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Modified scattering for the Vlasov–Poisson system

We study the asymptotic behavior of dispersing solutions to the Vlasov–Poisson system. Due to long interaction range, we do not expect linear scattering (Choi S-H and Ha S-Y 2011 SIAM J. Math. Anal. 43 2050–77). Instead, we prove a modified scattering result (or long range scattering result) of small and dispersing solutions. We find a quasi-free forward trajectory so that along the trajectory, the solution has an asymptotic limit. We extract the logarithmic growth part of the Duhamel term, and absorb it into the quasi-free trajectory, then the remaining part enjoys faster decay so as to obtain the asymptotic state. Throughout this paper, we simplify notation of norms when there is no confusion. For example, we denote Lp=Lx,vp for global norms. We will denote C as a generic constant which varies from lines to lines and is independent of large time T. We also use the symbol A≲B when there is a constant C such that A⩽CB.