Low regularity local well-posedness for the zero energy Novikov-Veselov equation

TL;DR

This paper establishes local well-posedness for the zero energy Novikov-Veselov equation in low regularity Sobolev spaces, using Fourier restriction norms and bilinear estimates, advancing understanding of its solution behavior.

Contribution

It proves local well-posedness for low regularity initial data in both nonperiodic and periodic cases, utilizing novel symmetrization and bilinear Strichartz estimates.

Findings

Well-posedness for $s > -3/4$ in nonperiodic case

Well-posedness for $s > -1/5$ in periodic case

Development of bilinear Strichartz-type estimate

Abstract

The initial value problem $u(x,y,0)=u_0(x,y)$ for the Novikov-Veselov equation $$\partial_tu+(\partial ^3 + \overline{\partial}^3)u +3(\partial (u\overline{\partial}^{-1}\partial u)+\overline{\partial}(u\partial^{-1}\overline{\partial}u))=0$$ is investigated by the Fourier restriction norm method. Local well-posedness is shown in the nonperiodic case for $u_0 \in H^s(\mathbb{R}^2)$ with $s > - \frac{3}{4}$ and in the periodic case for data $u_0 \in H^s_0(\mathbb{T}^2)$ with mean zero, where $s > - \frac{1}{5}$. Both results rely on the structure of the nonlinearity, which becomes visible with a symmetrization argument. Additionally, for the periodic problem a bilinear Strichartz-type estimate is derived.