The modelling of long distance effects in transport phenomena sometimes involves fractional diffusion operators. The goal of this course is to discuss various nonlinear PDE's involving the fractional laplacian, or more general operators, and study their qualitative properties. Sometimes the results are close to those obtained for standard diffusion models. In other situations, notable differences occur.

After motivating the course by deriving a general reaction-diffusion equation, we will focus on some more specific models: free boundary problems, minimal surfaces involving fractional Sobolev norms, and a version of the KPP equation with a nonlocal diffusion operator.

The theory of phase transition is well established, following Stefans pioneering work on mod- elling the freezing of sea ice in the 1890s. Phase change (or Stefan) problems are a specific form of moving boundary problem with a rich mathematical theory and numerous practical applications (e.g. melting and thawing, solidification of steel and chemical reactions). This course will deal with the modelling, theory and applications of Stefan problems.

Mathematical techniques (such as similarity solutions, perturbation methods and integral methods) for solving heat flow and phase change models will be introduced. The methods will then be applied in the analysis of the classical one-dimensional Stefan problems and, if time permits, extended to recent developments in the theory that involve a flowing liquid layer.

Throughout the course the focus will be on practical solutions and methods will be intro- duced primarily through physical applications.

Many partial differential equations arising in physics can be seen as infinite dimensional Hamiltonian systems. Main examples are the nonlinear wave and Schrödinger equations, the beam, the membrane and the Kirkhoff equations in elasticity theory, the Euler equations of hydrodynamics as well as their approximate models like the KdV, the Benijamin-Ono, the Boussinesq, the K-P equations, etc....

In the last years important mathematical progresses have been achieved in the study of these evolutionary Partial Differential Equations (PDEs) adopting the "dynamical systems philosophy". The analysis of the main structures of an infinite dimensional phase space such as periodic orbits, embedded invariant tori, center manifolds..., as well as the use of normal forms, is an essential change of paradigm in the study of hyperbolic equations which has allowed a better understanding of the complicated flow evolution.

The aim of this course is to present first classical bifurcation results for finite dimensional Hamiltonian systems concerning periodic and quasi-periodic solutions and then we shall discuss infinite dimensional extensions to PDEs of these phenomena

In this course we shall present Mather and Mañé's variational approaches to the study of convex Lagrangian (and Hamiltonian) systems, and discuss their connection with more classical results from KAM theory, Hamilton-Jacobi equation, symplectic geometry, etc...

References:

1. John N. Mather, "Action minimizing invariant measures for positive definite Lagrangian systems", Math. Z. 207 (1991), no. 2, 169-207..
2. John N. Mather, "Variational construction of connecting orbits", Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349-1386.
3. Ricardo Mañé, "Lagrangian flows: the dynamics of globally minimizing orbits", Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 141-153.
4. Gonzalo Contreras, Jorge Delgado and Renato Iturriaga, "Lagrangian flows: the dynamics of globally minimizing orbits. II", Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 155-196.
5. Albert Fathi, "Weak KAM theorem and Lagrangian dynamics", Cambridge University Press, forthcoming (2010).
6. Alfonso Sorrentino, "Lecture notes on Mather's theory for Lagrangian systems", Preprint (2010)6.