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ACMS Abstracts: Fall 2015

Li Wang (UCLA)

Singular shocks: From particle-laden flow to human crowd dynamics

In this talk, we will present two examples in which singular shock arises. The first example, gravity-driven thin film flow with a suspension of particles down an incline, is described by a system of conservation laws equipped with an equilibrium theory for particle settling and resuspension. Singular shock appears in the high particle concentration case that relates to the particle-rich ridge observed in the experiments. We analyze the formation of the singular shock as well as its local structure, and extend to the finite volume case, which leads to a linear relationship between the shock front with time to the one-third power. The second example, a panicking crowd with a spread of fear, is modeled via ``emotional contagion”. Singular shock happens in an extreme case whose continuum limit is a pressure less Euler equation. Such system is then modified with a nonlocal alignment to regularize the singularity. We will discuss the hierarchy of models and their mathematical properties. Novel numerical methods will be presented for both examples.

Wai Tong (Louis) Fan

Reflected diffusions with partial annihilations on a membrane

Mathematicians and scientists use interacting particle models to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. In this talk, I will introduce a class of interacting particle systems that can model the transport of positive and negative charges in solar cells. To connect the microscopic mechanisms with the macroscopic behaviors at two different scales, we obtain the hydrodynamic limits and the fluctuation limits for these systems. Proving these two types of limits represents establishing the law of large numbers and the central limit theorem, respectively, for the time-trajectory of the particle densities. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. This is joint work with Zhen-Qing Chen. This talk will focus on modeling methods and applications. A second talk on the probabilistic techniques involved in the proofs will be given in the Probability Seminar on Oct 15.

Wenjia Jing (Chicago)

Limiting distributions of random fluctuations in stochastic homogenization

In this talk, I will present some results on the study of limiting distributions of the random fluctuations in stochastic homogenization. I will discuss first a framework of such studies for linear equations with random potential. The scaling factor and the scaling limit of the homogenization error turn out to depend on the singularity of the Green’s function and the correlation structure of the random potential. I will also present some results that extend the scope of the framework to the setting of oscillatory differential operators and to some nonlinear equations. Such results find applications, for example, in uncertainty quantification and Bayesian inverse problems.

Arthur Evans (UW)

Ancient art and modern mechanics: using origami design to create new materials