MathMods :: Joint MSc

The course will cover some mathematical models currently used in the analysis of collective phenomena, such as vehicular and pedestrian traffic, flocking phenomena. Emphasis will be given to the mathematical treatment of specific problems coming from real world applications.

Macroscopic traffic models: LWR model, fundamental diagrams. The Riemann problem. Point-type constraints (the "Toll gate" problem). Second order models for traffic flow: Aw-Rascle model, shocks description, instabilities near vacuum. Basics on the theory of systems of conservation laws. Junction of roads, networks. Distribution rules along the roads, optimization of the flux. Solution to the Riemann problem at a junction. Pedestrian flow: normal and panic situation. Macroscopic models, conservation of "mass", eikonal equation. The Hughes model for pedestrian flow; in one space dimension: curve of turning points, Rankine-Hugoniot conditions. Macroscopic models for pedestrian flow that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain. Introduction to the theory of flocking. Cucker-Smale model for flocking, conservation of moment and decay of the kinetic energy. Asymptotic flocking. Singular communication rate. The kinetic limit. Introduction to synchronization: Kuramoto model, basic properties.

The aim of the course is to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type with a particular attention to biofluid dynamics.

At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for the analysis of other partial differential equations.

Derivation of the governing equations: Euler and Navier-Stokes
Eulerian and Lagrangian description of fluid motion; examples of fluid flows
Vorticity equation in 2D and 3D
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models
Existence of solutions for viscid and inviscid fluids
Fluid dynamic modeling in various fields: magnetohydrodynamics, combustion, astrophysics.
Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Basic notions of functional analysis and multi variable calculus, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.