Small data well-posedness for derivative nonlinear Schr\"odinger equations

TL;DR

This paper establishes local and global well-posedness results for a class of derivative nonlinear Schrödinger equations in Sobolev spaces, depending on the polynomial degree and derivative structure of the nonlinearity.

Contribution

It provides new well-posedness thresholds for derivative NLS equations based on polynomial degree and derivative terms, including global results for higher degrees.

Findings

Local well-posedness in H^{1/2} for degree ≥ 3 with single derivatives

Local well-posedness in H^{3/2} for equations with higher degree or multiple derivatives

Global well-posedness for degree ≥ 5, including in critical Sobolev spaces

Abstract

We study the generalized derivative nonlinear Schr\"odinger equation $i\partial_t u+\Delta u = P(u,\overline{u},\partial_x u,\partial_x \overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg } P\geq 3$, the equation is locally well-posed in $H^{\frac{1}{2}}$ when each term in $P$ contains only one derivative, otherwise we have a local well-posedness in $H^{\frac{3}{2}}$. If $\text{deg } P \geq 5$, the solution can be extended globally. By restricting to equations of the form $i\partial_t u+\Delta u = \partial_x P(u,\overline{u})$ with $\text{deg } P\geq5$, we were able to obtain the global well-posedness in the critical Sobolev space.