A Remark on the global existence of a third order dispersive flow into locally Hermitian symmetric spaces

TL;DR

This paper proves the global existence of solutions for a third order dispersive flow into compact locally Hermitian symmetric spaces, extending models of vortex filament motion with new analytical techniques.

Contribution

It establishes global solutions for a generalized third order dispersive flow, which is not necessarily integrable, using conservation laws and energy estimates.

Findings

Global existence of solutions proved

Conservation laws are key to the analysis

Extension beyond integrable cases

Abstract

We prove global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation we consider generalizes two-sphere-valued completely integrable systems modelling the motion of vortex filament. Unlike one-dimensional Schr\"odinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an energy which is not necessarily preserved in time but does not blow up in finite time.