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

Abstract

The initial value problem $u(x,y,0)=u_0(x,y)$ for the Novikov-Veselov equation $${\partial }_{t}u+({\partial }^3+{\overline{\partial }}^3)u+3(\partial (u{\overline{\partial }}^{-1}\partial u)+\overline{\partial }(u{\partial }^{-1}\overline{\partial }u))=0$$ 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 $u_0\in H_0^s({\mathbb{T}}^2)$ 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.

Fixed various typos caught by referees. Closed gap in proof of bilinear estimate