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Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting

Ioakeim Ampatzoglou, Charles Collot, Pierre Germain

Abstract

We examine the validity of the kinetic description of wave turbulence for a model quadratic equation. We focus on the space-inhomogeneous case, which had not been treated earlier; the space-homogeneous case is a simple variant. We determine nonlinearities for which the kinetic description holds, or might fail, up to an arbitrarily small polynomial loss of the kinetic time scale. More precisely, we focus on the convergence of the Dyson series, which is an expansion of the solution in terms of the random data.

Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting

Abstract

We examine the validity of the kinetic description of wave turbulence for a model quadratic equation. We focus on the space-inhomogeneous case, which had not been treated earlier; the space-homogeneous case is a simple variant. We determine nonlinearities for which the kinetic description holds, or might fail, up to an arbitrarily small polynomial loss of the kinetic time scale. More precisely, we focus on the convergence of the Dyson series, which is an expansion of the solution in terms of the random data.

Paper Structure

This paper contains 54 sections, 21 theorems, 375 equations.

Table of Contents

  1. Introduction
  2. The equation, the data, and the singular limit
  3. Background
  4. Derivation of the kinetic wave equation
  5. Derivation of related kinetic models
  6. Statement of the main result
  7. Strategy of the proof
  8. Failure of convergence on the kinetic time scale for $m(0)=1$ and $\omega_0=0$
  9. Difficulties: the belt and the inhomogeneous setting
  10. Application to some physical examples
  11. Notations
  12. Probability space
  13. Fourier transform
  14. Bourgain spaces
  15. Wigner transform and space correlation
  16. ...and 39 more sections

Key Result

Theorem \oldthetheorem

Let $a\in \mathcal{C}^\infty_0(\mathbb R^{2d})$ and $\nu>0$. Consider Equation nonlinschrod with initial data hedgehog and Then there exist $\epsilon^*>0$ and $\kappa>0$ such that for any $0<\epsilon<\epsilon^*$ and for any $0<T<\min\{\epsilon,\epsilon^\nu T_{kin}\}$, there exists a set $E$ of probability $\mathbb{P}(E)>1-\epsilon^{\kappa}$, such that on $E$, there exists a unique solution $u$ to

Theorems & Definitions (43)

  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem: Dirichlet kernel
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 33 more