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 $1$-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.