#### Vol. 8, No. 4, 2015

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Growth of Sobolev norms for the quintic NLS on $T^2$

### Emanuele Haus and Michela Procesi

Vol. 8 (2015), No. 4, 883–922
##### Abstract

We study the quintic nonlinear Schrödinger equation on a two-dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one discussed by Colliander et al. for the case of the cubic NLS. This requires an accurate combinatorial analysis.

##### Keywords
nonlinear Schrödinger equation, growth of Sobolev norms, Hamiltonian PDEs, weak turbulence
##### Mathematical Subject Classification 2010
Primary: 35B34, 35Q55, 37K45
##### Milestones
Received: 9 June 2014
Revised: 19 January 2015
Accepted: 6 March 2015
Published: 21 June 2015
##### Authors
 Emanuele Haus Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università di Napoli “Federico II” Via Cintia, Monte S. Angelo I-80126 Napoli Italy Michela Procesi Dipartimento di Matematica Università di Roma “La Sapienza” Piazzale Aldo Moro, 5 I-00185 Roma Italy