A Uniqueness Criterion for Unbounded Solutions to the Vlasov-Poisson System

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 grow 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.