On self-similar solutions to the incompressible Euler equations

Abstract

Recent numerical simulations have shown the existence of multiple self-similar solutions to the Cauchy problem for the 2-dimensional incompressible Euler equation, with initial vorticity in Llocp(R2), 1≤p<+∞. Toward a rigorous validation of these computations, in this paper we analytically construct self-similar solutions (i) on an outer domain of the form {|x|>R}, and (ii) in a neighborhood of the points where the solution exhibits a spiraling vortex singularity. The outer solution is obtained as the fixed point of a contractive transformation, based on the Biot-Savart formula and integration along characteristics. The inner solution is constructed using a system of adapted coordinates, following the approach of V. Elling (2016) [17].