Contour dynamics and singularities in incompressible flows & Non-local dynamics in incompressible fluids


CDSIF: The search of singularities in incompressible flows has become a major challenge in the area of non-linear partial differential equations and is relevant in applied mathematics, physics and engineering. The existence of such singularities would have important consequences for the understanding of turbulence. One way to make progress in this direction, is to study plausible scenarios for the singularities supported by experiments or numerical analysis. With the more sophisticated numerical tools now available, the subject has recently gained considerable momentum. The main goal of this project is to study analytically several incompressible fluid models. In particular solutions that involve the possible formation of singularities or quasi-singular structures.

NONFLU: he goal of this project is to pursue new methods in the mathematical analysis of non-local and non-linear partial differential equations. For this purpose we present several physical scenarios of interest in the context of incompressible fluids, from a mathematical point of view as well as for its applications: both from the standpoint of global well-posedness, existence and uniqueness of weak solutions and as candidates for blowup.

The equations we consider are the incompressible Euler equations, incompressible porous media equation and the generalized Quasi-geostrophic equation. This research will lead to a deeper understanding of the nature of the set of initial data that develops finite time singularities as well as those solutions that exist for all time for incompressible flows.