Local well-posedness for a fourth-order nonlinear dispersive system on the 1D torus

TL;DR

This paper establishes local well-posedness for a class of fourth-order nonlinear dispersive PDEs on the 1D torus, using energy methods, gauge transformations, and regularization techniques, with conditions linked to geometric analysis.

Contribution

It provides new sufficient conditions for local well-posedness of a complex fourth-order dispersive system, connecting PDE analysis with geometric structures.

Findings

Sufficient conditions for well-posedness are derived.

The proof combines energy methods with gauge transformations.

Conditions relate to geometric analysis on symmetric spaces.

Abstract

This paper is concerned with the initial value problem for a system of one-dimensional fourth-order dispersive partial differential equations on the torus with nonlinearity involving derivatives up to second order. This paper gives sufficient conditions on the coefficients of the system for the initial value problem to be time-locally well-posed in Sobolev spaces with high regularity. The proof is based on the energy method combined with the idea of a gauge transformation and the technique of Bona-Smith type parabolic regularization. The sufficient conditions can been found in connection with geometric analysis on a fourth-order geometric dispersive partial differential equation for curve flows on a compact locally Hermitian symmetric space.