In the study of an isolated vortex line for 3D Euler one is trying to make sense of the evolution of a curve, where the vorticity (a distribution in this case) is supported, and tangential to the curve.  This idealised vorticity generates a velocity field that is too singular (like the inverse of the distance to the curve and therefore not in L2) and making rigorous sense of the evolution of the curve remains a fundamental problem.    In the talk I will present examples of simple globally divergence-free  velocity fields for which an initial delta function in one point (in 2D,  with analogous results in 3D) becomes a delta supported on a set of Hausdorff dimension 2.  In this examples the velocity does not correspond to an active scalar equation.    I will also present a construction of an active scalar equation in 2D, with a milder singularity than that present in Euler for which there exists an initial data given by a point delta becomes a one dimensional set. These results are joint with C. Fefferman and B. Pooley.    These are examples in which we have non-uniqueness for the evolution of a singular "vorticity". For the Surface Quasi-Geostrophic equation, an equation with great similarities with 3D Euler, the evolution of a sharp front is the analogous scenario to a vortex line for 3D Euler. I will describe a geometric construction using "almost-sharp" fronts than ensure the evolution according to the equation derived heuristically.  This part is joint work with C. Fefferman.
June 2019
2pm - 3pm
Room 3472, Academic Building (near Lifts 25 - 26)
Department of Mathematics
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