Abstract
<abstract><p>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 form $ \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 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.</p></abstract>
Highlights
We consider the quantum hydrodynamic system on a d-dimensional irrational torus with d = 2, 3
The aim of this paper is to prove that, for generic irrational tori, the solution remains of size ε for longer times
We look for solutions u which stay irrotational for all times, i.e
Summary
The natural phase space for (QHD) is H0s(Tdν) × Hs(Tdν) where Hs(Tdν) := Hs(Tdν)/∼ is the homogeneous Sobolev space obtained by the equivalence relation ψ1(x) ∼ ψ2(x) if and only if ψ1(x) − ψ2(x) = c is a constant; H0s(Tdν) is the subspace of Hs(Tdν) of functions with zero average Despite this fact we prefer to work with a couple of variable (ρ, φ) ∈ H0s(Tdν) × Hs(Tdν) but at the end we control only the norm. The case (QHD), i.e. the system (EK) with the particular choice (1.6), reduces, for small solutions, to a semi-linear equation, more precisely to a nonlinear Schrodinger equation This is a consequence of the fact that the Madelung transform (introduced for the first time in the seminal work by Madelung [18]).
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