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

Abstract

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 $q<\infty$ but is not bounded. One solution is radially symmetric and stationary. The other is $3$-fold rotationally symmetric, is equimeasurable with $f$ at every time, and departs from $f$ in finite time; the two solutions are distinct in $L^\infty_t(L^1\cap L^q)_x$ on $[0,T]\times\mathbb{R}^2$ for every $q\in(4/3,\infty)$ and every sufficiently large $T$. In this first part we prove two ingredients. One is a compactness criterion: for a radial datum, a sequence of $3$-fold rotationally symmetric solutions with converging data, whose non-radial moment is bounded below at some bounded time, has a subsequence converging to a second distinct renormalized weak solution. The other is the linear analysis for the building blocks of our multiscale datum. We show that the linearization of the Euler equations around the basic building block plus a small point vortex has an unstable eigenvalue in a space of $3$-fold rotationally symmetric perturbations. This latter argument depends on rigorous interval arithmetic that we discuss more fully in Part II.