Time decay for solutions to the linearized Vlasov equation

Abstract

Time decay for solutions to the initial-value problem for the linearized Vlasov equation

is studied. Here E_x = ρ = ∫ gdv and f(v² ) ≥ 0 is to be sufficiently smooth and strictly decreasing. The initial value for g is to be suitably smooth and small at infinity. When f¹ (v² ) → 0 as |v| → ∞ at an algebraic rate, it is shown that ρ → 0 at an algebraic rate as t → ∞ in both the L² and maximum norms. When f is a Gaussian, the decay rate is logarithmic. The field E is also shown to decay in the maximum norm for both generic classes of f's. Similar results are obtained in three dimensions for spherically symmetric data. When f has compact support, no decay of the density in L ²(R¹) is possible for data of compact support.