2D homogeneous solutions to the Euler equation

Abstract

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp Ψ$, $Ψ(r,θ) = r^λ ψ(θ)$, for $λ>0$, we show that only trivial solutions exist in the range $0<λ<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $λ>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2λ}) \cap \mathbb{N}$. The case $λ= 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity 1/3.