Global well-posedness and stability of the inhomogeneous kinetic wave equation near vacuum

TL;DR

This paper establishes the global existence, uniqueness, and stability of solutions near vacuum for the 4-wave inhomogeneous kinetic wave equation in 2D and 3D, linking it to quantum Boltzmann-type equations.

Contribution

It proves well-posedness and stability of solutions near vacuum for the inhomogeneous kinetic wave equation, a novel result in this context.

Findings

Solutions remain non-negative for non-negative initial data

Global in time existence and uniqueness are established

Stability of solutions near vacuum is demonstrated

Abstract

In this paper, we prove global in time existence, uniqueness and stability of mild solutions near vacuum for the 4-wave inhomogeneous kinetic wave equation, for Laplacian dispersion relation in dimension $d=2,3$. We also show that for non-negative initial data, the solution remains non-negative. This is achieved by connecting the inhomogeneous kinetic wave equation to the cubic part of a quantum Boltzmann-type equation with moderately hard potential and no collisional averaging.