A Critical Density Estimate for the Vlasov--Poisson System: Energy Conservation and Moment Propagation

This paper establishes a critical space--time density estimate for the three-dimensional Vlasov--Poisson system in the whole space and on the torus.

This estimate has two consequences.

First, every nonnegative weak solution with finite initial kinetic energy in the bounded finite-energy weak-solution class conserves the total energy.

This gives a positive answer, within this class, to the energy-conservation question raised by Lions and Perthame.

Second, in the periodic repulsive case, it yields propagation of every velocity moment of order \(k>2\), thereby closing the previously unresolved interval \(2<k\leq3\).