A characterization of probability measure with finite moment and an application to the Boltzmann equation

TL;DR

This paper characterizes probability measures with finite moments using Fourier transform operators and applies this to prove regularity of solutions to the non-cutoff Boltzmann equation for Maxwellian molecules.

Contribution

It introduces a new Fourier-based characterization of probability measures with finite moments and applies it to analyze the regularity of Boltzmann equation solutions.

Findings

Proves continuity of the solution density in weighted L^1 spaces over time.

Establishes conditions on initial data for solution regularity.

Provides a new tool for analyzing probability measures with finite moments.

Abstract

We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity $f(t,v)\in C((0, \infty),L^1_{2k-2 +\alpha})$, where $f(t, v)$ stands for the density of unique measure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum $F_0$ satisfying \[ \int |v|^{2k-2+\alpha} dF_0(v) < \infty, 0\leq \alpha < 2,k= 2, 3, 4,\cdots \] provided that $F_0$ is not a single Dirac mass.