A Moser-Type Construction for the Liouville Equation

TL;DR

This paper extends Moser's volume form lemma to the kinetic Liouville equation, establishing conditions for connecting phase-space densities and constructing force fields, supported by numerical validation.

Contribution

It introduces a novel extension of Moser's lemma to the Liouville equation, providing explicit force field constructions under compatibility conditions.

Findings

Phase-space densities can be connected via the Liouville equation under a compatibility condition.

Explicit force fields are constructed through a weighted elliptic problem.

Numerical experiments validate the theoretical framework.

Abstract

We present a novel extension of Moser's volume form lemma to the kinetic Liouville equation. In particular, we show that two smooth, positive phase-space densities $f$ and $g$ can be connected in unit time by the Liouville equation if and only if a natural compatibility condition on velocity marginals is satisfied. Under this condition, an explicit family of force fields is constructed via a weighted elliptic problem in the velocity variable. Results of numerical experiments are presented to validate the theoretical framework.