Gevrey regularity for the Vlasov-Poisson system

We prove propagation of $\frac{1}{s}$-Gevrey regularity $(s\in(0,1))$ for the Vlasov-Poisson system on $\mathbb{T}^d$ using a Fourier space method in analogy to the results proved for the 2D Euler system in \cite{KV} and \cite{LO}. More precisely, we give a quantitative estimate for the growth in time of the $\frac{1}{s}$-Gevrey norm for the solution of the system in terms of the force field and the gradient in the velocity variable of the distribution of matter. As an application, we show global existence of $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\mathbb{T}^3$. Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in $\mathbb{R}^d$. In particular, this implies global existence of analytic $(s=1)$ and $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\mathbb{R}^3$.