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 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.Comment: Fixed various typos caught by referees. Closed gap in proof of bilinear estimat

A priori estimates for the derivative nonlinear Schrödinger equation

We prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive regularity index conditional upon small $L^2$ -norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip–Vişan–Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant