Global well-posedness for the derivative nonlinear Schr\"odinger equation

TL;DR

This paper proves the global well-posedness of the derivative nonlinear Schrödinger equation for initial data in the Sobolev space H^{1/2}, establishing boundedness of solutions and closing the well-posedness discussion in this space.

Contribution

It demonstrates the global well-posedness of the derivative nonlinear Schrödinger equation in H^{1/2} for the first time, utilizing profile decomposition and integrability properties.

Findings

Global well-posedness in H^{1/2}

Bounded H^{1/2} norm over time

Closes the well-posedness gap for s ≥ 1/2

Abstract

This paper is dedicated to the study of the derivative nonlinear Schr\"odinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $H^{\frac12}$ with mass strictly less than $4\pi$ or general initial conditions in the weighted Sobolev space $H^{2, 2}$. In this article, we prove that the derivative nonlinear Schr\"odinger equation is globally well-posed for general Cauchy data in $H^{\frac12}$ and that furthermore the $H^{\frac12}$ norm of the solutions remains globally bounded in time. One should recall that for $H^s$, with $s < 1 / 2 $, the associated Cauchy problem is ill-posed in the sense that uniform continuity with respect to the initial data fails. Thus, our result closes the discussion in the setting of the Sobolev spaces $H^s$. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.