This project is focused on solving problems in fluid mechanics which involve the possible formation of singularities. In particular, the principal investigator will study free boundary problems given by incompressible fluids with different properties modeled by the 2D surface quasi-geostrophic equation, Darcy's law and Euler's equation. The well-posedness of these models will be considered. In particular, the principal investigator will look for the existence of singularities or lack thereof giving global existence results. The associated physical scenarios will be presented that have interesting applications. Research in this direction requires the combination of analytic techniques, asymptotics, numerics and modeling.
This project could lead to important new methods for understanding and simulating fluid interface problems in incompressible flows. These are high-profile problems, so that the results obtained will be certain to attract attention throughout the mathematical and scientific community. Successful analysis of formation of singularities in incompressible flows would solve a major problem of mathematics and would establish a new method for addressing blow-up formation in non-linear PDE. A fluid dynamic understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics. This in turn could lead to important new methods for understanding and simulating turbulent flows, with potential for great impact throughout science and technology.
