Asymptotics and singularity formation in Nonlinear PDEs related to fluid dynamic, geophysical flows, quantum physics and optics.

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The research component of the proposal addresses fundamental mathematical problems invoking nonlinear partial differential equations PDEs of incompressible fluids, dispersive (conservative, wave-like) and kinetic models. Examples include the nonlinear Schroedinger and nonlinear wave equations, Euler, Navier-Stokes, primitive equations (also known as hydrostatic Euler), Prandtl equation, and the relativistic Vlasov-Maxwell system. They all arise throughout science and engineering as fundamental models of dynamical processes. Areas of application include quantum physics, general relativity, optics, plasma physics, fluid turbulence, geophysics, weather prediction and boundary layers. While this project is motivated by many classical questions which have been studied for over a century, the overall objective of this proposal is to obtain analytical information about a number of new problems that we identify. These problems are intrinsically interesting from both the mathematical and physical points of view relating issues of well-posedness, instabilities, long-time behaviour, and singularity formation. Moreover, the problems we are proposing will also shed light on some of the major classical problems in the field related to global well-posedness vs. blow-up for strong solutions and well-posedness vs. non-uniqueness and instability for weak solutions.