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Weakly turbulent solution to the Schrödinger equation on the two-dimensional torus with real potential decaying to zero at infinity

Ambre Chabert

Vol. 18 (2025), No. 8, 2061–2080
Abstract

We build a smooth time-dependent real potential on the two-dimensional torus, decaying as time tends to infinity in Sobolev norms along with all its time derivatives, and we exhibit a smooth solution to the associated Schrödinger equation on the two-dimensional torus whose Hs norms nevertheless grow logarithmically as time tends to infinity. We use Fourier decomposition in order to exhibit a discrete resonant system of interactions, which we are further able to reduce to a sequence of finite-dimensional linear systems along which the energy propagates to higher and higher frequencies. The constructions are very explicit, and we can thus obtain lower bounds on the growth rate of the solution.

Keywords
linear Schrödinger equation, weak turbulence, resonant system, forward cascade of energy, backward integration
Mathematical Subject Classification
Primary: 35B40
Secondary: 35Q41
Milestones
Received: 10 April 2024
Revised: 14 August 2024
Accepted: 20 September 2024
Published: 25 July 2025
Authors
Ambre Chabert
Département de Mathématiques et Applications
Ecole Normale Supérieure
Paris
France

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