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

TL;DR

This paper analyzes the inhomogeneous six-wave kinetic equation, proving existence, uniqueness, and scattering of solutions in exponentially weighted spaces, addressing the complexity of six-wave interactions in physical systems.

Contribution

It establishes the first well-posedness and scattering results for the inhomogeneous six-wave kinetic equation in weighted $L^ fty$ spaces.

Findings

Existence and uniqueness of solutions in weighted spaces

Solutions scatter and converge to transport solutions as time goes to infinity

Addresses geometric complexity of six-wave interactions

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.