Stability estimates for the Vlasov–Poisson system in pp‐kinetic Wasserstein distances

Abstract We extend Loeper's ‐estimate (Theorem 2.9 in J. Math. Pures Appl . (9) 86 (2006), no. 1, 68–79) relating the force 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 2‐Wasserstein stability estimate (Theorem 1.2 in J. Math. Pures Appl . (9) 86 (2006), no. 1, 68–79) and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance (Theorem 3.1 in Arch Rational Mech. Anal . 244 (2022), no. 1, 27–50) to kinetic Wasserstein distances of order .