Growth of Sobolev norms for the quintic NLS on T2

Haus, Emanuele; Procesi, Michela

doi:10.2140/apde.2015.8.883

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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

DOI: 10.2140/apde.2015.8.883

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

Vol. 8, No. 4, 2015