La ecuaci\'on de Keller-Segel

TL;DR

This paper studies the Keller-Segel equations modeling chemotaxis, analyzing their behavior in different mass regimes, and explores entropy methods for related equations, providing a comprehensive overview and bibliography.

Contribution

It offers an exhaustive summary of Keller-Segel equations, including probabilistic derivation, behavior analysis based on initial mass, and application of entropy techniques.

Findings

Mass remains constant over time.

Different behaviors depending on initial mass M: diffusion dominates if M<8π, aggregation if M>8π, blow-up at M=8π.

Entropy methods are developed for related equations.

Abstract

The purpose of this work is the study of \textit{chemotaxis} and how to model it through the equations of Keller-Segel. \textit{Chemotaxis} is a natural process which induces the organisms to direct their movement according to certain chemicals concentrated on their surroundings. This was observed in some myxamoebas, which experiment a random walk, spreading through space, looking for food, while they are affected by chemotaxis. As a result of combining both behaviours, one obtains a competition between diffusion and aggregation. This notes try to make a summary of the main results about these equations, providing an exhaustive bibliography about them. First of all, we would like to present the model in PDEs, we will use the model set by E.F. Keller and L.A. Segel and by C.S. Patlak. Following it, we will do a probabilistic deduction in order to check that these equations could model this situation. After that, we realise that the mass remains constant along time. In dimension $2$, on a slightly simplified model, we observe several behaviours according to the initial mass $M$: If $M<8 \pi$, the diffusion dominates, if $M>8 \pi$ (and finite initial second moment) there will be aggregation, and if $M=8 \pi$ (and finite initial second moment) we get a blow-up in infinite time. Finally, motivated by the use of entropy techniques on several parts of the work, we decide to further develop them on other type of equations.