Abstract

Let us suppose that we are a team of biologists studying a population of bacteria. If we watch the population through a microscope, we will have many information on the dynamics of the population (how they move, how they interact with each other) but if we want to study the front of propagation (how the population invades its environment), it is more cleaver to watch the Petri dish with the naked eye for a long time.

From a mathematical point of view, the observations though the microscope are modeled by a kinetic partial differential equation (PDE) with a parameter epsilon (the higher epsilon, the more we zoom in). To recover the naked eye observations on a large time-scale, we pass to the limit as epsilon goes to zero. The resulting equation is a useful tool to study the propagation front of the bacterial population. Doing this, we adopt the so-called geometric optics approach for front propagation.

In this presentation, we will show how it works on a very simple example. For this, we will need to make a short introduction to viscosity solutions of Hamilton-Jacobi equation, which are solutions of a PDE in a weak sense