Abstract
We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as trightarrow -infty to asymptotic dynamics as trightarrow +infty. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.
Highlights
The three-dimensional Vlasov–Poisson system describes the evolution of a particle distribution1 (t, x, v) ∶ R × R3 × R3 → R satisfying t + v ⋅ ∇x + ∇x ⋅ ∇v = 0, Δx (t, x) = (t, x), (t, x) = ∫R3 2(t, x, v)dv. (1.1)
This is a model for a continuum limit of a classical many-body problem with Newtonian self-interactions through a force field ∇x that can be attractive ( = −1 ) as in a galactic setting, or repulsive ( = 1 ) as in a plasma or ion gas, and which is generated by the spatial density (t, x) of the particle distribution
There has been progress in understanding the long time asymptotic behavior: sharp decay rates of the density and force field are known in some settings [17, 19, 29, 31, 36, 38], and it has been shown that for sufficiently small initial data 0 the problem (1.1) exhibits a modified scattering dynamic [6, 20] defined in terms of a limit distribution ∞ and an asymptotic force field E∞[ ∞], defined by inverting the roles of x and v: E∞[
Summary
(5) It is worth noting a curious fact: our proof can be adapted directly to the case of a plasma of two species (ions and electrons) In this case, using (ii), one can construct solutions for which the asymptotic electric field profile E∞ ≡ 0 vanishes and the solutions scatter linearly. The proof of part (i) shows how natural the pseudo-conformal inversion I is to study asymptotics of (1.1): working with only moments that are conserved in the linear evolution of (1.1) one directly obtains global solutions in a bootstrap argument. (iii) follows by combining (ii) (backwards in time) to go from pastasymptotic data to initial data and (i) to go from initial data to future asymptotic data While it may be less intuitive, using the pseudo-conformal transformation simplifies the presentation over the physical space analysis as in [20], and quickly leads to the natural modified scattering behavior. It sheds new light on some classical decay estimates like (1.13)
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