The course provides an introductory mathematical study of some peculiar time-dependent partial differential equations that describe transport and diffusion phenomena. The lectures will enlight the links between the physical properties of the systems and the mathematical properties of the corresponding models.

Lectures.

Written assessment.

Basic knowledge of functional analysis.

Introduction to kinetic modelling of transport phenomena. Mathematical and numerical study of linear transport equations.



a) Transport equations



Origin of the transport and diffusion equations: the random walk, the heat equation and the free transport equation.

The formalism of kinetic theory. Transport and diffusive scalings. Formal relationships between transport and diffusion.

Phenomena described by transport equations. An introduction to theVlasov-Poisson and the Vlasov-Maxwell systems.

The free transport equation: the Cauchy problem. The method of characteristics, estimates.

The initial-boundary value problem for the free transport equation. Incoming, outcoming and characteristic boundary. Backwards exit time, regularity.

The maximum principle for the transport equation.

The stationary transport equation: the existence and uniqueness theorem, the maximum principle.

The Cauchy problem for the linear Boltzmann equation: existence, uniqueness, estimates and positivity of the solution.

The initial-boundary value problem for the linear Boltzmann equation: specular reflection, diffuse reflection and mixed reflection. The Darrozes-Guiraud lemma. Existence and uniqueness of the solution.

The time asymptotics for the linear Boltzmann equation.

The diffusion limit for the linear Boltzmann equation. The diffusive scaling and the Hilbert series.

Finite difference methods for transport equations: Lax-Friedrichs and upwind schemes. The diamond method.

The discrete ordinate method and the Monte Carlo methods for the linear Boltzmann equation.

Introduction to the Boltzmann equation.



b) Diffusion equations

L.C. Evans: "Partial Differential Equations", American Mathematical Society, Providence (RI), 1998.



R.T. Glassey: "The Cauchy problem in kinetic theory", Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.



C. Villani: "A review of mathematical topics in collisional kinetic theory". Handbook of mathematical fluid dynamics, Vol. I,71-305, North-Holland, Amsterdam, 2002.



C. Banfi: "Introduzione alla meccanica dei continui", CEDAM (Padova), 1990.



M.E. Gurtin: "An Introduction to Continuum Mechanics", Academic Press (NY), 1981.



Lecture notes.