Hydrodynamic limit of gradient exclusion processes with conductances

Fix a strictly increasing right continuous with left limits function $W: \bb R \to \bb R$ and a smooth function $\Phi : [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le \Phi'\le b^{-1}$. 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 $\partial_t \rho = (d/dx)(d/dW) \Phi(\rho)$. We derive some properties of the operator $(d/dx)(d/dW)$ and prove uniqueness of weak solutions of the previous non-linear differential equation.