### Abstract

Original language | Undefined |
---|---|

Article number | 1303.4354 |

Journal | arXiv |

State | Published - Mar 18 2013 |

### Keywords

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

### Cite this

*arXiv*, [1303.4354].

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

Research output: Contribution to journal › Article

*arXiv*.

}

TY - JOUR

T1 - Nonlinear resonances with a potential

T2 - Multilinear estimates and an application to NLS

AU - Germain, Pierre

AU - Hani, Zaher

AU - Walsh, Samuel

N1 - 47 pages

PY - 2013/3/18

Y1 - 2013/3/18

N2 - 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.

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.

KW - math.AP

KW - math.CA

KW - 35Q30, 82C31, 76A05

M3 - Article

JO - arXiv

JF - arXiv

M1 - 1303.4354

ER -