22-26 May 2017

Organisers José A. Carrillo (Imperial), José L. Rodrigo (Warwick) , Juan Luis Vázquez (UAM - Madrid)

Scientific summary

The workshop will have four primary topics in the study of non-local equations and fractional diffusion:

(1) Elliptic theory with nonlocal operators: The theory involving nonlocal operators is now well established in a number of directions, like semilinear equations and obstacle problems. There is a strong interest in connections with geometry, like fractional Yamabe problems and fractional curvature problems. Interesting connections with probabilistic approaches are being developed.

(2) Nonlinear diffusion involving nonlocal operators: Up to now this research has involved mainly fractional Laplacian operators combined with nonlinearities of the porous medium, fast diffusion and logarithmic types. Main problems are (i) regularity (boundedness, Cα regularity, Harnack inequalities), (ii) long time asymptotic behaviour and entropy methods, (iii) existence or no existence of free boundaries. Future research will involve more general non local operators and nonlinearities (including Monge Ampere models), the precise study of free boundary behaviour, and the possible occurrence of blow up in some models. The last item makes a link with very active research that is taking place in other topics, like geostrophic flows. The interest in systems is already in progress.

(3) Nonlocal Aggregations: this research has been focused on obtaining conditions for blowup or not of the densities and the equilibrium states due to the balance between attractive and repulsive effects. There are still many open problems such as typical profiles of blow-up, qualitative properties of the minimizers or steady states, conditions for their existence. These problems are related to fractional diffusion and obstacle problems when the repulsion is very strong at the origin.

(4) Balance between nonlinear diffusions and Nonlocal attraction: The classical Keller-Segel model is the archetypical example in this topic. It exhibits a critical mass dichotomy dividing the blow-up case from the global time existence of solutions. Understanding this behaviour for a general class of equations in which repulsion is modelled by nonlinear diffusion (fractional or porous medium like) and attraction modelled by nonlocal operators is still quite open. Long time asymptotics are widely open in many cases.

Confirmed Speakers


Titles and abstracts

Currently available titles and abstracts can be found here.

Programme

The current programme can be found here.