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

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 Hs-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form εsmall initial conditions, remain bounded in Hs for a time scale of order O(ε−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 Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form 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 prevents 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 high/low frequencies analysis, which gives an a priori estimate on the solutions.