Tuesday, May 27
9:20-10:10 M Zhiwu Lin, Fudan University
The Onset of Instability for Zonal Stratospheric Flows
We investigate some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant for stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet, which corresponds to choosing the Legendre polynomial of degree n as the stream function. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We confirm conjectures based on numerical simulations by proving that there exist positive and negative critical rotation rates for the linear instability of 3-jet. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotating rate goes to infinity, nearby traveling waves gradually change from a cats’s eyes streamline pattern to a profile with no stagnation points. This is a joint work with Adrian Constantin, Pierre Germain and Hao Zhu.
10:10-10:40 AM Morning Tea Break
10:40-11:30 AM Wei Wang, Zhejiang University
Existence of Kayaking Solutions for the Doi-Onsager Equation in Shear Flow
The periodic motions of nematic liquid crystals in shear flow have long been predicted through experiments and numerical simulations. In this talk, we will show the existence of a specific type of periodic solution, known as kayaking, under small shear rates.
11:30 AM-1:30 PM Lunch Break
1:30-2:20 PM Congming Li, Shanghai Jiao Tong University
Qualitative Analysis on Nonlinear PDEs of Elliptic Type
We present our research works on qualitative analysis of some nonlinear elliptic type PDEs. It is mainly on maximum principles, Liouville type theorems and classification of solutions. We focus on the Hardy-Littlewood-Sobolev type systems and curvatures related geometric equations. We also report some recent work on steady Euler equations when time permit.
2:20-2:50 PM Afternoon Tea Break
2:50-3:40 PM Zhiyuan Geng, Purdue University
Triple Junction Solution for the Allen-Cahn System
In this talk, I will discuss recent results on the 2D Allen-Cahn system with a triple-well potential. By analyzing the blow-up limit of solutions near the junction of three phases, we construct an entire minimizing solution that asymptotically converges at infinity to a unique triple junction, corresponding to a planar minimal cone. We further establish the almost 1D symmetry of the solution along the sharp interface. A crucial estimate in our blow-down analysis is the sharp energy lower and upper bounds, which enables the localization of the diffuse interface within a small neighborhood of the limiting interface. The results don't rely on any symmetry assumptions. This is joint work with Nicholas Alikakos.
3:50-4:40 PM Yifu Zhou, Wuhan University
Bubbling Solutions for the $H$-system and its Heat Flow
In this talk, we will report some construction of bubbling solutions for the $H$-system and its heat flow in two dimensions, modeling surfaces with constant mean curvature. We construct type II finite-time blow-up solution with degree 1 for the heat flow, and multi-bubble solutions with degree 2 for its elliptic counterpart. The latter gives a partial answer to a question raised by Brezis-Coron and Chanillo-Malchiodi concerning the possible limiting configuration of spheres in the degree 2 case. The key ingredients include the non-degeneracy of the $H$-bubble with any degree, a decoupling property for the linearized system, and the dealing with extra modulation parameters. This is based on joint works with X. Fang, Y. Sire, J. Wei, and Y. Zheng.