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Stability Estimates in Kinetic Wasserstein Distances for the Vlasov-Poisson System with Yudovich Density

Jonathan Junné, Alexandre Rege

Abstract

We investigate the stability of solutions to the Vlasov-Poisson system using the unifying framework of the kinetic Wasserstein distance, introduced by Iacobelli in (Section 4 in Arch. Ration. Mech. Anal. 244 (2022), no. 1, 27-50). This allows us to treat both macroscopic densities that lie in a Yudovich space, as recently considered by Crippa et al. (Theorem 1.6 in Nonlinearity 37 (2024), no. 9, 095015) for the $1$-Wasserstein distance, and higher order Wasserstein distances, for which only bounded macroscopic densities were treated by Iacobelli and the first author (Theorem 1.11 in Bull. Lond. Math. Soc. 56 (2024), 2250-2267). First, we establish an $L^p$-estimate on the difference between two force fields in terms of a suitable nonlinear quantity that controls the kinetic Wasserstein distance between their macroscopic densities. Second, we use this estimate in order to derive a closable Osgood-type inequality for the kinetic Wasserstein distance between two solutions. This enables us to prove our main theorem; for $1 \le p < +\infty$ we show the $p$-Wasserstein stability of solutions to the Vlasov-Poisson system with macroscopic densities belonging to a Yudovich space.

Stability Estimates in Kinetic Wasserstein Distances for the Vlasov-Poisson System with Yudovich Density

Abstract

We investigate the stability of solutions to the Vlasov-Poisson system using the unifying framework of the kinetic Wasserstein distance, introduced by Iacobelli in (Section 4 in Arch. Ration. Mech. Anal. 244 (2022), no. 1, 27-50). This allows us to treat both macroscopic densities that lie in a Yudovich space, as recently considered by Crippa et al. (Theorem 1.6 in Nonlinearity 37 (2024), no. 9, 095015) for the -Wasserstein distance, and higher order Wasserstein distances, for which only bounded macroscopic densities were treated by Iacobelli and the first author (Theorem 1.11 in Bull. Lond. Math. Soc. 56 (2024), 2250-2267). First, we establish an -estimate on the difference between two force fields in terms of a suitable nonlinear quantity that controls the kinetic Wasserstein distance between their macroscopic densities. Second, we use this estimate in order to derive a closable Osgood-type inequality for the kinetic Wasserstein distance between two solutions. This enables us to prove our main theorem; for we show the -Wasserstein stability of solutions to the Vlasov-Poisson system with macroscopic densities belonging to a Yudovich space.

Paper Structure

This paper contains 9 sections, 9 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. The Vlasov--Poisson system
  3. Yudovich spaces and assumptions
  4. Stability estimates
  5. An Lp-estimate via interpolation for Yudovich macroscopic densities
  6. Stability estimates in p-kinetic Wasserstein distances
  7. Insufficiency of Loeper's approach in the Yudovich setting
  8. Proof of the main result
  9. Beyond the Vlasov--Poisson system

Key Result

Theorem 1.9

Let $1 \le p < +\infty$. Let $f_1, f_2$ be two Lagrangian weak solutions to sys:Vlasov--Poisson and $\Theta$ a growth function such that ass:L^1-boundass:varphi_theta-continuousass:Theta-non-drecreasing-and-concaveass:Theta-two-inequalities-for-lambdaass:Osgood hold. Then there is a constant $c_{\Th where $\Phi^{-1}_{p,\Theta}$ is the inverse of the map $s \mapsto s/\sqrt{\lvert*\rvert{\log s}\The

Theorems & Definitions (19)

  • Definition 1.1: Wasserstein distances
  • Definition 1.2: kinetic Wasserstein distance
  • Definition 1.3
  • Theorem 1.9
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4: iacobelli_stability_2024
  • Theorem 2.5: Gangbo-McCann gangbo_geometry_1996
  • Lemma 2.6: crippa_existence_2024
  • ...and 9 more