Sobolev norms explosion for the cubic NLS on irrational tori. (July 2022)

• Record Type: Journal Article
• Authors: Giuliani, Filippo; Guardia, Marcel
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: Transfers of energy -- Growth of Sobolev norms -- Irrational tori -- NLS; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112865
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the cubic nonlinear Schrödinger equation on 2-dimensional irrational tori. We construct solutions which undergo growth of Sobolev norms. More concretely, for every s > 0, s ≠ 1 and almost every choice of spatial periods we construct solutions whose H s Sobolev norms grow by any prescribed factor. Moreover, for a set of spatial periods with positive Hausdorff dimension we construct solutions whose Sobolev norms go from arbitrarily small to arbitrarily large. We also provide estimates for the time needed to undergo the norm explosion. Note that the irrationality of the space periods decouples the linear resonant interactions into products of 1-dimensional resonances, reducing considerably the complexity of the resonant dynamics usually used to construct transfer of energy solutions. However, one can provide these growth of Sobolev norms solutions by using quasi-resonances relying on Diophantine approximation properties of the space periods.