Partial Differential Equations applied to fluid mechanics and related problems

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This project is aimed at investigating regularity and possible blow-up of solutions to partial differential equations of fluid mechanics and related mathematical models. Understanding of properties of these models is a fundamental challenge both for mathematics and science. The regularity and stability of the solutions are strongly related to complex multiscale physical phenomena such as turbulence and phase transitions. A principal goal of this project is developing new techniques to tackle the problem of global regularity for super-critical systems including reaction-diffusion systems or the 3D Navier-Stokes system.This research program will make some significant progress in the fundamental understanding of the mathematics involved in fluid mechanics and chemistry. It will help also to train new graduate students in this important field. The discovery of mathematical structure in physical models will lead to a deeper understanding of the physical phenomena themselves. It will also provide some information about the limits of validity of the theoretical physical models that are used to describe physical phenomena.