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 (dissertation).

Basic knowledge of calculus, linear algebra, mechanics, and functional analysis.

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 the Vlasov-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.


b) Diffusion equations

Introduction to Continuum Mechanics. Lagrangian (spatial) ed Eulerian (material) descriptions. Deformation and motion. Transport and balance equations. Thermodynamical quantities and constitutive equations.
Classical materials: perfect fluids, incompressible fluids,
barotropic fluids; Euler equations; Newtoniani fluids and Navier-Stokes equations. Uniqueness and stability for solutions of a viscous flow problem.

Heat equation as a paradigm of diffusion.
Boundary conditions: Dirichlet, Neumann, Robin, mixed.
Uniqueness of the solution by energy methods. Weak and strong maximum (minimum) principle; corollaries. Parabolico riscalaing. Fundamental solution. Usage of the fundamental solution in a homogeneous or an inhomogeneous Cauchy problem.

Standard porous medium equation
(non-linear heat equation) (PME).
Finite speed of propagation:
steady solutions, separable-variable solutions, wave solutions, Barenblatt fundamental solutions.
Incompressible fluid in a porous medium.
Stefan flow and Stefan-Maxwell diffusion; applications.

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.

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

S. Salsa: Partial Differential Equations in Action: From Modelling to Theory", Springer (Milan), 2009.

J. L. Vazquez: "The porous medium equation : mathematical theory" (XXII - Oxford mathematical monographs) Clarendon Press (Oxford), 2007.

Lecture notes.