Calculus of Variations and Geometric Measure Theory

D. Finco - L. Tentarelli - A. Teta

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

created by tentarelli on 04 Oct 2022

[BibTeX]

preprint

Inserted: 4 oct 2022
Last Updated: 4 oct 2022

Year: 2022

ArXiv: 2209.13504 PDF

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.