A fourth-order dispersive flow equation for closed curves on compact Riemann surfaces

TL;DR

This paper extends a fourth-order dispersive flow equation from the sphere to general compact Riemann surfaces, establishing local existence and uniqueness of solutions under constant sectional curvature using geometric energy methods.

Contribution

It generalizes the dispersive flow equation to Riemann surfaces and proves local well-posedness with a novel geometric approach to handle derivative loss.

Findings

Proved local existence and uniqueness of solutions.

Handled derivative loss using geometric energy methods.

Extended the model from spheres to general Riemann surfaces.

Abstract

A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main results, time-local existence and the uniqueness of a solution to the initial value problem is established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the absence of the local smoothing effect. The proof is based on the geometric energy method combined with a kind of gauge transformation to eliminate the loss of derivatives. Specifically, to show the uniqueness of the solution, the detailed geometric analysis of the solvable structure for the equation is presented.