Nonlinear Stochastic PDEs

This IMA Volume in Mathematics and its Applications NONLINEAR STOCHASTIC PDEs: HYDRODYNAMIC LIMIT AND BURGERS' TURBULENCE is based on the proceedings of the period of concentration on Stochas tic Methods for Nonlinear PDEs which was an integral part of the 1993- 94 IMA program on "Emerging Applications of Probability." We thank Tadahisa Funaki and Wojbor A. Woyczynski for organizing this meeting and for editing the proceedings. We also take this opportunity to thank the National Science Foundation and the Army Research Office, whose financial support made this workshop possible. A vner Friedman Willard Miller, Jr. xiii PREFACE A workshop on Nonlinear Stochastic Partial Differential Equations was held during the week of March 21 at the Institute for Mathematics and Its Applications at the University of Minnesota. It was part of the Special Year on Emerging Applications of Probability program put together by an organizing committee chaired by J. Michael Steele. The selection of topics reflected personal interests of the organizers with two areas of emphasis: the hydrodynamic limit problems and Burgers' turbulence and related models. The talks and the papers appearing in this volume reflect a number of research directions that are currently pursued in these areas.

I. Hydrodynamic Limit, Random Media, and Related Problems.- Hydrodynamic limit for lattice gas reversible under Bernoulli measures.- Equilibrium fluctuations of nongradient reversible particle systems.- The reversible measures of a conservative system with finite range interactions.- Diffusion in disordered media.- Reaction-diffusion equations in the random media: localization and intermittency.- Approximation of a one-dimensional stochastic PDE by local mean field type lattice systems.- Sharp asymptotics of diffusion processes with small parameter and applications to metastable behavior.- II. Burgers' Turbulence.- Intermediate asymptotics of statistical solutions of Burgers' equation.- On a stochastic PDE related to Burgers' equation with noise.- Shock density in Burgers' turbulence.- Model description of passive tracer density fields in the framework of Burgers' and other related model equations.- Evaluation of spectral behavior for large ensembles of exact solutions to Burgers' equation for Thomas initial conditions.- III. Stochastic Navier-Stokes Equation.- Stationary solutions of two-dimensional Navier-Stokes equations with random perturbation.- Nonlinear filtering of stochastic Navier-Stokes equation.- Mesoscopic modelling and stochastic simulations of turbulent flows.- Algebraic energy spectra in stochastic problems for the incompressible Navier-Stokes equation; relation to other nonlinear problems.