Long-time stability of the quantum hydrodynamic system on irrational tori

TL;DR

This paper proves long-time boundedness of solutions to the quantum hydrodynamic system on irrational tori, extending the stability time scale using a Birkhoff normal form and controlling small divisors.

Contribution

It introduces a novel analysis of the quantum hydrodynamic system on irrational tori, extending stability results via a Birkhoff normal form approach and small divisor estimates.

Findings

Solutions remain bounded in Sobolev norms over extended time scales.

The irrationality of the torus affects eigenvalue separation and energy transfer.

A new method for controlling derivative loss in nonlinear PDEs on irrational domains.

Abstract

We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a "non trivial" time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving from $\varepsilon$-small initial conditions, remain bounded in $H^s$ for a time scale of order $O(\varepsilon^{-1-1/(d-1)+})$, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schr\"odinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising from three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes.This is due to the irrationality of the torus which prevent to have "good separation" properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable \emph{high/low} frequencies analysis, which gives an \emph{a priori} estimate on the solutions.