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 C² 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.