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

Nahmod, 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 -