Low Regularity Local Well-Posedness for the Zero Energy Novikov–Veselov Equation

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 Fourier restriction norm method. Local well-posedness shown in nonperiodic case $u_0 \in H^s(\mathbb{R}^2)$ with $s > - \frac{3}{4}$ and periodic data H^s_0(\mathbb{T}^2)$ mean zero, where \frac{1}{5}$. Both results rely on structure of nonlinearity, which becomes visible a symmetrization argument. Additionally, bilinear Strichartz-type estimate derived.