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

Hyung Ju Hwang, Alan D. Rendall, and Juan J.L. Velazquez
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Submission date: 24. Jan. 2007
Pages: 40
MSC-Numbers: 82B40
Keywords and phrases: Vlasov-Poisson, gradient estimates
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Abstract:
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>2 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.