Hydrodynamic limit for perturbation of a hyperbolic equilibrium point in two-component systems

Abstract

We consider one-dimensional, locally finite interacting particle systems with two conservation laws. The models have a family of stationary measures with product structure and we assume the existence of a uniform bound on the inverse of the spectral gap which is quadratic in the size of the system. Under Eulerian scaling the hydrodynamic limit for the macroscopic density profiles leads to a two-component system of conservation laws. The resulting pde is hyperbolic inside the physical domain of the macroscopic densities, with possible loss of hyperbolicity at the boundary. We investigate the propagation of small perturbations around a \emph{hyperbolic} equilibrium point. We prove that the perturbations essentially evolve according to two \emph{decoupled} Burgers equations. The scaling is not Eulerian: if the lattice constant is $n^{-1}$, the perturbations are of order $n^{-\beta}$ then time is speeded up by $n^{1+\b}$. Our derivation holds for $0<\beta< \frac15$. The proof relies on Yau's relative entropy method, thus it applies only in the regime of smooth solutions.