Self-Similar Solutions of the Two-Dimensional Incompressible Euler Equation from Large Initial Data

Let $\frac{1}{3}<a<1$ and let $u_0$ be a $C^1$, divergence-free, $(-a)$-homogeneous vector field on $\mathbb{R}^2\setminus\{0\}$. We construct a forward-in-time self-similar solution of the two-dimensional incompressible Euler equations, \[ u(t,x)=t^{-\frac{a}{1+a}} U\left(\frac{x}{t^{{\frac{1}{1+a}}}}\right), \] with initial datum $u_0$. No smallness or sign assumption is imposed on the initial datum. The construction is a vanishing-dissipation limit of the hypodissipative self-similar profiles. The key is a vorticity profile estimate, uniform in the dissipation parameter, in the critical Lorentz space $L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)$.
The resulting Euler solution $u$ has locally finite energy and $\nabla u \in L^\infty_t L^{\frac{2}{1+a},\infty}_x$. Indeed, the velocity is continuous in $L^2_{\text{loc}}$, and the vorticity converges weak-* in $L^{\frac{2}{1+a},\infty}$ at the initial time.