The schedule of the conference


Each main lecturer gives a series of four lectures, each 70 minutes long.

The participants can present their results in the framework of
Short Communications (10 minutes each, including discussion)
and within the reprint and preprint exhibition.


The detailed (updated) program of the school


programme



Main speakers and the titles and abstract of their lecture courses


  1. Miroslav Bulíček

    Mathematical Institute of Charles University
    Faculty of Mathematics and Physics
    Charles University
    Sokolovska 83
    186 75 Prague
    Czech Republic
    email: mbul8060 [at] karlin.mff.cuni.cz
    home: http://www.karlin.mff.cuni.cz/~mbul8060/

    On implicitly constituted incompressible fluids

    Abstract:

    Recently, the implicit constitutive theory was developed to describe a very general class of materials. In the series of lectures we focus on the mathematical aspects of this theory in the context of incompressible fluids (as the prototype examples one can consider the Bingham fluid and the Herschel–Bulkley fluid). We describe the existence theory for a cascade of simplified models, i.e., starting from the generalized Stokes system and generalized steady and unsteady Navier-Stokes system, and end up with the generalized Navier-Stokes-Fourier system, where the heat transfer is taken into account. We also decribe how such results depend on the choice of boundary conditions and shortly discuss further qualitative properties of a solution. Finally, we complete the series of lectures by a short overwiev of the most important open problems.


  2. Jean-Michel Coron

    Laboratoire Jacques-Louis Lions
    Université Pierre et Marie Curie
    4, place Jussieu
    75005 Paris
    France
    email: coron [at] ann.jussieu.fr
    home: http://www.ann.jussieu.fr/~coron/

    Control, Nonlinearities and Fluid Mechanics

    Abstract:

    A control system is a dynamical system on which one can act by using suitable actuators (or controls). There are a lot of problems that appear when studying a control system. The most common ones include the controllability problem and the stabilization problem. The first one deals with the question: Is it possible to reach a desired target? The second one deals with the question: Can we stabilize by means of feedback laws an equilibrium which is otherwise unstable?

    In these lectures we present some methods to deal with these two problems and show how to apply them on flow controls modeled by means of various equations, as the Euler equations, the Navier-Stokes equations, the Saint-Venant (shallow water) equations, and the Korteweg-de Vries equations. A special emphasis is put on the role played by the nonlinearities for these two problems.


  3. Clément Mouhot

    University of Cambridge
    Wilberforce Road
    Cambridge CB3 0WA
    United Kingdom
    email: C.Mouhot [at] dpmms.cam.ac.uk
    home: http://www.dpmms.cam.ac.uk/~cm612/

    Many-particle systems and kinetic theory

    Abstract:

    Many-particle systems (gas, plasmas, galaxies, system of vortices...) are in general too hard to study through microscopic models, that attempt at following all the trajectories in phase space. Hence the need for a many-particle limit and a statistical kinetic viewpoint. In the case of long-range interactions this limit is based on the self-induced mean-field of the particles. We shall introduce this problematic and its connexion with the problem of irreversibility and the entropy, then review some classical works on this approach, and present a program of research devised by Kac in 1956 and some recent results related to this program.


  4. Armen Shirikyan

    Département de Mathématiques
    Université de Cergy-Pontoise
    Site de Saint Martin
    2, avenue Adolphe Chauvin
    95302 Cergy-Pontoise Cedex
    France
    email: Armen.Shirikyan [at] u-cergy.fr
    home: http://shirikyan.u-cergy.fr/

    Mathematical results in statistical hydrodynamics

    Abstract:

    The aim of this course is to give a self-contained introduction to the mathematical theory of statistical hydrodynamics, which has its roots in the pioneering articles of Hopf, Foias, and Vishik-Fursikov-Komech. Our focus will be on two-dimensional Navier-Stokes equations in a bounded domain, subject to the no-slip boundary condition and a random external force. The following questions will be considered in detail: the existence and uniqueness of a stationary measure, ergodicity of the flow, and the property of exponential mixing. We shall also discuss various limit theorems on the time-averages of physically relevant functionals, such as the energy, enstrophy, and correlation tensors. The theory we develop in these lectures applies equally well to other randomly forced PDE's, such as the complex Ginzburg-Landau equation and 3D viscous primitive equations of large-scale ocean dynamics.


  5. László Székelyhidi

    Lehrstuhl für Angewandte Mathematik
    Universität Leipzig
    Leipzig
    Germany
    email: szekelyhidi [at] math.uni-leipzig.de
    home: http://www.math.uni-leipzig.de/~szekelyhidi

    Convex Integration and Onsager's Conjecture

    Abstract:

    Convex Integration is a method to deal with "soft PDE", developed by Gromov, following Nash's groundbreaking work on rough isometric immersions. Although Gromov's approach is inherently geometric, with the main emphasis on global topological issues, his ideas have lead to an entirely new, constructive approach to certain PDE systems. An example where this method has been applied quite successfully is the incompressible Euler equations. In particular a famous conjecture of Onsager states a sharp relation between the minimal smoothness of weak solutions and conservation of the kinetic energy. This conjecture is also closely related to homogeneous isotropic turbulence in 3d and the Kolmogorov K41 theory. In the lectures I will discuss the basic ideas of convex integration and then concentrate on Onsager's conjecture and weak solutions of the Euler equations.