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 language | English (US) |
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Pages (from-to) | 751-775 |
Number of pages | 25 |
Journal | SIAM Journal on Applied Mathematics |
Volume | 61 |
Issue number | 3 |
DOIs | |
State | Published - 2000 |
Keywords
- Aggregation
- Chemotaxis equations
- Diffusion approximation
- Transport equations
- Velocity-jump processes