## The program of the conference

Each main lecturer gives 4 comprehensive lectures, 70 minutes each (incl. discussion)

There will be one "guest one-lecture speaker" on Monday evening

SC = Short Communications of participants (the length depending on number of them)

## Abstracts of main lectures

DIEGO CORDOBAInstituto de Matematicas y Fisica Fundamental

Consejo Superior de Investigaciones Cientificas

C/Serrano, 123, 28006 Madrid, Spain

email: dcg@imaff.cfmac.csic.es

home: http://www.mat.csic.es/investigadores.php

On the search for singularities in incompressible flowsAbstract:The main topic of these lectures will be to discuss the possible formation of singularities in different models of incompressible flows.

VOLKER JOHNFR 6.1 - Mathematik

Universitaet des Saarlandes

Postfach 15 11 50, 66041 Saarbruecken, Germany

email: john@math.uni-sb.de

home: http://www-ian.math.uni-magdeburg.de/~john/

Numerical simulation of turbulent flowsAbstract:The numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD). In general, a Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory) and the use of a turbulence model becomes necessary. The lectures will present Large Eddy Simulation (LES) and Variational Multiscale (VMS) models. Topics which will be addressed are the analysis of commutation errors in LES models, the derivation and the finite element discretization of LES and VMS models, finite element error estimates, issues on the implementation of these models as well as results of numerical simulations.

GIUSEPPE R. MINGIONEDipartimento di Matematica

Universita degli Studi di Parma

Parco Area delle Scieze 53/a (Campus)

43100 PARMA, Italy

email: giuseppe.mingione@unipr.it

home: http://www2.unipr.it/~mingiu36/

Regularity topics for elliptic problems: standard and non- standardAbstract:I will give a general review of regularity theory for elliptic, possibly variational, problems, starting from the standard partial regularity in the vectorial case and everywhere regularity in the scalar one. I will then pass to examine so called problems with non-standard growth conditions, especially those coming up in non-newtonian fluid-dynamics. Finally I will concentrate on the problem of estimate the size of the singular sets of such problems, talking about positive results and counterexamples.

References:

- Acerbi, E. and Mingione, G.: Regularity results for stationary electro-rheological fluids.
Arch. Ration. Mech. Anal.164(2002), 213--259.- Fonseca I., Maly J., and Mingione G.: Scalar minimizers with fractal singular sets -
Arch. Ration. Mech. Anal.172(2004) 295-312- Giusti:
Direct Methods in the Calculus of Variations. World Scietific 2003.- Marcellini P.: Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions.
J. Differential Equations90(1991), 1-30.- Mingione G.: Bounds for the singular set of solutions to non linear elliptic systems.
Calc. Var.18(2003) 373-400.- Sverak, V. and Yan, X.: Non-Lipschitz minimizers of smooth uniformly convex functionals.
Proc. Natl. Acad. Sci. USA99(2002), 15269-15276.

PING ZHANGAcademy of Mathematics and System Sciences

Chinese Academy of Sciences

Beijing 100080, P.R. China

email: zp@math03.math.ac.cn

The methods of Young measures and mollifiersAbstract:We study an elegant nonlinear wave equation which arises beautifully in a simplified liquid crystal model through the variational principle. The equation appears in several other disciplines as well. A distinctive feature of the equation is that the wave speed depends on the wave amplitude, in addition to its nonconservative appearance that is a natural consequence of the variational principle. We present recent results on the existence and uniqueness of weak solutions for the initial value problems for the equation. Our emphasis is to show the reader the various interesting phenomena that the equation possesses and the modern methods of Young measures, mollifiers, and related techniques. We first present simple asymptotic equations for weakly nonlinear and unidirectional waves of the equation. The asymptotic equations to the nonlinear variational wave equation is what the Burgers equation to hyperbolic systems of conservation laws. We show the existence of two types of weak solutions to the initial value problem for the (first-order) asymptotic equation for data in the natural space of square integrable functions, and we establish the uniqueness of weak solutions for both the dissipative and conservative types.

Enlarged abstract to be found here.

... and a guest one-lecture speaker

VLADIMIR SVERAKSchool of Mathematics

127 Vincent Hall, 206 Church St. S.E.

Minneapolis, MN 55455, USA

email: sverak@math.umn.edu

home: http://www2.itdean.umn.edu/faculty/detail.jsp?facultyID=349

Some recent results in mathematical theory of incompressible flows