A priori estimates and weak solutions for the derivative nonlinear Schr\"{o}dinger equation on torus below $H^{1/2}$

TL;DR

This paper establishes a priori estimates and demonstrates the existence of weak solutions for the derivative nonlinear Schrödinger equation on a torus in low regularity Sobolev spaces below H^{1/2}.

Contribution

It introduces new a priori estimates for weak solutions of DNLS on the torus in low regularity spaces and discusses error estimates from finite-dimensional approximations.

Findings

Existence of solutions in H^s for s<1/2

A priori estimates for weak solutions with small L^2 data

Remarks on error estimates from finite-dimensional approximations

Abstract

We propose a priori estimates for a weak solution to the derivative nonlinear Schr\"odinger equation (DNLS) on torus with small $L^2$-norm datum in low regularity Sobolev spaces. These estimates allow us to show the existence of solutions in $H^s(\mathbb{T})$ with some $s<1/2$ in a relatively weak sense. Furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of DNLS using the Fourier-Lesbesgue type as auxiliary spaces, despite the fact that Nahmod, Oh, Rey-Bullet and Staffilani \cite{nors} have already seen them.