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

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.

40 pages. Keywords: NLS equation, concentrated nonlinearity, unit sphere, well-posedness, spherical harmonics. Minor changes have been made with respect to the previous version

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• A general review on the NLS equation with point-concentrated nonlinearity