Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

TL;DR

This paper establishes local and global well-posedness results for the three-dimensional nonlinear Schrödinger equation with nonlinearity concentrated on a sphere, addressing challenges posed by the support's dimension.

Contribution

It proves well-posedness for sphere-concentrated NLS, extending techniques beyond point-concentrated models and handling the geometric complexity.

Findings

Local well-posedness for any $C^2$ power-nonlinearity

Global well-posedness for small data or in the defocusing case

New analytical methods for support of nonlinearity on a sphere

Abstract

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local well-posedness is proved for any $C^2$ power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas.