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Stability estimates for the Vlasov-Poisson system in $p$-kinetic Wasserstein distances

Mikaela Iacobelli, Jonathan Junné

Abstract

We extend Loeper's $L^2$-estimate relating the electric fields to the densities for the Vlasov-Poisson system to $L^p$, with $1 < p < +\infty$, based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical Loeper's $2$-Wasserstein stability estimate and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance to kinetic Wasserstein distances of order $1 < p < +\infty$.

Stability estimates for the Vlasov-Poisson system in $p$-kinetic Wasserstein distances

Abstract

We extend Loeper's -estimate relating the electric fields to the densities for the Vlasov-Poisson system to , with , based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical Loeper's -Wasserstein stability estimate and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance to kinetic Wasserstein distances of order .
Paper Structure (8 sections, 8 theorems, 98 equations)

This paper contains 8 sections, 8 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Definitions and main results
  4. A new $L^p$-estimate via the Helmholtz-Weyl decomposition for $1 < p < +\infty$
  5. Proof of the $L^p$-estimate
  6. Stability estimates revisited for Wasserstein-like distances
  7. Loeper's estimate revisited
  8. Kinetic Wasserstein distance revisited

Key Result

Theorem 1.4

The Helmholtz-Weyl decomposition holds for $L^p(\mathcal{X})$, for any $1 < p < +\infty$; that is, Moreover, when $p = 2$, this decomposition is orthogonal.

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: Helmholtz-Weyl decomposition
  • Definition 1.5
  • Definition 1.6
  • Lemma 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 8 more