Hydrodynamic limit of gradient exclusion processes with conductances

Fix a strictly increasing right continuous with left limits function W : R → R and a smooth function Φ : [l, r] → R, defined on some interval [l, r] of R, such that 0 < b ≤ Φ′ ≤ b^−1. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by W, is described by the weak solutions of the non-linear differential equation ∂tρ = (d/dx)(d/dW)Φ(ρ). We derive some properties of the operator (d/dx)(d/dW) and prove uniqueness of weak solutions of the previous non-linear differential equation.