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

TL;DR

This paper investigates a specific nonlinear PDE related to the Euler-Arnold equation on contactomorphism groups, analyzing solutions' existence, blowup, and global behavior through ODE reformulation in Banach spaces.

Contribution

It introduces a new one-parameter PDE model derived from contactomorphism groups and provides a rigorous analysis of solution existence, blowup criteria, and global solutions.

Findings

Established local existence of solutions.

Identified conditions for finite-time blowup.

Described scenarios for global existence.

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.