One-parameter solutions of the Euler-Arnold equation on the contactomorphism group

Stephen C. Preston; Alejandro Sarria

paper

One-parameter solutions of the Euler-Arnold equation on the contactomorphism group

arXiv:1405.4339

Abstract

We study solutions of the equation $g_{t} - g_{t y y} + 4 g^{2} - 4 g g_{y y} = y g g_{y y y} - y g_{y} g_{y y}, y \in R,$ which arises by considering solutions of the Euler-Arnold equation on a contactomorphism group when the stream function is of the form $f (t, x, y, z) = z g (t, y)$ . The equation is analogous to both the Camassa-Holm equation and the Proudman-Johnson equation. We write the equation as an ODE in a Banach space to establish local existence, and we describe conditions leading to global existence and conditions leading to blowup in finite time.