Well-posedness for a generalized derivative nonlinear Schrödinger equation

TL;DR

The article addresses local and global well-posedness for a generalized derivative nonlinear Schrödinger equation with Dirichlet boundary conditions in one dimension. It develops a Yosida-regularization framework to construct approximate solutions in $H^2$, establishes uniform a priori bounds, and passes to the limit without compactness arguments to obtain true solutions in $H^2$ and, via gauge transforms and Strichartz estimates, in $H^1$. It proves a sharp local well-posedness result in $H^2$ for σ≥1/2 and in $H^1$ for σ≥1, with global existence in the energy space under small-data conditions, and provides a Yudovich-type uniqueness argument in $H^{3/2}$. For 0<σ<1, it achieves global existence in a weak sense through compactness, highlighting a nuanced difference in behavior across σ. These results advance understanding of derivative nonlinear Schrödinger dynamics under Dirichlet boundary conditions and offer a robust, non-compactness-based construction method.

Abstract

We study the Cauchy problem for a generalized derivative nonlinear Schrödinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space $H^1$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4