Scattering theory for the Inhomogeneous Kinetic Wave Equation

TL;DR

This paper develops a comprehensive scattering theory for the inhomogeneous kinetic wave equation, establishing global solutions that conserve key physical quantities and demonstrating their dispersive behavior using novel analytical techniques.

Contribution

It introduces a new scattering framework for the inhomogeneous KWE, employing physical space methods and novel bounds to analyze solutions and their long-term behavior.

Findings

Existence of global strong dispersive solutions conserving mass, momentum, and energy.

Proof of scattering and bijective wave operators in a suitable topology.

Development of new trilinear bounds and a collisional averaging estimate.

Abstract

In this paper we construct global strong dispersive solutions to the space inhomogeneous kinetic wave equation (KWE) which propagate $L^1_{xv}$ -- moments and conserve mass, momentum and energy. We prove that they scatter, and that the wave operators mapping the initial data to the scattering states are 1-1, onto and continuous in a suitable topology. Our proof is carried out entirely in physical space, and combines dispersive estimates for the free transport with new trilinear bounds for the gain and loss operators of the KWE on weighted Lebesgue spaces. A fundamental tool in obtaining these bounds is a novel collisional averaging estimate. Finally we show that the nonlinear evolution preserves positivity forward in time. For this, we use the Kaniel-Shinbrot iteration scheme \cite{KS}, properly initialized to ensure the successive approximations are dispersive.