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Bilinear Estimates in the Presence of a Large Potential and a Critical NLS in 3D
About this Title
Fabio Pusateri and Avraham Soffer
Publication: Memoirs of the American Mathematical Society
Publication Year:
2024; Volume 299, Number 1498
ISBNs: 978-1-4704-7067-8 (print); 978-1-4704-7876-6 (online)
DOI: https://doi.org/10.1090/memo/1498
Published electronically: July 11, 2024
Table of Contents
Chapters
- 1. Introduction
- 2. Main ideas and strategy
- 3. Linear Spectral Theory
- 4. Preliminary bounds: Linear estimates and high frequencies
- 5. Analysis of the NSD I: structure of the leading order
- 6. Bilinear estimates for the leading order of the NSD
- 7. Weighted estimates for leading order terms
- 8. Analysis of the NSD II: Lower order terms
- 9. Weighted estimates for lower order terms
Abstract
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions.
One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-\Delta +V$, which we employ at a nonlinear level. As a first step we analyze the spatial integral of the product of three generalized eigenfunctions of $H$, and determine the precise structure of its singularities. This leads to study bilinear operators with certain singular kernels, for which we derive product estimates of Coifman-Meyer type. This analysis can then be combined with multilinear harmonic analysis tools and the study of oscillations to obtain (distorted Fourier space analogues of) weighted estimates for dispersive and wave equations.
As a first application we consider the nonlinear Schrödinger equation in $3$d in the presence of large, decaying and generic potential with no bound states, and with a $u^2$ non-linearity. The main difficulty is that a quadratic nonlinearity in $3$d is critical with respect to the Strauss exponent; moreover, this nonlinearity has non-trivial fully coherent interactions even when $V=0$. We prove quantitative global-in-time bounds and scattering for small solutions.
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