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

Stephen C. Preston; Alejandro Sarria

doi:10.3934/dcds.2015.35.2123

Article Contents

2015, Volume 35, Issue 5: 2123-2130. Doi: 10.3934/dcds.2015.35.2123

This issue Previous Article Unbounded regime for circle maps with a flat interval Next Article Local integration by parts and Pohozaev identities for higher order fractional Laplacians

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

Stephen C. Preston^1, and
Alejandro Sarria^1,

1.
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395

Received: April 30, 2014

Revised: July 31, 2014

Published: May 2017

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Abstract

We study solutions of the equation $$ g_t-g_{tyy} + 4g^2 - 4gg_{yy} = y gg_{yyy}-yg_yg_{yy}, \qquad y\in\mathbb{R},$$ which arises by considering solutions of the Euler-Arnold equation on a contactomorphism group when the stream function is of the form $f(t,x,y,z) = zg(t,y)$. 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.

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Mathematics Subject Classification: Primary: 35B65, 53C21, 58D05; Secondary: 35Q35.

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