.zoom.us/j/91064241084?pwd=VW0vT3FDZHFubkFWcm5KaXgzNXFOZz09
Abstract
We consider the 2D Euler equations on $\mathbb{R}^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$.
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 show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise.
Joint work with Michele Coghi.