Fifth Winter SchoolMATHEMATICAL THEORY IN FLUID MECHANICS
PASEKY (Czech Republic)

It is supposed that every lecturer will give five onehourlong comprehensive lectures and lead two seminars focused on the interaction between the lecturer and the audience.The participants can present their results in the framework of short communications and/or they can exhibit their papers and preprints (the number of papers/preprints is not limited, they may be related to other scientific area than fluid mechanics; 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).
The Boltzmann or more generally the kinetic equations play an important role in the understanding of the coherence of the laws of physic from a microscopic description (at the level of particles) to a macroscopic description using the NavierStokes or Euler equations. These relations are also useful for many applications involving rarefied gases of particles, for instance the problem of reentry of a space vehicle in the atmosphere, or the behaviour of the electric current in a microscopic semiconductor as used in modern computers.Therefore the lectures will emphasize the relations which can be derived at the certain level of mathematical knowledge, first between particles and kinetic equations, second between the Boltzmann equations and the NavierStokes equations, and finally the similarity and the difference between the fluid limit of kinetic equations and the modelling of turbulence for NavierStokes equation will be briefly mentioned.
In some cases complete mathematical proofs (when they are available) will be given, in other cases only formal derivations will be explained. The role of the entropy and the effect of the nonlinearity will be stressed.
The course will discuss problems lying on the interface between continuum physics and the theory of hyperbolic systems of conservation laws whose solution may reveal something about the physics and, in return, the underlying physical structure may direct and drive the analysis. Issues to be discussed will include the difficulties posed by systems in several space dimensions, the dissipative effects of fading memory, the role of entropy, and methods of redistributing damping.
We will investigate the steady compressible isothermal or isentropic NavierStokes equations by using the method of decomposition which splits the flow field into its compressible and incompressible parts. First part of the course will be devoted to the questions related to the compactness of the equations and to the existence of weak solutions for arbitrary large data. In the second part, we will investigate compressible flows near the equilibrium past a threedimensional body.
The following topics will be considered and illustrated on concrete free boundary problems: Boundary value problems for the NavierStokes equations in domains with angular points and wedges on the boundary. Estimates of solutions in weighted spaces. Behaviour of solutions at infinity. The problem of dynamic contact angle; slip boundary conditions. Motion of two liquids with a free interface.