The Rayleigh model of nonlinear Brownian motion is revisited in which the heavy particle of mass M interacts with ideal gas molecules of mass m5M via instantaneous collisions. Using the van Kampen method of expansion of the master equation, nonlinear corrections to the Fokker–Planck equation are obtained up to sixth order in the small parameter l 1⁄4 ffiffiffiffiffiffiffiffiffiffiffi m=M p ; im...

A growing interface subject to noise is described by the KardarParisi-Zhang equation or, equivalently, the noisy Burgers equation. In one dimension this equation is analyzed by means of a weak noise canonical phase space approach applied to the associated Fokker-Planck equation. The growth morphology is characterized by a gas of nonlinear soliton modes with superimposed linear diffusive modes. ...

This paper is concerned with the dynamics for nonlinear one-dimensional beam equations. We consider a nonlinear beam equation with viscosity or with a lower order damping term instead of the viscosity, and we establish the existence of global attractors for both systems. Keyword: beam equations; global attractor; in nite-dimensional dynamics MSC 2000: 35G25; 35B41

We report on some recent results concerning the dynamics of Bose-Einstein condensates, obtained in a series of joint papers [5, 6] with L. Erdős and H.-T. Yau. Starting from many body quantum dynamics, we present a rigorous derivation of a cubic nonlinear Schrödinger equation known as the Gross-Pitaevskii equation for the time evolution of the condensate wave function.

My research focuses on scientific computing and kinetic equations. More specifically I develop and analyze numerical methods for solving the Boltzmann transport equation and similar kinetic models, as well as their application to high performance computing resources. My work is supported by NSF-DMS Grant 1217254 “Accurate high performance computing for nonlinear collisional kinetic theory”, (20...

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolicelliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynam...

We use an "equation-free," coarse-grained computational approach to accelerate molecular dynamics-based computations of demixing (segregation) of dissimilar particles subject to an upward gas flow (gas-fluidized beds). We explore the coarse-grained dynamics of these phenomena in gently fluidized beds of solid mixtures of different densities, typically a slow process for which reasonable continu...