Deriving hydrodynamic equations for lattice systems

Abstract

We study the dynamics of lattice systems in ℤd, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ0ɛ, ɛ > 0}. The measures μ0ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ−1) and inhomogeneous under shifts of the order ɛ−1. Moreover, correlations of the measures μ0ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝd, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛκ at points close to [r/ɛ] ∈ ℤd. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type.