Lattice Gases, Large Deviations, and the Incompressible Navier-Stokes Equations

Abstract

We study the incompressible limit for a class of stochastic particle systems on the cubic lattice Zd, d = 3. For initial distributions corresponding to arbitrary macroscopic L2 initial data, the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore, explicit exponential rates for the convergence (large deviations) are obtained. The probability to violate the divergence-free condition decays at rate at least exp{-E-d+l}, while the probability to violate the momentum conservation equation decays at rate exp{fE-d+2} with an explicit rate function given by an HI1 norm.