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A Moser-Type Construction for the Liouville Equation

Alfio Borzì, Marco Caponigro, Arianna Vicari

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.

A Moser-Type Construction for the Liouville Equation

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 and 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.
Paper Structure (5 sections, 5 theorems, 49 equations, 1 table)

This paper contains 5 sections, 5 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Analytical and numerical validation
  4. Numerical validation
  5. Conclusions

Key Result

Theorem 2.1

Let $f,g\in C^\infty(\overline\Omega)$ be such that $f>0$, $g>0$ on $\overline{\Omega}$. A necessary and sufficient condition for the existence of a field $a \in C^\infty(\overline{\Omega}\times [0,1];{\mathbb R}^n)$ in eq:Liouville-proof such that $\rho(1)=g$, is given by the following compatibilit More precisely, if eq:compat holds, then $a$ is given by where $U_t$ solves, for any $x,t$ fixed,

Theorems & Definitions (16)

  • Theorem 2.1
  • Lemma 2.1: Comoving transform
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['thm:main-compact']}
  • ...and 6 more