**PROGRAM TOPIC**

Mathematical modeling of turbulent flow is a theme with great scientific and practical interest, with applications in industrial and biological flows, and with particular interest for the large scale flows that occur in geophysics and astrophysics, in metereology, climatology, oceanography, magma dynamics, planetary and stellar dynamics. The turbulence models used in applications are either based on ad hoc closure of the small scales, called large eddy simulations in computational modeling or on phenomenological models of statistical nature based on Kolmogorovâ€™s theory of homogeneous turbulence. There is no rational theory of turbulence, which would derive the observed behavior of turbulent flows from fundamental physical principles, in particular, those expressed in the Navier-Stokes system, in the small viscosity limit. The search for a rational theory of turbulence is primarily a mathematical problem, but one with great physical and practical relevance. Progress on the development of a rigorous treatment for turbulence involves confronting several open problems in the field of partial differential equations: the mathematical treatment of boundary layers, the statistical behavior of solutions to the Navier-Stokes equations, uniqueness of weak solutions for the Euler equations and the problem of singularity for the Navier-Stokes equations (Clay Millenium Problem), among others. Recent developments in this field have been based on the use of modern mathematical analysis techniques such as weak convergence methods, harmonic analysis, convex analysis, optimal transportation theory, stochastic analysis and geometric measure theory.

From the broadest viewpoint, the purpose of this Thematic Program on Incompressible Fluid Dynamics is to contribute to progress along this line of investigation by introducing young researchers to the contemporary themes and techniques, by promoting collaboration and by encouraging the development and consolidation of this area of research in Brazil.