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 = ∇⊥Ψ, Ψ(r, θ) = r λψ(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 . 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.