Existence of solutions of the hyperbolic Keller-Segel model
By Benoit Perthame, Anne-Laure Dalibard. 2006
82 citations recorded by the source corpus.
We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This model lacks strong compactness or contraction properties. Our purpose is to prove the existence of an entropy solution obtained, as usual, in passing to the limit in a sequence of solutions to the parabolic approximation. The method consists in the derivation of a kinetic formulation for the weak limit. The specific structure of the limiting kinetic equation allows for a `rigidity theorem', which identifies some property of the solution (which might be nonunique) to this kinetic equation. This is enough to deduce a posteriori the strong convergence of a subsequence.
Subfields: Biomedical Engineering, Modeling and Simulation, Molecular Biology
View Paper: https://www.semanticscholar.org/paper/5b85cbe92e9a4e76a5d5ee1d6ef1c41b862331e5
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In Biomedical Engineering
In Modeling and Simulation
In Molecular Biology
Citations in the corpus
- Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions (2008, background): “In [8] it has been shown that (2.”
- Control of the Continuity Equation with a Non Local Flow (2009, background): “Other analytically similar situations are found in [5], where a kinetic model for pedestrians is presented, and in [23], which concerns the Keller-Segel model.”