## Main lectures of the

## Sixth School

## MATHEMATICAL THEORY IN FLUID MECHANICS

## PASEKY (Czech Republic)

September 19 - 26, 1999

It is supposed that every main lecturer will give comprehensive lectures of the total length of about 5 hours.

See the layout of the conference for the conference schedule.

The participants of the conference can present their results in the form of very short communications (10-15 minutes) and/or they can exhibit their papers, preprints and books (the number of these is not limited, and they may be related to other scientific areas; the idea is that there will be an exhibition place located within the hotel, on which the papers and preprints will be exhibited during the whole school).

Marco CANNONE

Marco.Cannone@gauss.math.jussieu.fr

(University of Denis Diderot, Paris VII, France):

Viscous flows in Besov spacesThe aim of these lectures is to present an introductory survey to the existence and uniqueness results of strong solutions to the Navier-Stokes equations. These equations govern the time evolution of an incompressible viscous fluid which is assumed here to fill the whole three dimensional space.A topic which will be extensively treated concerns the space of initial conditions for which there exists a unique local (in time) solution or a global small solution. For sufficiently small Reynolds numbers, the conditional stability in the sense of Lyapunov will also be studied.

The definition of a somewhat esoteric Besov space (in terms of the Littlewood-Paley dyadic decomposition) will play a crucial role throughout the lectures.

Eduard FEIREISL

feireisl@math.cas.cz

(Czech Academy of Sciences, Prague, Czech Republic):

On the long-time behaviour of solutions to the Navier-Stokes equations of compressible flowWe give a survey on recent development of the mathematical theory of viscous compressible fluids. The questions addressed will be: The global existence of weak solutions in several space dimensions, the existence of time-periodic solutions under the condition that the fluid is driven by a periodic external force and, last but not least, the long-time stabilization towards a stationary state for conservative external forces.We shall also discuss the role of integrability of the density for the isentropic model which is intimately related to the problem of existence.

As for the mathematical tools employed, we shall only need the classical Sobolev spaces as well as some elements of the modern functional analysis, in particular, the Lebesgue spaces endowed with the weak topology.

Michael GRIEBEL

griebel@iam.uni-bonn.de

(University of Bonn, Germany):

Multilevel techniques for the fast numerical solution of the Navier-Stokes equationsMultiscale techniques can help to obtain a numerical approximation to the solution of PDEs in various ways. First, they directly allow to use a fast solver of the linear equation systems which arise after linearization and time- and space-discretization. Second, they also allow to use an adaptive nonlinear approximation of the solution itself. Here, the discretization is locally refined to resolve steep gradients or singularities of the solution. Moreover, information obtained from a multilevel resolution of the actually computed approximation can guide further refinement or coarsening of the discretization. Third, multiscale techniques allow to use certain norm equivalences which, in turn, give a priori and a posteriori error estimates.These benefits of a multiscale approach which are well known for scalar elliptic PDEs can also be exploited for nonlinear PDE systems to some extent. Here we consider the Navier-Stokes equations. We discuss the advantages of certain multiscale bases (multigrid, wavelets) for the adaptive discretization, the multigrid solution and the multiscale error estimation for incompressible viscous flow problems with moderate Reynolds number. We also address the benefits of a multiscale technique for turbulence modeling. Furthermore, we present the results of numerical experiments obtained with a multiscale wavelet discretization and solution method for various laminar and turbulent flow problems.

Nader MASMOUDI

masmoudi@dmi.ens.fr

(University of Paris-Dauphine, France):

Some asymptotic problems in fluid mechanicsWe will discuss various asymptotic problems arising in fluid mechanics and specially the compressible-incompressible limit which corresponds to a Mach number going to zero. More precisely we will show various results establishing the convergence, as the density becomes constant and the Mach number goes to zero, towards solutions of incompressible models (Navier-Stokes or Euler).Other problems may be discussed as the Navier-Stokes-Euler limit corresponding to a viscosity going to zero as well as the the study of rotating fluids at high frequency.

Serguei A. NAZAROV

serna@snark.ipme.ru

(St. Petersburg University, Russia):

Weighted spaces with detached asymptotics in application to Navier-Stokes problemsLinear elliptic problems in domains with cylindrical and conical outlets to infinity gain the Fredholm property if posed in function spaces with weighted norms of exponential and power types. With the help of such norms, asymptotics of solutions in the outlet can be described as well. A correct formulation of limiting conditions at infinity (like flux and pressure conditions) needs a certain generalization of the spaces mentioned above, and it gives rise to weighted spaces with detached asymptotics. These spaces consist of functions with a prescribed asymptotic form while their norms reflect properties of both, the detached terms and the remainder. However, the most applicable ability of those spaces becomes the direct maintenance of nonlinear problems in the case that linear and nonlinear differential expressions are of the same asymptotic power at infinity. The 3D exterior problem, the 2D aperture problem, the problem of quiet flows are to be presented as applications of the developed mathematical tools for Navier-Stokes equations.

Athanasios E. TZAVARAS

tzavaras@math.wisc.edu

http://www.math.wisc.edu/~tzavaras

(University of Wisconsin, Madison, USA):

Topics in the theory of hyperbolic systems of conservation lawsThese lectures will discuss topics in the theory of hyperbolic systems of conservation laws focusing on the equations of gas dynamics. First, we outline the structure of the equations as dictated by the second law of thermodynamics (in the form of Clausius-Duhem inequality) and by the issue of fluid-mechanic limits of the Boltzman equation. Then we will survey certain topics in the theory of weak solutions for conservation laws: These are the kinetic formulation of conservation laws, and the approximation of conservation laws via relaxation or via discrete-velocity models.