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…