Mathematics Colloquia and Seminars

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The two-dimensional incompressible Euler and transport equations play a fundamental role in physics, with applications ranging from modeling cyclones and hurricanes to studying heat, mass and electric charge transfer. The uniqueness of the Cauchy problem is a basic requirement for any deterministic model to be considered physically acceptable. In the first part of this talk, I will explore central questions surrounding the uniqueness of solutions of these equations. I will present novel techniques and flow mechanisms that fully resolve these issues for the transport equation and lead to important progress in the context of the Euler equation. In the second part, I shift my focus to turbulent flows, which are characterized by enhanced heat, mass and momentum transport, intense mixing and high dissipation rates. An active area of study from a PDE perspective is of design flows that capture these essential features of turbulence. I will introduce a novel mechanism based on branching flows to achieve efficient heat and momentum transport. These flow designs are not merely mathematical constructs but can, in principle, be realized through the design of a mechanical apparatus.

Reception at 3:45pm