# Finite Time Aggregation In Some Models Of Chemotaxis

Wednesday, September 9, 1998 - 3:30pm - 4:30pm

Keller 3-180

Michel Rascle (Université de Nice Sophia Antipolis)

We consider a few models of chemotactism, inspired from the classical Keller-Segel model, namely

begin{equation} left { begin{array}{lll} partial_t u+ text{div}(u chi(s)nabla s) & = & ; muDelta u,\ partial_t s - nu Delta s & = & pm ; k(s,u) end{array} right. label{1} end{equation}

where $nabla$ denotes the gradient, u the concentration in predators, s the concentration in substrate. I will discuss a few different examples in which the population of predators concentrates in finite time to a delta-function : aggregation. I will mainly focus on cartoons, where the diffusion is neglected. In the unstable case (the + case), aggregation is linked with a severe pathology in the structure of the underlying system of conservation laws, which is typically mixed type. However, I will show how one can mathematically solve the problem, and even describe such a singular solution after aggregation. Such solutions seem to be numerically very stable!

begin{equation} left { begin{array}{lll} partial_t u+ text{div}(u chi(s)nabla s) & = & ; muDelta u,\ partial_t s - nu Delta s & = & pm ; k(s,u) end{array} right. label{1} end{equation}

where $nabla$ denotes the gradient, u the concentration in predators, s the concentration in substrate. I will discuss a few different examples in which the population of predators concentrates in finite time to a delta-function : aggregation. I will mainly focus on cartoons, where the diffusion is neglected. In the unstable case (the + case), aggregation is linked with a severe pathology in the structure of the underlying system of conservation laws, which is typically mixed type. However, I will show how one can mathematically solve the problem, and even describe such a singular solution after aggregation. Such solutions seem to be numerically very stable!