One-Dimensional hard-rod caricature of hydrodynamics: "Navier-Stokes correction" for local equilibrium initial states *

A one-dimensional system, consisting of identical hard-rod particles of length a is studied in the hydrodynamical limit. A "Navier-Stokes correction" to the Euler equation is found for an initial local equilibrium family of states P^∈, ∈ > 0, of constant density. The correction is given, at t ∼ 0, by a non-linear second order differential operator acting on f (q, v), the hydrodynamical density at a point q ∈ R¹ of the "species" of fluid with velocity v ∈ R¹.