Euler-Arnold equations and Teichmüller theory

In this paper we prove that for s>3/2, all Hs solutions of the Euler-Weil-Petersson equation, which describes geodesics on the universal Teichmüller space under the Weil-Petersson metric, will remain in Hs for all time. This extends the work of Escher-Kolev for strong Riemannian metrics to the borderline case of H3/2 metrics. In addition we show that all Hs solutions of the Wunsch equation, a variation of the Constantin-Lax-Majda equation which also describes geodesics on the universal Teichmüller curve under the Velling-Kirillov metric, must blow up in finite time due to wave breaking, extending work of Castro-Córdoba and Bauer-Kolev-Preston. Finally we illustrate these phenomena numerically.