Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Guardia, Marcel, and Kaloshin, Vadim. "Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation." Journal of the European Mathematical Society 017.1 (2015): 71-149. <http://eudml.org/doc/277324>.

@article{Guardia2015,
abstract = {We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s>1$. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $\mathcal \{K\}\gg 1$ we find a solution $u$ and a time $T$ such that $\Vert u(T)\Vert _\{H^s\}\ge \mathcal \{K\} \Vert u(0)\Vert _\{H^s\}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<\mathcal \{K\}^C$.},
author = {Guardia, Marcel, Kaloshin, Vadim},
journal = {Journal of the European Mathematical Society},
keywords = {Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points; Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points},
language = {eng},
number = {1},
pages = {71-149},
publisher = {European Mathematical Society Publishing House},
title = {Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation},
url = {http://eudml.org/doc/277324},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Guardia, Marcel
AU - Kaloshin, Vadim
TI - Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 1
SP - 71
EP - 149
AB - We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s>1$. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $\mathcal {K}\gg 1$ we find a solution $u$ and a time $T$ such that $\Vert u(T)\Vert _{H^s}\ge \mathcal {K} \Vert u(0)\Vert _{H^s}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<\mathcal {K}^C$.
LA - eng
KW - Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points; Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points
UR - http://eudml.org/doc/277324
ER -