Dispersion estimates for the two-dimensional Vlasov–Yukawa system with small data

We present dispersion estimates for the two-dimensional Vlasov–Yukawa system with small data. When the initial data are sufficiently regular and small, we show that the local mass density and the Yukawa force field decay to zero algebraically fast in time. These dispersion estimates are not known for the two-dimensional Vlasov–Poisson system. For the dispersion estimates, we effectively use the short-range character of the Yukawa potential and the optimal gradient estimates introduced by Hwang, Rendall and Velázquez for the three-dimensional Vlasov–Poisson system