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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.

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