Global well-posedness of the derivative nonlinear Schr\"odinger equation with periodic boundary condition in $H^{\frac12}$

TL;DR

This paper proves the global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary conditions in the Sobolev space $H^{1/2}$ for initial data with mass less than $4 extpi$, matching recent non-periodic results.

Contribution

It establishes the global well-posedness in $H^{1/2}$ for periodic DNLS with a mass threshold, extending non-periodic results and analyzing solution map continuity below $H^{1/2}.

Findings

Global well-posedness in $H^{1/2}$ for mass < 4π.

Matching recent non-periodic mass threshold results.

Failure of uniform continuity of the solution map below $H^{1/2}$.

Abstract

We establish the global well-posedness of the derivative nonlinear Schr\"odinger equation with periodic boundary condition in the Sobolev space $H^{\frac12}$, provided that the mass of initial data is less than $4\pi$. This result matches the one by Miao, Wu, and Xu and its recent mass threshold improvement by Guo and Wu in the non-periodic setting. Below $H^{\frac12}$, we show that the uniform continuity of the solution map on bounded subsets of $H^s$ does not hold, for any gauge equivalent equation.