A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system

TL;DR

This paper establishes a uniqueness criterion for solutions to the Vlasov-Poisson system in 2D and 3D, accommodating solutions with certain singularities, and provides explicit examples illustrating the criterion.

Contribution

It introduces a new uniqueness condition based on the growth of $L^p$ norms of the density, allowing for solutions with logarithmic singularities.

Findings

Proves uniqueness under specified growth conditions.

Constructs explicit initial data satisfying the uniqueness criterion.

Connects solutions to radially symmetric steady states in 2D gravitational case.

Abstract

We prove uniqueness for the Vlasov-Poisson system in two and three dimensions under the condition that the $L^p$ norms of the macroscopic density growth at most linearly with respect to $p$. This allows for solutions with logarithmic singularities. We provide explicit examples of initial data that fulfill the uniqueness condition and that exhibit a logarithmic blow-up. In the gravitational two-dimensional case, such states are intimately related to radially symmetric steady solutions of the system. Our method relies on the Lagrangian formulation for the solutions, exploiting the second-order structure of the corresponding ODE.