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We study the small data global regularity problem of the $3D$ Vlasov-Poissonsystem for both the relativistic case and the non-relativistic case. The maingoal of this paper is twofold. (i) Based on a Fourier method, which workssystematically for both the relativistic case and the non-relativistic case, wegive a short proof for the global regularity and the sharp decay estimate forthe $3D$ Vlasov-Poisson system. Moreover, we show that the nonlinear solutionscatters to a linear solution in both cases. The result of sharp decayestimates for the non-relativistic case is not new, seeHwang-Rendall-Vel\'azquez and Smulevici. (ii) The Fourier method presented inthis paper serves as a good comparison for the study of more complicated $3D$relativistic Vlasov-Nordstr\"om system and $3D$ relativistic Vlasov-Maxwellsystem.

Decay estimates for the $3D$ relativistic and non-relativistic Vlasov-Poisson systems