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On the optimal local well-posedness of the wave kinetic equation in $L^r$

Ioakeim Ampatzoglou, Tristan Léger

TL;DR

This work delivers a unified local well-posedness theory for the 3D wave kinetic equation with Laplacian dispersion in almost critical weighted $L^r$ spaces, valid for all $2\le r\le \infty$ and small $\delta$-types near criticality. The authors develop a kinetic-only toolkit based on Bobylev variables, geometric inequalities, collisional averaging, and an involutional change of variables, to obtain moment-preserving gain and loss estimates. By organizing the collision operator into gain/loss components and six multilinear pieces, they prove a contraction in a weighted $L^r$-space ball, yielding existence, uniqueness, continuous dependence, and positivity of strong solutions on a local time interval. The paper also provides a precise resonant-manifold parametrization and a unified approach that recovers prior $L^2$ and $L^ inity$ results while extending to all almost critical regimes, contributing foundational results for the kinetic-WKE theory.

Abstract

In this paper, we give a unified treatment of the local well-posedness for the wave kinetic equation in almost critical weighted $L^r$ spaces with $2 \leq r \leq \infty.$ The proof builds on ideas from our earlier works \cite{AmLe24, AmLemain25}. Our approach is based solely on kinetic tools, with no appeal to Fourier theory.

On the optimal local well-posedness of the wave kinetic equation in $L^r$

TL;DR

This work delivers a unified local well-posedness theory for the 3D wave kinetic equation with Laplacian dispersion in almost critical weighted spaces, valid for all and small -types near criticality. The authors develop a kinetic-only toolkit based on Bobylev variables, geometric inequalities, collisional averaging, and an involutional change of variables, to obtain moment-preserving gain and loss estimates. By organizing the collision operator into gain/loss components and six multilinear pieces, they prove a contraction in a weighted -space ball, yielding existence, uniqueness, continuous dependence, and positivity of strong solutions on a local time interval. The paper also provides a precise resonant-manifold parametrization and a unified approach that recovers prior and results while extending to all almost critical regimes, contributing foundational results for the kinetic-WKE theory.

Abstract

In this paper, we give a unified treatment of the local well-posedness for the wave kinetic equation in almost critical weighted spaces with The proof builds on ideas from our earlier works \cite{AmLe24, AmLemain25}. Our approach is based solely on kinetic tools, with no appeal to Fourier theory.

Paper Structure

This paper contains 21 sections, 10 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results obtained
  4. Parametrization of the resonant manifolds
  5. Collisional operators
  6. Organization of the paper
  7. Notations
  8. Kinetic toolbox
  9. Bobylev variables
  10. Geometric identities
  11. Collisional averaging estimates
  12. An involutionary change of variables
  13. Gain operators estimates
  14. Estimate for $\mathcal{G}_0$
  15. Estimate for $\mathcal{G}_1$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $r\geq 2$ and $0<\delta<1/r$ if $r<\infty$, $\delta > 0$ if $r = \infty.$ Let $f_0 \in {\langle} k {\rangle}^{-2 + \frac{3}{r} -\delta} L^r.$ Then there exists $T=T(\|{\langle} v{\rangle}^k f_0\|_{L^r})>0$ such that the initial value problem KWE has a unique strong solution $f(t) \in \mathcal{C} Moreover the solution depends continuously on the initial data: let $f_0, g_0 \in {\langle} k {\ran

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 10 more