Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data

Details Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data

2006-06-16

arxiv.org/abs/math/0606389v1

Mathematics

Analysis of PDEs

37 pages

Scientific paper

The Vlasov-Poisson system describes interacting systems of collisionless particles. For solutions with small initial data in three dimensions it is known that the spatial density of particles decays like $t^{-3}$ at late times. In this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay so that in $N$ dimensions for $N\ge 3$ the derivative of the density of order $k$ decays like $t^{-N-k}$. An asymptotic formula for the solution at late times is also obtained.

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