On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schroedinger equations

TL;DR

This paper establishes local well-posedness for a class of derivative nonlinear Schrödinger equations in multiple dimensions and provides a global result in one dimension for a specific nonlinearity, using Fourier restriction techniques.

Contribution

It introduces new well-posedness results for derivative NLS equations in higher dimensions and a global result in one dimension for a particular nonlinearity.

Findings

Local well-posedness for s > n/2 - 1/(m-1) in n dimensions

Global L^2 result for 1D case with specific nonlinearity

Use of Fourier restriction norm method for proofs

Abstract

The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - i\Delta u = (\nabla \bar{u})^{\beta}, |\beta|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s > s_c := \frac{n}{2} - \frac{1}{m-1}$, $s \ge 0$. In the special case of space dimension $n=1$ a global $L^2$-result is obtained for NLS with the nonlinearity $N(u)= \partial_x (\bar{u} ^2)$. The proof uses the Fourier restriction norm method.