Shnirelman–Šverák relaxation conjecture for two-dimensional Euler flows

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, let $S_t:L^\infty(\Omega)\to L^\infty(\Omega)$ denote the two-dimensional Euler solution operator acting on vorticity, and define the weak limit set

$$\mathcal{A}_\infty:=\left\{f\in L^\infty(\Omega): S_{t_n}(\omega_0)\rightharpoonup f\text{ for some }\omega_0\in L^\infty(\Omega)\text{ and }t_n\to\infty\right\}.$$

Shnirelman–Šverák relaxation conjecture. The set $\mathcal{A}_\infty$ consists of precisely those $f\in L^\infty(\Omega)$ for which $\{\mathcal{S}_t(f)\}_{t\in\mathbb{R}}$ is pre-compact in $L^2(\Omega)$; moreover, for generic $f\in L^\infty(\Omega)$, the orbit $\{S_t(f)\}_{t\in\mathbb{R}}$ is not pre-compact in $L^2(\Omega)$. This conjecture describes the expected long-time simplification of bounded-vorticity solutions of the two-dimensional Euler equation, distinguishing data with pre-compact dynamics from generic data whose trajectories fail to be pre-compact. The source attributes it to Shnirelman and Šverák; no resolution is supplied here.