# 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 $\mathbb{Z}^d$, $d = 3$. For initial distributions corresponding to arbitrary macroscopic $L^2$ 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 $\exp\{-\varepsilon^{-d+1}\}$, while the probability to violate the momentum conservation equation decays at rate $\exp\{-\varepsilon^{-d+2}\}$ with an explicit rate function given by an $H_{-1}$ norm.

Jeremy Quastel

Horng-Tzer Yau