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

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Rendall,  Alan D.
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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0606389.pdf
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ArchRatMechAnal200_313.pdf
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Citation

Hwang, H. J., Rendall, A. D., & Velazquez, J. J. L. (2011). Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data. Archive for Rational Mechanics and Analysis, 200 (1), 313-360. doi:10.1007/s00205-011-0405-3.


Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-5F9F-5
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 $N\ge 3$ 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.