The DOI matches its Crossref record; 17 of the 17 references it cites were found in a registry or an index.
The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes
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.
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Cites
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Diffusion by Continuous Movements
Taylor, G. I. (1922) · Proceedings of the London Mathematical Society · doi:10.1112/plms/s2-20.1.196 · Cited by 2,922
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Random walk with persistence and external bias
Patlak, C. S. (1953) · The Bulletin of Mathematical Biophysics · doi:10.1007/bf02476407 · Cited by 1,008
Models of dispersal in biological systems
Othmer, H. G., Dunbar, S. R., & Alt, W. (1988) · Journal of Mathematical Biology · doi:10.1007/bf00277392 · Cited by 858
Self-citationON DIFFUSION BY DISCONTINUOUS MOVEMENTS, AND ON THE TELEGRAPH EQUATION
GOLDSTEIN, S. (1951) · The Quarterly Journal of Mechanics and Applied Mathematics · doi:10.1093/qjmam/4.2.129 · Cited by 751
Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks
Stevens, A., & Othmer, H. G. (1997) · SIAM Journal on Applied Mathematics · doi:10.1137/s0036139995288976 · Cited by 704
Self-citationA stochastic model related to the telegrapher's equation
Kac, M. (1974) · Rocky Mountain Journal of Mathematics · doi:10.1216/rmj-1974-4-3-497 · Cited by 608
Cited by
A user’s guide to PDE models for chemotaxis
Hillen, T., & Painter, K. J. (2009) · Journal of Mathematical Biology · doi:10.1007/s00285-008-0201-3 · Cited by 1,650
Self-citationCodling, E. A., Plank, M. J., & Benhamou, S. (2008) · Journal of The Royal Society Interface · doi:10.1098/rsif.2008.0014 · Cited by 1,482
Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues
Bellomo, N., Bellouquid, A., Tao, Y., & Winkler, M. (2015) · Mathematical Models and Methods in Applied Sciences · doi:10.1142/s021820251550044x · Cited by 1,077
Cates, M. E. (2012) · Reports on Progress in Physics · doi:10.1088/0034-4885/75/4/042601 · Cited by 481
The Diffusion Limit of Transport Equations II: Chemotaxis Equations
Othmer, H. G., & Hillen, T. (2002) · SIAM Journal on Applied Mathematics · doi:10.1137/s0036139900382772 · Cited by 347
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Metadata
- Authors
- Othmer, H. G., & Hillen, T.
- Venue
- SIAM Journal on Applied Mathematics
- Published
- 2000-01
- Volume
- 61
- Issue
- 3
- Pages
- 751-775
- ISSN
- 0036-1399 (ISSN-L)
- 1095-712X
- OpenAlex
- W1973496993
- Cited by
- 383
- References
- 17
- FWCI
- 8.29
- Type
- journal article
- Language
- en
- Open access
- closed
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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