Nonlinear resonances with a potential: Multilinear estimates and an application to NLS

Pierre Germain, Zaher Hani, Samuel Walsh

Research output: Contribution to journalArticle

Abstract

This paper considers the question of global in time existence and asymptotic behavior of small-data solutions of nonlinear dispersive equations with a real potential $V$. The main concern is treating nonlinearities whose degree is low enough as to preclude the simple use of classical energy methods and decay estimates. In their place, we present a systematic approach that adapts the space-time resonance method to the non-Euclidean setting using the spectral theory of the Schroedinger operator $-\Delta+V$. We start by developing tools of independent interest, namely multilinear analysis (Coifman-Meyer type theorems) in the framework of the corresponding distorted Fourier transform. As a first application, this is then used to prove global existence and scattering for a quadratic Schroedinger equation.
Original languageUndefined
Article number1303.4354
JournalarXiv
StatePublished - Mar 18 2013

Keywords

  • math.AP
  • math.CA
  • 35Q30, 82C31, 76A05

Cite this

Nonlinear resonances with a potential : Multilinear estimates and an application to NLS. / Germain, Pierre; Hani, Zaher; Walsh, Samuel.

In: arXiv, 18.03.2013.

Research output: Contribution to journalArticle

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AB - This paper considers the question of global in time existence and asymptotic behavior of small-data solutions of nonlinear dispersive equations with a real potential $V$. The main concern is treating nonlinearities whose degree is low enough as to preclude the simple use of classical energy methods and decay estimates. In their place, we present a systematic approach that adapts the space-time resonance method to the non-Euclidean setting using the spectral theory of the Schroedinger operator $-\Delta+V$. We start by developing tools of independent interest, namely multilinear analysis (Coifman-Meyer type theorems) in the framework of the corresponding distorted Fourier transform. As a first application, this is then used to prove global existence and scattering for a quadratic Schroedinger equation.

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