**October 24, 2013 at 11:30**

**Speaker(s):**
**Carlos PARÉS, Juan Luis VÁZQUEZ**

**Center(s): **Universidad de Málaga, Málaga, Spain, Universidad Autónoma de Madrid, Madrid, Spain

Basque Colloquium in Mathematics and its Applications

8th edition on the occasion of the 5th anniversary of BCAM

October 24, 2013

BCAM-Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain

11:30 Carlos PARÉS, Universidad de Málaga, Málaga, Spain

NUMERICAL METHODS FOR SHALLOW FLOWS: DIFFICULTIES AND APPLICATIONS

Many geophysical flows can be modelled by means of hyperbolic PDE systems with non-conservative products and/or source term: in particular this is the case for many models based on the shallow water hypothesis. In the talk some shallow water models will be presented which are useful to model sedimentary flows, turbidity currents, floods, tsunamis, avalanches, marine flows, etc. Next, a general methodology for developing high order well-balanced numerical schemes for this kind of systems will be presented and the main difficulties will be discussed. Finally, some applications to real flows will be shown.

12:30 Juan Luis VÁZQUEZ, Universidad Autónoma de Madrid, Madrid, Spain

RECENT PROGRESS IN THE THEORY OF FRACTIONAL HEAT AND POROUS MEDIUM EQUATIONS

We will report on recent research in the area of elliptic and parabolic equations, aimed at understanding the effect of replacing the Laplace operator, and its usual variants, by a fractional Laplacian operator or other similar nonlocal operators, which represent long distance interactions. Linear and nonlinear models are involved.

The lecture will describe some of the progress made by the author and collaborators on the topic of nonlinear fractional heat equations, in particular when the nonlinearities are of porous medium and fast diffusion type. The results cover existence and uniqueness of solutions, regularity and continuous dependence, positivity and Harnack estimates and symmetrization. Special attention is given to the construction of fractional Barenblatt solutions.

Collaborators: C. Caffarelli, F. Soria, S. Serfaty, M. Bonforte, A. de Pablo, F. Quirós, A. Rodríguez, D. Stan, F. del Teso, B. Volzone.

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