The search session has expired. Please query the service again.

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia; Vadim Kaloshin

Journal of the European Mathematical Society (2015)

  • Volume: 017, Issue: 1, page 71-149
  • ISSN: 1435-9855

Abstract

top
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 𝒦 1 we find a solution u and a time T such that u ( T ) H s 𝒦 u ( 0 ) H s . Moreover, the time T satisfies the polynomial bound 0 < T < 𝒦 C .

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.