I will discuss some theoretical and numerical results on the stability properties of a dynamical system modeled by a nonlinear wave equation with a "degenerate" damping. The coefficient of the dissipation term is proportional to the amplitudes, hence its support depends on the geometry of the solution. This feature substantially complicates the stability analysis even…

I will explain some interesting selection problems in nonlinear PDEs. The basic question is about how to select one good solution out of many reasonable ones. A question of this type led to the whole theory of viscosity solutions in 1980s. Then I will focus on the vanishing discount problem and describe the main results,…

We visit several models of polymer flooding in reservoir simulation. A special common feature shared by the models, i.e., the thermo-dynamics is decoupled from the hydro-dynamics, leads to a scalar conservation law with discontinuous flux. We discuss solution of Riemann problems as the vanishing viscosity limit. In particular, we show by counter examples that there exists…

In this talk we introduce new characterizations of spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order s in (0,1) with nonzero Dirichlet and Neumann boundary conditions. Here the domain Omega…

In this talk, we propose a model describing the growth of tree stems and vine, taking into account also the presence of external obstacles. The system evolution is described by an integral differential equation which becomes discontinuous when the stem hits the obstacle. The stem feels the obstacle reaction not just at the tip, but…

In this talk, I will first introduce the Monge-Ampere eigenvalue problem on general bounded convex domains and related analysis. Then I will discuss the recent resolution, in joint work with Ovidiu Savin, of global smoothness of the eigenfunctions of the Monge-Ampere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient…

Modeling of fluid flows through porous deformable media is relevant for many applications in biology, medicine and bioengineering, including blood flow through human tissues and fluid flow inside cartilages and bones. These fluid-structure mixtures are described mathematically by nonlinear poro-visco-elastic systems in bounded domains, with mixed boundary conditions. In this talk, I will present well-posedness…

Coupled array of Chua's circuits has been studied for many years. Some experimental and numerical study of traveling waves were conducted by Perez and Chua in 1993. Existence of periodic traveling waves were proved by S.-N. Chow, M. Jiang and X.-B. Lin in 2013. This talk is based on a joint work with F. Geng…

As one of the oldest nonlinear PDE systems, the compressible Euler equations has been studied by many outstanding mathematicians. However, some basic questions, such as the global existence of classical solution v.s. finite time blowup, are still open even in one space dimension. In this lecture, we will report our recent progress in this direction, including a complete understanding on…

Semilinear control systems in infinite dimensional Banach spaces are the natural framework for the description of several control problems governed by PDEs. In many models, some constraints for the state of the system may be present. In this talk, a Mayer problem associated to such systems will be discussed. In particular, a simple proof of…

Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions which lead to the emergence of global behaviors: aggregation and flocking. I will discuss the models in different scales: from microscopic agent-based…

In many practical control problems (e.g. in the power grid/traffic flow) agents are not able to effectively coordinate their actions. One classical method proposed by economists for solving such decentralized control problems is known as the ficticious play algorithm. This talk will discuss some recent work which establishes converge rates for ficticious play. In particular,…

The phaseless coefficient inverse problem arises in imaging of nano-structures. One uses light source whose wavelength is in the range of hundreds of nanometers to illuminate those structures and collects the intensity of the complex scattering field on a plane behind such structures. Due to the smallness of the wavelength, measuring the phase of the…

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system which was introduced to describe the dynamics of urban crime.  In bounded intervals I will first discuss the existence of global classical solutions for all reasonably regular non-negative initial data. Next I will address the issue of determining the qualitative behavior of solutions.  Finally, I will conclude with some numerical simulations exploring possible effects…

This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations which were derived from models of nonlinear wave with constant frequency. Some open questions will be discussed.

We propose a model of sovereign debt management, where a state trade some bonds to service the debt with a pool of risk-neutral competitive foreign investors. At each instant of time, the government decides which fraction of the GDP (subject to random fluctuations) must be used to repay the debt, by paying a social cost.…