One-parameter solutions of the Euler-Arnold equation on the contactomorphism group
Abstract
We study solutions of the equation gt-gtyy + 4g2 - 4ggyy = y ggyyy-ygygyy, y∈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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