Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

TL;DR

This paper constructs an invariant measure for the periodic derivative nonlinear Schrödinger equation and proves almost sure global well-posedness for initial data in specific Fourier-Lebesgue spaces.

Contribution

It introduces a new invariant weighted Wiener measure for the equation and establishes global well-posedness for a broad class of initial data.

Findings

Invariant measure is constructed for the equation.

Almost sure global well-posedness is proved for data in certain Fourier-Lebesgue spaces.

The measure's invariance under the flow is demonstrated.

Abstract

In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schr\"odinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space ${\mathcal F}L^{s,r}(\T)$ with $s \ge \frac{1}{2}$, $2 < r < 4$, $(s-1)r <-1$ and scaling like $H^{\frac{1}{2}-\epsilon}(\T),$ for small $\epsilon >0$. We also show the invariance of this measure.