The diffusion limit of transport equations derived from velocity-jump processes

Thomas Hillen, Hans G. Othmer

Research output: Contribution to journalArticle

255 Scopus citations

Abstract

In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak-Keller-Segel-Alt model for chemotaxis.

Original languageEnglish (US)
Pages (from-to)751-775
Number of pages25
JournalSIAM Journal on Applied Mathematics
Volume61
Issue number3
DOIs
StatePublished - 2000

Keywords

  • Aggregation
  • Chemotaxis equations
  • Diffusion approximation
  • Transport equations
  • Velocity-jump processes

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