Title: On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem

Authors: Hyung Ju Hwang and Juan J. L. Velazquez

Issue: Volume 58 (2009), Issue 6, 2623-2660

Abstract: In this paper we prove the existence of a large class of periodic solutions of the Vlasov-Poisson in one space dimension that decay exponentially as $t \to \infty$. The exponential decay is well known for the linearized version of the Landau damping problem and it has been proved in [E. Caglioti and C. Maffei, \emph{Time asymptotics for solutions of Vlasov-Poisson equation in a circle}, J. Stat. Phys. \textbf{92} (1998), number 1/2] for a class of solutions of the Vlasov-Poisson system that behaves asymptotically as free streaming solutions and are sufficiently flat in the space of velocities. The results in this paper enlarge the class of possible asymptotic limits, replacing the flatness condition in [op. cit.] by a stability condition for the linearized problem.