# Research Information

Much (but not all) of my research has revolved around the central
theme of understanding how large-scale behaviors emerge from dynamics
or structures on small-scales. This includes the classical question
of statistical physics about the macroscpic desciption of systems of
large numbers of particles given known microscopic physics. It also
includes studies of semiclassical limits of nonlinear wave equations,
convergence of numerical schemes, turbulence modeling, derivations of
shallow water systems, derivations of fluid dynamical systems from
kinetic theories, radiation transport through random media, and many
other areas. These problems all fall into the what is now called the
class of "multiscale" problems.
My research work breaks down into the following areas:

### Establishing Fluid Dynamical Limits from Boltzmann Equations

In 1988 I began a collaboration with Claude Bardos and Francois
Golse that sought to establish the global validity of fluid
dynamical approximations to the Boltzmann equation starting from
the (new at the time) DiPerna-Lions theory of global solutions of
the Boltzmann equation. This effort has been joind by others ---
most notably Pierre-Louis Lions, Nader Masmoudi, and Laure
Saint-Raymond. Results have since been established for acoustic
limits, incompressible Stokes limits, incompressible
Navier-Stokes limits, incompressible Euler limits, and
compressible Stokes approximations. A benchmark in this program
was reached in 2004 when Francois Golse and Laure Saint-Raymond
established a Navier-Stokes limit for the case of bounded
collision kernels satisfying a Grad cutoff. My recent work with
Nader Masmoudi has extended their result.

### Semiclassical Limits of Nonlinear Wave Equations

My dissertation work was on the zero-dispersion limit of the
Korteweg-deVries (KdV) equation. In 1983 the details of this
work and some extensions appeared in a series of papers
co-authored with my supervisor, Peter D. Lax. Later I turned to
the related problems of the semiclassical limits for the
defocusing cubic Schroedinger hierarchy in one spatial dimension
(with David W. McLaughlin and Shan Jin), and the focusing
modified KdV hierarchy (with Nicholas M. Ercolani).

### Hyperbolic Systems

### Derivation of Transition Regime Models from Kinetic Equations

### Continuum Limits of Lattice Dynamics

### Numerical Schemes for Partial Differential Equations

### Radiation Transport through Radom Media

### Shallow Water Systems

### Damped-Driven Stochastic Systems

### Damped-Driven Chaotic Systems

Under construction