Regularity and blow up in models of fluid mechanics

Regularity and blow up in models of fluid mechanics

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Alexander Kiselev, Duke University
Fine Hall 314

I will discuss a family of modified SQG equations that varies between 2D Euler and SQG with patch-like initial data defined on half-plane. The family is modulated by a parameter that sets the degree of the kernel in the Biot-Savart law. The main result I would like to describe is the phase transition in the behavior of solutions that happens right beyond the 2D Euler case. Namely, for the 2D Euler equation the patch solution stays globally regular, while for a range of nearby models there exist regular initial data that lead to finite time blow up. The geometry of the blow up example involves hyperbolic fixed point of the flow at the boundary. If time permits, I will discuss some other recent works designed to better understand vorticity growth in a similar setting for solutions of the 3D Euler equation.