Aside from the extension of the already existing collaboration between the applicants in frames of diploma and PhD. thesis supervision, one of the main expected benefits of the center is the establishment of a close collaboration between various teams working in mathematical analysis, that is, purely theoretic problems, and applied mathematicians represented mostly by experts on development of numerical algorithms and subsequent calculations which will result in joint publications

As an example of the expected outcome let us describe the following situation. The suggested scientific team is specialized in the field of fluid and gas flux, and its members belong to the world top for example in the field of mathematical analysis and numerical simulation of the so-called complete Navier-Stokes-Fourier system of differential equations describing temperature-dependent flux of compressible one-component gases. It is needless to say that in spite of obtained results the area still contains a countless amount of fundamental open problems. Here is the list of the main ideas of the project:

• Based on the results obtained by the members of the team already, it is reasonable to expect that results which will be obtained by computation will contribute to the development of theoretical conjectures about the behaviour of the system, while the purely theoretic results will motivate numerical simulations.

• Navier-Stokes-Fourier system is a basic model part of the theory of multi-component substances, in models describing interaction of the solid state and the liquid state of matter, in fluid and gas flux affected by electromagnetic fields, etc. Such models are useful in physical, biological, technical and industrial applications. The main asset of the project will be the development of mathematical theories in these fields.

• Hyperbolic conservation laws can be in many cases obtained as a certain limit of a fax more complicated model, for example the above-mentioned Navier-Stokes-Fourier system. From this point of view the hyperbolic conservation laws can be understood as ideal (from practical point of view unreachable) object which contains in itself all the difficulties of the original far more complicated system. On the other hand, a natural question is the investigation of those properties which are conserved under the passage from the dissipative system to the limiting one. The gain of knowledge as complete as possible about the limiting mechanisms and a progress in understanding of them would then constitute a significant asset of the project.

• It is also possible to understand the Navier-Stokes-Fourier system as a limit of kinetic equations and further discrete (atomistic) systems. Here, too, one can expect that some features of these mathematically quite different structures can be preserved even for their limiting states. One of the benefits of the project will consist of the understanding of such mechanisms.

The above-mentioned examples illustrate necessity and at the same time also the substantial efficiency of collaboration of experts in the field of kinetic equations, hyperbolic systems and basic equations of the continuum mechanics such as the Navier-Stokes system. One of the expected benefits of the suggested project is the creation of appropriate conditions for such collaboration.

The same goes for other fields considered in frames of the center.