Mathematics Department - Joint Princeton-Rutgers Seminar on Geometric PDE's - Spring 2017

Joint Princeton-Rutgers Seminar on Geometric PDE's - Spring 2017


Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum


Past Talks

Tuesday, April 18th

Matthew Gursky, University of Notre Dame

"Some existence and non-existence results for Poincare-Einstein metrics"

Time: 3:00 PM
Location: Hill 705
Abstract: I will begin with a brief overview of the existence question for conformally compact Einstein manifolds with prescribed conformal infinity. After stating the seminal result of Graham-Lee, I will discuss a non-existence result (joint with Qing Han) for certain conformal classes on the 7-dimensional sphere. I will also mention some ongoing work (with Gabor Szekelyhidi) on a version of "local existence" of Poincare-Einstein metrics.

Tuesday, April 18th

Philip Isett, MIT

"A Proof of Onsager’s Conjecture for the Incompressible Euler Equations"

Time: 1:40 PM
Location: Hill 705
Abstract: In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.

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