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

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.