Results motivated by the the study of the evolution of isolated vortex lines for 3D Euler
报告题目(Title)：Results motivated by the the study of the evolution of isolated vortex lines for 3D Euler
报告人(Speaker)： Professor José Luis Rodrigo ( University of Warwick, U.K.)
地点(Place)：Zoom Id: 62055020210; Password: 801235
时间(Time)：2021. 1. 20，17:00-18:00
邀请人(Inviter)：Calvin Khor; 许孝精
报告摘要
In the study of an isolated vortex line for 3D Euler one is trying tomake sense of the evolution of a curve, where the vorticity (adistribution 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". In the second part of the tallk I will describe work on the Surface Quasi-Geostrophic equation (and some related models), 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 for SQG, and with C. Khor for the more singular models.
