DECAY ESTIMATES FOR THE 3D RELATIVISTIC AND NON-RELATIVISTIC VLASOV-POISSON SYSTEMS

Abstract

We study the small data global regularity problem of the 3D Vlasov-Poisson system for both the relativistic case and the non-relativistic case. The main goal of this paper is twofold. (i) Based on a Fourier method, which works systematically for both the relativistic case and the non-relativistic case, we give a short proof for the global regularity and the sharp decay estimate for the 3D Vlasov-Poisson system. Moreover, we show that the nonlinear solution scatters to a linear solution in both cases. The result of sharp decay estimates for the non-relativistic case is not new, see Hwang-Rendall-Velázquez [9] and Smulevici [23]. (ii) The Fourier method presented in this paper serves as a good comparison for the study of more complicated 3D relativistic Vlasov-Nordström system in [24] and 3D relativistic Vlasov-Maxwell system in [25].