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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Mathematical tools for kinetic equations
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by Benoît Perthame PDF
Bull. Amer. Math. Soc. 41 (2004), 205-244 Request permission


Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, their range of application has been considerably extended. First introduced as a means to unify various perspectives on fluid mechanics, they are now used in plasma physics, semiconductor technology, astrophysics, biology.... They all are characterized by a density function that satisfies a Partial Differential Equation in the phase space. This paper presents some of the simplest tools that have been devised to study more elaborate (coupled and nonlinear) problems. These tools are basic estimates for the linear first order kinetic-transport equation. Dispersive effects allow us to gain time decay, or space-time $L^p$ integrability, thanks to Strichartz-type inequalities. Moment lemmas gain better velocity integrability, and macroscopic controls transform them into space $L^p$ integrability for velocity integrals. These tools have been used to study several nonlinear problems. Among them we mention for example the Vlasov equations for mean field limits, the Boltzmann equation for collisional dilute flows, and the scattering equation with applications to cell motion (chemotaxis). One of the early successes of kinetic theory has been to recover macroscopic equations from microscopic descriptions and thus to be able theoretically to compute transport coefficients. We also present several examples of the hydrodynamic limits, the diffusion limits and especially the recent derivation of the Navier-Stokes system from the Boltzmann equation, and the theory of strong field limits.
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Additional Information
  • Benoît Perthame
  • Affiliation: Département de Mathématiques et Applications, CNRS UMR 8553, École Normale Supérieure, 45, rue d’Ulm, 75230 Paris Cedex 05, France
  • Email:
  • Received by editor(s): January 15, 2003
  • Received by editor(s) in revised form: November 6, 2003
  • Published electronically: February 2, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 41 (2004), 205-244
  • MSC (2000): Primary 35F10, 35L60, 35Q35, 76P05, 82B40
  • DOI:
  • MathSciNet review: 2043752