Monday, May 26
8:30-9:10 AM Registration
9:10-9:20 AM Opening Remarks
9:20-10:10 AM Juncheng Wei, The Chinese University of Hong Kong & University of British Columbia
Uniqueness and Multiplicity for Semilinear Elliptic Problems in Unbounded Domains
We study the influence of geometry on semilinear elliptic equations of bistable type in unbounded domains. First we prove all stable solutions of nonlinear equations with bistable nonlinearity other than Allen-Cahn in the whole space are constants. Then we study the problem on epigraphs. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of aperture greater than 𝜋: the former admit at most one positive bounded solution, while the latter support infinitely many. Nonetheless, we show that every epigraph admits at most one strictly stable solution. To prove uniqueness, we strengthen the method of moving planes by decomposing the domain into one region where solutions are stable and another where they enjoy a form of compactness. Our construction of many solutions exploits a connection with Delaunay surfaces in differential geometry, and extends to all domains containing a suitably wide cone, including exterior domains. (Joint work with Y. Liu, Kelei Wang, Ke Wu; and Berestycki, Graham.)
10:10-10:40 AM Morning Tea Break
10:40-11:30 AM Changyou Wang, Purdue University
Variational Approximation of Heat Flow of Harmonic Maps into Non-Positively Curved Manifolds
We introduce a new approach to construct (weak) heat flow of harmonic maps between manifolds based on the Weighted-Energy-Dissipation (WED) approach, which involves a variational functional with a small parameter. For smooth target manifolds, we recover the well-known theorems by Eells-Sampson (on NPC target manifolds) through Dynamical Variational Principle (DCP) as well as PDE convergence approach. This is a joint work with Fanghua Lin, Antonio Segatti, and Yannick Sire.
11:30-11:40 AM Group Photo
11:40 AM-1:30 PM Lunch Break
1:30-2:20 PM Alexey Cheskidov, Westlake University
Energy Cascade in Fluids: from Convex Integration to Mixing
In the past couple of decades, mathematical fluid dynamics has made significant strides with numerous constructions of solutions to fluid equations that exhibit pathological or wild behaviors. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations.
2:20-2:50 PM Afternoon Tea Break
2:50-3:40 PM Hao Jia, University of Minnesota
Relaxation Mechanisms in Incompressible Fluid Flows and Related Models
In this talk we will present several mechanisms for asymptotic stability in the two dimensional incompressible fluid equations, including inviscid damping, vorticity depletion, and enhanced dissipation. We will also discuss an important stability mechanism due to the balance between convection and vorticity stretching in the De Gregorio model for the three dimensional incompressible Euler equations. These stability mechanisms play an essential role in the long-time behavior of smooth solutions to incompressible fluid equations, and have been used to prove asymptotic stability of important steady solutions. Numerical results and open questions about dynamical behavior of general solutions will also be presented.
3:50-4:40 PM Jiajun Tong, Peking University
Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation
It is of great mathematical and physical interest to study traveling wave solutions to the 2D incompressible Euler equation in the form of a touching pair of symmetric vortex patches with opposite signs. Such a solution was numerically illustrated by Sadovskii in 1971, but its rigorous existence was left as an open problem. In this talk, we will rigorously construct such a solution by a novel fixed-point approach that determines the patch boundary as a fixed point of a nonlinear map. Smoothness and other properties of the patch boundary will also be characterized. This is based on a joint work with De Huang.