Non-Uniqueness of Weak Solutions to the Unforced Two-Dimensional Euler Equations with Vorticity in $L^1\cap L^p$, Part I: A Compactness Criterion and the Linear Analysis

This paper and its companion Part II prove that the incompressible Euler equations in the plane, without external force, admit two distinct renormalized weak solutions with the same initial vorticity $f$, where $f$ is radial and nonnegative and lies in $(L^1\cap L^q)(\mathbb{R}^2)$ for every $q<\inf