Yudovich's conjecture on generic vorticity-gradient growth

Let $u$ be an inviscid incompressible flow on $\mathbb{R}^2$ with scalar vorticity $\omega=\partial_1u^2-\partial_2u^1$, recovered from $\omega$ by the Biot–Savart law $u=\nabla^{\perp}\Delta^{-1}\omega$. The vorticity is transported by the flow, so its $L^p$ norms are conserved, although its derivatives may grow. Yudovich's conjecture. There is a “substantial set” of inviscid incompressible flows whose vorticity gradients grow without bound. This set is dense enough to cause a loss of smoothness for arbitrarily small perturbations of any steady flow. This conjecture concerns the possibility of derivative growth despite conservation of all vorticity $L^p$ norms, and predicts generic small-scale creation and loss of smoothness near steady flows.