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 (Download here)


programme



Main speakers and the titles and abstract of their lecture courses


  1. Dorin Bucur

    Laboratoire de Mathématiques CNRS UMR 5127
    Université de Savoie
    Campus Scientifique
    73 376 Le-Bourget-Du-Lac
    France
    email: dorin.bucur [at] univ-savoie.fr
    home: http://www.lama.univ-savoie.fr/~bucur/

    Geometric domain perturbations for PDEs

    Abstract:

    The main purpose of these lectures is the study of the qualitative behaviour of the solutions of certain partial differential equations in domains with varying boundaries. Several classical examples are recalled: the strange term "coming from somewhere else" of Cioranescu-Murat, Babuska's paradox, the Courant-Hilbert eigenvalue problem and the rugosity effect in fluid dynamics. After a short introduction to variational techniques (Gamma-convergence, capacity estimates...), some classical and recent results on shape stability and relaxation are presented, with a focus on the Stokes and Navier-Stokes equations.


  2. Lars Diening

    Mathematisches Institut der Universität München
    Theresienstr. 39
    D-80333 München
    Germany
    email: Lars.Diening [at] mathematik.uni-muenchen.de
    home: http://www.mathematik.uni-muenchen.de/~diening/

    On the analysis and numerical analysis of power law fluids

    Abstract:

    We present an overview over the techniques that are needed for the study of power law fluids. Different from Newtonian fluids the viscosity depends non-linearly on the shear rate. We address questions of existence, regularity and numerical analysis. We will discuss in particular, tools like the Lipschitz truncation, Korn's inequality and the Scott-Zhang interpolation operator on Orlicz spaces. If time permits, we will also discuss the case of electrorheological fluids. These fluids have the special properties that their viscosity changes significantly, if an electric field is applied.


  3. Ansgar Jüngel

    Institute for Analysis and Scientific Computing
    Vienna University of Technology
    Wiedner Hauptstr. 8-10
    1040 Wien
    Austria
    email: juengel [at] asc.tuwien.ac.at
    home: http://www.asc.tuwien.ac.at/~juengel/

    Entropy dissipation methods for nonlinear diffusion equations

    Abstract:

    Entropy-dissipation methods have been developed recently to investigate the qualitative behavior of solutions to nonlinear partial differential equations and to derive explicit or optimal constants in convex Sobolev inequalities. It turned out that these methods also help for the global existence analysis and the design of stable numerical schemes. In this lecture series, we will highlight some of the aspects of entropy methods, in particular for linear and nonlinear Fokker-Planck equations, cross-diffusion systems from biology, and Maxwell-Stefan systems for multicomponent fluid mixtures. New analytical tools are the boundedness-by-entropy principle and systematic integration by parts.


  4. Chun Liu

    Department of Mathematics
    The Penn State University
    University Park, PA 16802
    U.S.A.
    email: liuc [at] psu.edu
    home: http://www.personal.psu.edu/cxl41/

    Introduction of Mathematical Theories of Complex Fluids

    Abstract:

    Complex fluids is ubiquitous in our daily life, from the food we eat, the things we use to the exact materials we are made of. It is also important in many industrial, physical and biological applications. Studying these materials requires a wide range of tools and techniques for different disciplinary, even within mathematics. Here I will present a narrow glimpse of the area. Some of the topics that I wish to cover in the course:
    0. Basic mechanics.
    1. General energetic variational framework for complex fluids:
       a) least action principle and maximum dissipation principle.
       b) Navier-Stokes equations and elasticity.
       c) viscoelastic materials: nonlinear elasticity, incompressible elasticity, viscoelasticity.
       d) generalized diffusion, nonlocal diffusion.
    2. Free interface motion in the mixture of different fluids:
       a) conventional description: sharp interface description, water wave, vortex sheet, surface tensions.
       b) diffusive interface description: microscopic background (self-consistent field theory), Flory-Higgins theory, sharp interface limit, dynamics.
       c) slippery boundary conditions and systems on (or near) surfaces.
       d) Helfrich elastic bending energy and application to vesicle membranes.
    3. Multiscale modeling and analysis:
       a) basis of stochastic differential equations: Fokker-Planck equations, diffusion, Smoluchowski coagulation equations, variational formulations, kinetic theory.
       b) micro-macro models for polymeric materials.
       c) moment closure methods, Mori-Zwanzig formulation and other coarse grain methods.
    4. Ionic fluids and ion channels:
       a) electroeheological (ER) fluids: Poisson-Boltzman fluids, like charge attaction (LCA) and charge inversion, steric effects of ion particles, density function theory of Rosenfeld, equation of states.
       b) basic physiology of ion channels and protein structures.
       c) ionic fluids in ion channels.
       d) ionic osmosis in biological systems.


  5. Arghir Dani Zarnescu

    Pevensey III
    University of Sussex, Falmer, BN1 9QH, UK
    and
    "Simion Stoilow" Institute of Mathematics of the Romanian Academy
    email: A.Zarnescu [at] sussex.ac.uk
    home: http://www.sussex.ac.uk/Users/az73/

    Liquid crystal hydrodynamics: between Leslie-Ericksen and De Gennes

    Abstract:

    Some of the most useful and striking achievements in the realm of modern technology are based on the use of materials with complex microstructure, that exhibit remarkable behaviours. A paradigmatic example and entry point into this world of complex fluids are liquid crystals, material that is ubiquitous nowadays around us, especially in various displays. Despite their enormous technological impact, the materials are poorly understood at a fundamental level and there exist several major competing theories that attempt to describe them. The equations used to model the material couple a forced incompressible Navier-Stokes system describing the underlying fluid motion, with a parabolic-type system describing the motion of the "directors", that is the rigid rod-like liquid crystal molecules. The equations describing the motion of the directors come with specific geometric and topological constraints, that add up to the difficulties inherent to the Navier-Stokes system. The talks will survey two of most relevant models and the interplay between them, focusing on the specific non-Newtonian aspects of these fascinating and mysterious materials.