An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces

We revisit Yudovich's well-posedness result for the $2$-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set $\Omega\subset\mathbb{R}^2$ or on the torus $\Omega=\mathbb{T}^2$. We construct global-in-time weak solutions with vorticity in $L^1\cap L^p_{\mathrm{ul}}$ and in $L^1\cap Y^\Theta_{\mathrm{ul}}$, where $L^p_{\mathrm{ul}}$ and $Y^\Theta_{\mathrm{ul}}$ are suitable uniformly-localized versions of the Lebesgue space $L^p$ and of the Yudovich space $Y^\Theta$ respectively, with no condition at infinity for the growth function $\Theta$. We also provide an explicit modulus of continuity for the velocity depending on the growth function $\Theta$. We prove uniqueness of weak solutions in $L^1\cap Y^\Theta_{\mathrm{ul}}$ under the assumption that $\Theta$ grows moderately at infinity. In contrast to Yudovich's energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calder\'on-Zygmund theory or Littlewood-Paley decomposition, and actually applies not only to the Biot-Savart law, but also to more general operators whose kernels obey some natural structural assumptions.