PDE Geometric Analysis seminar

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

PDE GA Seminar Schedule Spring 2018

date	speaker	title	host(s)
January 29, 3-3:50PM, B341 VV.	Dan Knopf (UT Austin)	Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons	Angenent
February 5, 3-3:50PM, B341 VV.	Andreas Seeger (UW)	Singular integrals and a problem on mixing flows	Kim & Tran
February 12	Sam Krupa (UT-Austin)	Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case	Lee
February 19	Maja Taskovic (UPenn)	TBD	Kim
February 26	Ashish Kumar Pandey (UIUC)	Instabilities in shallow water wave models	Kim & Lee
March 5	Khai Nguyen (NCSU)	Burgers Equation with Some Nonlocal Sources	Tran
March 12	Hongwei Gao (UCLA)	TBD	Tran
March 19	Huy Nguyen (Princeton)	TBD	Lee
March 26		Spring recess (Mar 24-Apr 1, 2018)
April 2		TBD
April 9	Jeff Calder (Minnesota)	TBD	Tran
April 21-22 (Saturday-Sunday)	Midwest PDE seminar		Angenent, Feldman, Kim, Tran.
April 25 (Wednesday)	Hitoshi Ishii (Wasow lecture)	TBD	Tran.

Abstracts

Dan Knopf

Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons

Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.

Andreas Seeger

Title: Singular integrals and a problem on mixing flows

Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.

Sam Krupa

Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case

Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.

Ashish Kumar Pandey

Title: Instabilities in shallow water wave models

Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.