Seminar on partial differential equations

Chemotaxis-consumption model and the importance of the boundary conditions

Marcel Braukhoff

Vienna University of Technology

Tuesday, 26. November 2019 - 10:15 to 11:15

in IM, rear building, ground floor

In the talk we discuss the behavior of the concentration of some bacteria swimming in water (for example of the species Bacillus subtilis), whose otherwise random motion is partially directed towards higher concentrations of a signaling substance (oxygen) they consume. After a transition phase, the system can be described using a chemotaxis-consumption model on a bounded domain. Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed.

Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, this talks considers no-flux boundary conditions for the bacteria density and the physically meaningful Robin boundary conditions for the signaling substance and Dirichlet boundary conditions for the flow.

In the talk, we study the existence of a global (weak) solution. Moreover, we discuss how to show that there exists a unique stationary solution for any

given mass assuming that the flow vanishes. This solution is non-constant. In the radial symmetric case, the densities are strictly convex.

Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, this talks considers no-flux boundary conditions for the bacteria density and the physically meaningful Robin boundary conditions for the signaling substance and Dirichlet boundary conditions for the flow.

In the talk, we study the existence of a global (weak) solution. Moreover, we discuss how to show that there exists a unique stationary solution for any

given mass assuming that the flow vanishes. This solution is non-constant. In the radial symmetric case, the densities are strictly convex.