"The ``FLUX'' project is devoted to the mathematical analysis of an array of problems originating from different domains of real world applications. They include: fluid mechanics, aerodynamics, geophysics, phase transitions and image processing, meteorology. The language for rigorous formulation of models of these phenomena is the theory of partial differential equations. In addition, we face free boundary problems.Our goal is the complete analysis of the model equations. This begins with establishing global in time existence of properly defined solutions. This is a well-known fact that the notion of solution has to be adjusted to the specific problem we study. Once the first task is complete, the regularity and asymptotic behavior will be tackled.Although these equations describe different phenomena, they share common features and hence similar techniques can be applied to analyze the solutions. The cooperation within the project should result in the development of new mathematical tools to deal with problems appearing in the mentioned above applications. The wide exchange on the level of PhD students and young researchers, that we are planning, will lay down foundations of a long lasting cooperation. Many researchers from the external institutions are world-renowned experts in their fields. Their participation in the project should clearly benefit the European partners.This project is a revised and greatly improved version of the ``TOUCH'' proposal, which we submit aware of the fact, that the previous proposal constitutes a lower bound for evaluation of this one. We have thoroughly studied all pointed out weaknesses and corrected them by: reducing the number of both involved nodes and of packages and, simultaneously, creating comparatively small network of nodes with possibly numerous interactions; specifying previously lacking managerial and financial aspects, as well as giving more precise justification for detailed shape of the proposal."