4:00 pm Wednesday, November 15, 2017
Geometry-Analysis Seminar : Stability results of generalized Beltrami fields and applications to vortex structures in the Euler equations
by David Poyato (University of Granada) in HBH 227Strong Beltrami fields, that is, 3D velocity fields whose vorticity is the product of itself by a constant factor, are particular solutions to the Euler equations that have long played a key role in Fluid Mechanics. Its importance relies on the Lagrangian theory of turbulence as they were expected to exhibit chaotic configurations. Very recently, this ancient conjecture of Lord Kelvin was positively answered by Alberto Enciso and Daniel Peralta-Salas (ICMAT, Spain). Specifically, there are strong Beltrami fields exhibiting any type of linked vortex lines and tubes of arbitrarily complicated topology. Nevertheless, such existence result is quite tight in the sense that Beltrami fields with non-constant factor (generalized Beltrami fields) are “rare”. Thus, the existence of turbulent configurations is limited to Beltrami fields with a constant factor. The aim of this talk is twofold. First, we will review the state of the art in this topic. Second, we will show that although a full stability result is not possible, there are certain privileged ways to perturb a strong Beltrami field and obtain Beltrami fields with a non-constant factor that even realize arbitrarily complicated vortex structures. This partial stability will be captured in terms of an "almost global” and a “local" stability theorem. The proof relies on analyzing the well-posedness and propagation of compactness and regularity of an innovative iterative scheme of Grad-Rubin type inspired by some numerical methods coming from Astrophysics. Host Department: Rice University-MathematicsSubmitted by ctan@rice.edu