We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as $t\rightarrow -\infty$ to asymptotic dynamics as $t\rightarrow +\infty$. 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.
In this paper, we classify Möbius invariant differential operators of second order in two-dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations. The equations are naturally associated with a continuous family of convex cones $\Gamma _p$ in $\mathbb R^2$, with parameter $p\in [1, 2]$, joining the half plane $\Gamma _1:=\{ (\lambda _1, \lambda _2):\lambda _1+\lambda _2>0\}$ and the first quadrant $\Gamma _2:=\{ (\lambda _1, \lambda _2):\lambda _1, \lambda _2>0\}$. Chen and C. M. Li established in 1991 a Liouville type theorem corresponding to $\Gamma _1$ under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville type theorem we establish in this paper for $\Gamma _p$, $1<p\le 2$, does not require any additional assumption on the solution as for $\Gamma _1$. This is reminiscent of the Liouville type theorems in dimensions $n\ge 3$ established by Caffarelli, Gidas and Spruck in 1989 and by A. B. Li and Y. Y. Li in 2003–2005, where no additional assumption was needed either. On the other hand, there is a striking new phenomena in dimension $n=2$ that $\Gamma _p$ for $p=1$ is a sharp dividing line for such uniqueness result to hold without any further assumption on the solution. In dimensions $n\ge 3$, there is no such dividing line.