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Optimal Gradient Estimates and Asymptotic Behaviour for the Vlasov–Poisson System with Small Initial Data

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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 as 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 \({N \geqq 3}\) the derivative of the density of order k decays as t −N-k. An asymptotic formula for the solution at late times is also obtained.

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Authors and Affiliations

  1. Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, Korea

    Hyung Ju Hwang

  2. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, 14476, Potsdam, Germany

    Alan D. Rendall

  3. ICMAT (CSIC-UAM-UC3M-UCM), C/ Nicolas Cabrera 15, Cantoblanco, 28049, Madrid, Spain

    Juan J. L. Velázquez

Authors
  1. Hyung Ju Hwang
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  2. Alan D. Rendall
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  3. Juan J. L. Velázquez
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Correspondence to Juan J. L. Velázquez.

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Communicated by Y. Brenier

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Hwang, H.J., Rendall, A.D. & Velázquez, J.J.L. Optimal Gradient Estimates and Asymptotic Behaviour for the Vlasov–Poisson System with Small Initial Data. Arch Rational Mech Anal 200, 313–360 (2011). https://doi.org/10.1007/s00205-011-0405-3

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