# Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod; Gigliola Staffilani

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 7, page 1687-1759
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topNahmod, Andrea R., and Staffilani, Gigliola. "Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space." Journal of the European Mathematical Society 017.7 (2015): 1687-1759. <http://eudml.org/doc/277609>.

@article{Nahmod2015,

abstract = {We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set $\Sigma _T$, $\mathbb \{P\}(\Sigma _T) > 0$ such that any data $\phi ^\{\omega \}(x) \in H^\{\gamma \}(\mathbb \{T\}^3), \gamma <1, \omega \in \Sigma _T$, evolves up to time $T$ into a solution $u(t)$ with $u(t) - e^\{it\Delta \} \phi ^\{\omega \} \in C([0,T]; H^s(\mathbb \{T\}^3))$, $s=s(\gamma )>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1(\mathbb \{T\}^3)$, that is in the supercritical scaling regime.},

author = {Nahmod, Andrea R., Staffilani, Gigliola},

journal = {Journal of the European Mathematical Society},

keywords = {supercritical nonlinear Schrödinger equation; almost sure well-posedness; random data; supercritical nonlinear Schrödinger equation; almost sure well-posedness; random data},

language = {eng},

number = {7},

pages = {1687-1759},

publisher = {European Mathematical Society Publishing House},

title = {Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space},

url = {http://eudml.org/doc/277609},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Nahmod, Andrea R.

AU - Staffilani, Gigliola

TI - Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 7

SP - 1687

EP - 1759

AB - We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set $\Sigma _T$, $\mathbb {P}(\Sigma _T) > 0$ such that any data $\phi ^{\omega }(x) \in H^{\gamma }(\mathbb {T}^3), \gamma <1, \omega \in \Sigma _T$, evolves up to time $T$ into a solution $u(t)$ with $u(t) - e^{it\Delta } \phi ^{\omega } \in C([0,T]; H^s(\mathbb {T}^3))$, $s=s(\gamma )>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1(\mathbb {T}^3)$, that is in the supercritical scaling regime.

LA - eng

KW - supercritical nonlinear Schrödinger equation; almost sure well-posedness; random data; supercritical nonlinear Schrödinger equation; almost sure well-posedness; random data

UR - http://eudml.org/doc/277609

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.