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.
Categories: math.AP