Energy cascade and Sobolev norms inflation for the quantum Euler equations on tori

TL;DR

This paper demonstrates the existence of solutions to quantum Euler equations on tori that exhibit energy transfer to high Fourier modes and Sobolev norm inflation, revealing insights into quantum turbulence and instability mechanisms.

Contribution

It introduces a novel Sobolev instability result for plane waves of the cubic nonlinear Schrödinger equation, linking it to quantum Euler equations via the Madelung transform.

Findings

Existence of solutions with energy cascade and Sobolev norm inflation.

Polynomially fast growth of Sobolev norms above finite-energy level.

Uniform timing of Sobolev norm inflation near the semiclassical limit.

Abstract

In this paper we prove the existence of solutions to the quantum Euler equations on $\mathbb{T}^d$, $d\geqslant 2$, with almost constant mass density, displaying energy transfers to high Fourier modes and polynomially fast-in-time growth of Sobolev norms above the finite-energy level. These solutions are uniformly far from vacuum, suggesting that weak turbulence in quantum hydrodynamics is not necessarily related to the occurrence of vortex structures. In view of possible connections with instability mechanisms for the classical compressible Euler equations, we also keep track of the dependence on the semiclassical parameter, showing that, at high regularity, the time at which the Sobolev norm inflations occur is uniform when approaching the semiclassical limit. Our construction relies on a novel result of Sobolev instability for the plane waves of the cubic nonlinear Schr\"odinger equation (NLS), which is connected to the quantum Euler equations through the Madelung transform}. More precisely, we show the existence of smooth solutions to NLS, which are small-amplitude perturbations of a plane wave and undergo a polynomially fast $H^s$-norm inflation for $s>1$. The proof is based on a partial Birkhoff normal form procedure, involving the normalization of non-homogeneous Hamiltonian terms.