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 $u = \nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda >0$, we show that only trivial solutions exist in the range $0<\lambda<1/2$, 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 $\lambda>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\lambda}) \cap \mathbb{N}$. The case $\lambda = 2/3$ 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.