Well-posedness and ill-posedness results for the Novikov-Veselov equation

TL;DR

This paper investigates the well-posedness of the Novikov-Veselov equations in Sobolev spaces, establishing conditions for local solutions and highlighting ill-posedness in certain regimes.

Contribution

It provides new results on local well-posedness thresholds for the Novikov-Veselov and modified Novikov-Veselov equations in Sobolev spaces.

Findings

Well-posedness for Novikov-Veselov when s > 1/2

Well-posedness for modified Novikov-Veselov when s > 1

Ill-posedness issues in supercritical regimes

Abstract

In this paper we study the Novikov-Veselov equation and the related modified Novikov-Veselov equation in certain Sobolev spaces. We prove local well-posedness in H^s (R2) for s > 1/2 for the Novikov-Veselov equation, and local well-posedness in H^s (R2) for s > 1 for the modified Novikov-Veselov equation. Finally we point out some ill-posedness issues for the Novikov- Veselov equation in the supercritical regime.