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Nonlocal maximum principles for active scalars

4:00 pm Thursday, September 15th, 2011
Alexander Kisilev (University of Wisconsin - Madison)

Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler equation describing two-dimensional ideal flow, Burgers equation modeling flow of gas, and surface quasi-geostrophic (SQG) equation describing temperature evolution on the surface of the Earth. Active scalar equations are difficult to analyze since they are nonlinear and usually nonlocal. There remain many open questions about regularity and properties of solutions to active scalar equations. In particular, there has been much recent research focusing on the SQG equation, perhaps motivated by the fact that it is the simplest-looking PDE of fluid mechanics for which global regularity of solutions is open. I will review recent developments in studying properties of solutions to the active scalar equations. In particular, I will describe a novel Fourier analysis-inspired nonlocal maximum principle approach that allows to prove global regularity of the critical SQG equation.

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