Global solution for a kinetic chemotaxis model with internal dynamics and its fast adaptation limit

Abstract

A nonlinear kinetic chemotaxis model with internal dynamics incorporating signal transduction and adaptation is considered. This paper is concerned with: (i) the global solution for this model, and, (ii) its fast adaptation limit to Othmer-Dunbar-Alt type model. This limit gives some insight to the molecular origin of the chemotaxis behaviour.First, by using the Schauder fixed point theorem, the global existence of weak solution is proved based on detailed a priori estimates, under quite general assumptions. However, the Schauder theorem does not provide uniqueness, so additional analysis is required to be developed for uniqueness. Next, the fast adaptation limit of this model is derived by extracting a weak convergence subsequence in measure space. For this limit, the first difficulty is to show the concentration effect on the internal state. Another difficulty is the strong compactness argument on the chemical potential, which is essential for passing the nonlinear kinetic equation to the weak limit.

On Entropy Dynamics for Active “Living” Particles

“…It is worth mentioning that the aforementioned micro-macro derivation generates challenging analytic problems related to the qualitative analysis of the solutions to mathematical problems at different scales-say at the level of kinetic equations [51] or at that of tissues [52]. Hence, one also has to look for a multiscale validity of the entropy function by looking at it as a tool that can contribute to the qualitative analysis of mathematical problems [53].…”

Kinetic models of chemotaxis towards the diffusive limit: asymptotic analysis

“…A proof of the global existence of the parabolic system (5) is given in this section providing that D i , i D 1, 2 are symmetric positive, definite, and satisfy (25). The proof is based on known techniques, but it is given for the sake of completeness as well as with tutorial aims.…”

“…Moreover, integrating (47) with respect to t from 0 to T yields Then, integrating (59) with respect to t and using (25), and summing with respect to˛, yields…”

“…This approach has been recently extended to derive hyperbolic models of vehicular traffic and crowd dynamics . Interesting results have been recently obtained by hybrid systems mixing the microscopic scale to the macroscopic one . The reader interested to more general approaches to multiscale derivation is referred to the book .…”

“…This approach has been recently extended to derive hyperbolic models of vehicular traffic [23] and crowd dynamics [24]. Interesting results have been recently obtained by hybrid systems mixing the microscopic scale to the macroscopic one [25]. The reader interested to more general approaches to multiscale derivation is referred to the book [26].An important investigation, not yet treated in the literature, reviewed and critically analyzed in [9], is the derivation of models of the chemotactic behavior when the diffusion term dominates, such as the following Keller-Segel:T 0…”