Transport-noise well-posedness conjecture for the 2D Euler equations

Let $W$ be a noise satisfying Assumption

, and let $\omega_0\in L^1_x\cap L^p_x\cap \dot H^{-1}_x$ for some $p<\infty$. The stochastic 2D Euler equations

{\rm d} \omega + \operatorname{curl}^{-1} \omega\cdot \nabla \omega, {\rm d} t + \circ {\rm d} W\cdot \nabla \omega = 0

are considered in the absence of forcing. **Transport-noise well-posedness conjecture.** There \exists a noise $W$ satisfying Assumption

such that these stochastic 2D Euler equations are well-posed in law for all such initial data $\omega_0$. The conjecture asks whether sufficiently active Kraichnan transport noise can extend weak well-posedness results from log-Euler and hypodissipative Navier–Stokes equations to the 2D Euler equations, where the corresponding deterministic low-regularity theory remains inadequate and uniqueness in $L^2_x$ can fail for related equations.