Riv. Mat. Univ. Parma, Vol. 3, No. 1, 2012

François James and Nicolas Vauchelet

On the hydrodynamical limit for a one dimensional kinetic model of cell aggregation by chemotaxis

Pages: 91-113
Received: 15 October 2010   
Accepted in revised form: 24 November 2010
Mathematics Subject Classification (2000): 92C17, 35L65.

Keywords: Chemotaxis, hydrodynamic limit, scalar conservation laws, aggregation.

Abstract: The hydrodynamic limit of a one dimensional kinetic model describing chemotaxis is investigated. The limit system is a conservation law coupled to an elliptic problem for which the macroscopic velocity is possibly discontinuous. Therefore, we need to work with measure-valued densities. After recalling a blow-up result in finite time of regular solutions for the hydrodynamic model, we establish a convergence result of the solutions of the kinetic model towards solutions of a problem limit defined thanks to the flux. Numerical simulations illustrate this convergence result.

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