Results motivated by the the study of the evolution of isolated vortex lines for 3D Euler
Analysis and Geometry Seminar
6th June 2019, 3:15 pm – 4:15 pm
Howard House, 4th Floor Seminar Room
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 $L^2$) 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 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.
(This will be a colloquium style talk.)