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Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces

Nataša Pavlović, Maja Tasković, Luisa Velasco

Abstract

Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin waves in superfluids). In this paper, we initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation. More precisely, we obtain the existence and uniqueness of non-negative mild solutions to this equation in exponentially weighted $L^\infty_{xv}$ spaces. This is accompanied by an analysis of the long-time behavior of such solutions - we prove that the solutions scatter, that is, they converge to solutions of the transport equation in the limit as $t \to \pm \infty$. Compared with the study of four-wave kinetic equations, the main challenge we face is to address the increased complexity of the geometry of the six-wave interactions.

Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces

Abstract

Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin waves in superfluids). In this paper, we initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation. More precisely, we obtain the existence and uniqueness of non-negative mild solutions to this equation in exponentially weighted spaces. This is accompanied by an analysis of the long-time behavior of such solutions - we prove that the solutions scatter, that is, they converge to solutions of the transport equation in the limit as . Compared with the study of four-wave kinetic equations, the main challenge we face is to address the increased complexity of the geometry of the six-wave interactions.
Paper Structure (18 sections, 20 theorems, 159 equations)

This paper contains 18 sections, 20 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Function spaces and main results
  3. Function Spaces
  4. Main results
  5. Preparations
  6. Representation Lemma
  7. Decomposition of the collision operator
  8. A priori estimates
  9. A key mapping
  10. Global well-posedness
  11. Kaniel-Shinbrot iteration scheme
  12. Iteration set up
  13. Associated Linear Problem
  14. Kaniel-Shinbrot Theorem
  15. Proof of Theorem \ref{['thm:KSnonnegative']}
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $\alpha,\beta > 0$, $d = 1$ and $0 < R_e \leq \frac{\alpha^{1/8}}{2^{\frac{7}{2}}C_{1,\beta}^{1/4}}$, where $C_{1,\beta} > 0$ is given by conv_constant. where $T^{-t}$ is defined in eq:transport.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Global well-posedness
  • Theorem 2.4: Existence and uniqueness of non-negative solutions
  • Theorem 2.5: Kaniel-Shinbrot
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 36 more