Local well-posedness of the periodic nonlinear Schr\"odinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces

TL;DR

This paper proves local well-posedness for the quadratic nonlinear Schrödinger equation with low regularity initial data on tori by modifying the standard function spaces, overcoming known bilinear estimate failures.

Contribution

The authors establish local well-posedness for the quadratic NLS in negative Sobolev spaces by introducing modified $X^{s, b}$-spaces, extending the understanding of low regularity solutions.

Findings

Proved local well-posedness for quadratic NLS in negative Sobolev spaces.

Developed modified $X^{s, b}$-spaces to handle bilinear estimate failures.

Extended low regularity analysis to 1D and 2D tori.

Abstract

We study low regularity local well-posedness of the nonlinear Schr\"odinger equation (NLS) with the quadratic nonlinearity $\overline{u}^2$, posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the $X^{s, b}$-space is known to fail when the regularity $s$ is below some threshold value, we establish local well-posedness for such low regularity by introducing modifications on the $X^{s, b}$-space.