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

Contact the author: Please use for correspondence this email.
Submission date: 24. Jan. 2007
Pages: 40
published in: Archive for rational mechanics and analysis, 200 (2011) 1, p. 313-360 
DOI number (of the published article): 10.1007/s00205-011-0405-3
Bibtex
MSC-Numbers: 82B40
Keywords and phrases: Vlasov-Poisson, gradient estimates
Download full preprint: PDF (299 kB)

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.