### Results motivated by the the study of the evolution of isolated vortex lines for 3D Euler.

Analysis and Geometry Seminar

7th February 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.)

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