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We revisit Yudovich's well-posedness result for the $2$-dimensional Eulerequations 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 $L^1\capL^p_{\mathrm{ul}}$ and in $L^1\cap Y^\Theta_{\mathrm{ul}}$, where$L^p_{\mathrm{ul}}$ and $Y^\Theta_{\mathrm{ul}}$ are suitableuniformly-localized versions of the Lebesgue space $L^p$ and of the Yudovichspace $Y^\Theta$ respectively, with no condition at infinity for the growthfunction $\Theta$. We also provide an explicit modulus of continuity for thevelocity depending on the growth function $\Theta$. We prove uniqueness of weaksolutions 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 nouse of either Sobolev spaces, Calder\'on-Zygmund theory or Littlewood-Paleydecomposition, and actually applies not only to the Biot-Savart law, but alsoto more general operators whose kernels obey some natural structuralassumptions.

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