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

Filippo Giuliani, Raffaele Scandone

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 Schrödinger 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.

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

Abstract

In this paper we prove the existence of solutions to the quantum Euler equations on , , 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 Schrödinger 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 -norm inflation for . The proof is based on a partial Birkhoff normal form procedure, involving the normalization of non-homogeneous Hamiltonian terms.

Paper Structure

This paper contains 11 sections, 25 theorems, 302 equations.

Table of Contents

  1. Introduction and main results
  2. Connection with NLS equation and fast norm inflation close to plane waves
  3. Open problems and perspectives
  4. Proof of the main results
  5. Growth of Sobolev norms for the NLS equation
  6. The set
  7. Normal form
  8. Effective finite dimensional model
  9. Approximation argument
  10. Conclusion of the proof of Theorem \ref{['thm:weak']}
  11. Estimates in Wiener algebra and Sobolev spaces

Key Result

Theorem 1.1

Let $d\geqslant 2$, $m,\sigma >0$, $\varepsilon\in(0,1]$, $s>1$. There exists $C>0$, independent on $\varepsilon$, such that the following holds. For every $\mathcal{K}>0$ large enough, there exists a smooth solution $(\rho,v)$ to system eq:QHD, defined on $[0,T]\times\mathbb{T}^d$ for some satisfying the estimates

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Remark 2
  • Proposition 2.1
  • Remark 3
  • proof : Proof of Proposition \ref{['pr:equiv1']}
  • Proposition 2.2
  • Remark 4
  • ...and 60 more