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

Domenico Finco, Lorenzo Tentarelli, Alessandro Teta

TL;DR

This work establishes well-posedness for a 3D nonlinear Schrödinger equation with a nonlinear concentration on the unit sphere $\\mathbb{S}^2$. It develops a spherical-harmonics based framework to formulate and analyze a nonlinear charge equation $q+i(\\Lambda u(q))=F_0$ governing the trace on the sphere, proving local existence and mass/energy conservation, and deriving global results under small-data or defocusing-growth regimes. The analysis hinges on detailed properties of the operator $\\Lambda$, trace regularity on $\\mathbb{S}^2$, and a boundary-integral formulation for the linearized problem, highlighting novel techniques needed when the nonlinearity is supported on a manifold of codimension one. The results provide a solid foundation for studying manifold-concentrated nonlinearities and open directions for long-time behavior, scattering, and optimality questions in higher codimension settings.

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.

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

TL;DR

This work establishes well-posedness for a 3D nonlinear Schrödinger equation with a nonlinear concentration on the unit sphere . It develops a spherical-harmonics based framework to formulate and analyze a nonlinear charge equation governing the trace on the sphere, proving local existence and mass/energy conservation, and deriving global results under small-data or defocusing-growth regimes. The analysis hinges on detailed properties of the operator , trace regularity on , and a boundary-integral formulation for the linearized problem, highlighting novel techniques needed when the nonlinearity is supported on a manifold of codimension one. The results provide a solid foundation for studying manifold-concentrated nonlinearities and open directions for long-time behavior, scattering, and optimality questions in higher codimension settings.

Abstract

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on . Precisely, local well-posedness is proved for any 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.
Paper Structure (19 sections, 13 theorems, 275 equations, 1 figure)

This paper contains 19 sections, 13 theorems, 275 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The linear case
  3. Setting and main results
  4. Organization of the paper
  5. Basics of analysis on $\mathbb{S}^2$
  6. Sobolev spaces on $\mathbb{S}^2$
  7. Function spaces on $I\times\mathbb{S}^2$
  8. Bessel functions and Bessel--Fourier transform
  9. Preliminary results
  10. Properties of the operator $\Lambda$
  11. Regularity of the trace on $\mathbb{S}^2$ of functions in $\mathcal{D}(\mathcal{H})$
  12. Regularity of the source term of (\ref{['eq-charge']})
  13. Local well--posedness: proof of Theorem \ref{['main']} -- item (i)
  14. Conservation laws: proof of Theorem \ref{['main']} -- item (ii)
  15. Global well--posedness: proof of Theorem \ref{['main2']}
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $\beta\in\mathbb{R}$, $\sigma\geqslant1/2$ and $\psi_0=\phi_0^\lambda-\mathcal{G}^{\lambda} \nu(q_0) \in \mathcal{D}(\mathcal{H})$. Then:

Figures (1)

  • Figure 1: The path $\Gamma_{R,\varepsilon}.$

Theorems & Definitions (46)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3: Local Well--Posedness
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Global Well--Posedness
  • Remark 1.8
  • Remark 1.9
  • Remark 2.1
  • ...and 36 more