Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction

We give a Lagrangian construction of forward self-similar solutions of the $2$D incompressible Euler equation. For every $1<α<2$ and every real $h\in L^{2/α}(\mathbb{S}^1)$, the construction produces a flow with initial vorticity $|x|^{-α}h(\arg x)$, without smallness or sign assumptions. The argument is based on a fixed-point formulation of backward particle transport from time $t=1$ to $t=0$. Mollifying the Biot–Savart velocity in self-similar variables preserves exact vorticity transport, while symmetrization yields the unforced Euler equation in velocity form even when the ordinary vorticity flux is not locally integrable. The limiting solution admits an almost-everywhere invertible, measure-preserving regular Lagrangian flow from time zero, with strong local continuity and preservation of signed value distributions and absolutely integrable Casimirs.