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

Abstract

Let $0<a<1$ 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 hypodissipative self-similar profiles built directly in the critical vorticity space. The main estimate is a uniform critical Lorentz bound $\|\operatorname{curl} U\|_{L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)}$. The resulting Euler solution $u$ belongs to $C([0,\infty);L^2_{\mathrm{loc}}(\mathbb{R}^2))$ and has vorticity uniformly bounded in the critical space $L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)$. Its velocity converges strongly to $u_0$ in $L^2_{\mathrm{loc}}$, while its vorticity converges weak-star to $\omega_0=\operatorname{curl} u_0$ in $L^{\frac{2}{1+a},\infty}$ as $t\downarrow0$.