Thematic Program: Nonlinear Flows
14.04.2016 - 30.06.2016
Optimal transport methods for hamiltonian PDEs
01.06.2016 - 2.06.2016 (10-12)
Functional inequalities and evolution equations
Abstract: Functional inequalities are an essential tool for understanding the behavior of solutions of PDE's and other sorts of evolution equations. At the same time, monotonicity along various evolution processes can often be used to prove functional inequalities. This course will focus on some recent examples in which this interplay between evolution processes and functional inequalities has been fruitful, and it will also introduce some open problems in the field. There will a special emphasis on stability results for sharp inequalities.
Compressible fluid flows, new results and perspectives
Abstract: We discuss several new results concerning well/ill posedness questions in the context of compressible fluid flows. We identify a large class of problems describing the behavior of inviscid flows for which there exist infinitely many solutions or even infinitely many
dissipative solutions satisfying the energy inequality. We relate these results to the theory of viscous fluids and identify certain problems in the inviscid theory as the vanishing viscosity limit. Finally, we discuss a proper choice of suitable admissibility criteria
that would imply well-posedness of the problem.
21.06.2016 - 23.06.2016 (10-12)
Sobolev and BV functions in metric measure spaces
Abstract: In the lectures I will describe recent development of calculus in metric measure spaces, revisiting old and new points
of view in the theory of Sobolev and BV functions. I will show how tools from Calculus of Variations, Optimal Transport
and PDE allow to estabilish powerful equivalence results between "Eulerian" and "Lagrangian" notions, even in the
broad context of metric measure spaces. I will mostly rely on papers written in collaboration with Simone Di Marino,
Nicola Gigli and Giuseppe Savaré.
30.06.2016 (9.30-11.00 and 11.30-13.00)
J. A. Carrillo
Minimizing interaction energies: Nonlocal Potentials and Nonlinear Diffusions
Variational motion in heterogeneous media
Abstract: I will consider the problem of defining an effective variational motion for oscillating energies.
The minimizing-movement scheme or implicit Euler scheme is a commonly used method
to define gradient-flow type dynamics in a variational framework and can be adapted to
this problem, obtaining effective motions that in general depend on the interaction between
time and space scales. I will review general theorems that link those effective motions to
that of the Gamma-limit of the oscillating energies. After a simple example in one dimension,
I will concentrate on geometric motion of interfaces as gradient-flow type dynamics for
(continuous and discrete) perimeter energies, highlighting different ways in which local
minima of heterogeneous perimeter energies affect the corresponding effective motion.
06.07.2016 - 07.07.2016 (10-12)
Fluid models for collective dynamics
Abstract: Collective dynamics appears ubiquitely in nature, from bird flocks to the
swimming of sperm. Collective dynamics creates emergent patterns at a scale
orders of magnitude larger than those of the individual agents. The
deciphering of the mechanisms underpinning the emergence of these large
scale structures requires the ability to coarse-grain the models from the
particle scale to the population scale. In this series of lectures we will
explore some of the mathematical difficulties that arise when trying to
coarse-grain particle models of collective dynamics and that contribute to
the forging of new mathematical tools in kinetic theory.