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