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

Abstract

Given $0<a<1$ and any $C^1$, divergence-free, $(-a)$-homogeneous vector field $u_0$ on the punctured plane, we construct a forward self-similar solution of the two-dimensional incompressible Euler equations on the plane with initial datum $u_0$. No smallness or sign condition is required. The solution is obtained as a vanishing-dissipation limit of hypodissipative self-similar profiles. The key estimate is a uniform vorticity profile bound in the critical Lorentz space $L^{\frac{2}{1+a},\infty }$. The resulting velocity has locally finite energy and converges strongly to $u_0$ locally in $L^2$ as time approaches zero. The vorticity preserves its distribution function and converges strongly to the initial vorticity locally in $L^{\frac{2}{1+a}-}$ as time approaches zero.

arXiv comment34 pages, main theorem and proof unchanged, added Proposition 4.3 regarding renormalization property