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The diffusion limit of transport equations derived from velocity-jump processes. (English) Zbl 1002.35120

Summary: 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.

MSC:

35Q80 Applications of PDE in areas other than physics (MSC2000)
92E20 Classical flows, reactions, etc. in chemistry
60J60 Diffusion processes
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations
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