The Department is delighted to welcome Professor Juan Luis Vázquez (Universidad Autónoma de Madrid, Spain) as Nelder Fellow in 2019.

Course abstract

Juan Luis will deliver a mini-course on "Theories of local and nonlocal diffusion: the state of the art". In this course he will describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic partial differential equations gives a foundation to the probabilistic study of diffusion. Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research, we use the porous medium  and fast diffusion equations as  case examples.

A large part of recent research has been devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear). Recent progress in which the author is involved will be reported.

Lectures

• Session 1: "Nonlinear Degenerate Equations of Porous Medium type"
  Thursday 21st February, 4pm-6pm
  Contents: it covers weak solutions, regularity, free boundaries, p-Laplacians, the mesa limit, Hele-Shaw flows, summary of asymptotics. View the slides for this session: Juan Luis Vazquez: Session 1 slides (PDF)

• Session 2: "Elliptic and Parabolic Equations driven by nonlocal operators"
  Thursday 7th March, 4pm-6pm
  Contents: Motivation. Main models. Basic theory. Existence/nonexistence of free boundaries. Asymptotics. Related problems. Open problems.

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Speaker biography

Juan Luis Vazquez's main interest are nonlinear partial differential equations of elliptic and parabolic type. His studies bear on the theoretical side, taking also into account the physical motivation and applications. Main research topics: Diffusion equations. Filtration equations in porous media.   Mathematical study of processes with free boundaries. Special solutions, selfsimilarity and asymptotic methods. Blow-up problems and singularities.  Nonlocal diffusion and fractional Laplacian operators.  PDEs in Riemannian Geometry. Entropy methods and functional inequalities. Due attention to numerical aspects. Juan Luis likes writing books and giving expository talks.

View Professor Vazquez's webpage