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

Abstract

We extend Loeper's $L^2$-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 $L^p$, with $1<p<+\infty$, based on the Helmholtz–Weyl decomposition. This allows us to generalize both the classical Loeper's 2-Wasserstein stability estimate (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 $1<p<+\infty$.