Generalized Young measures and the hydrodynamic limit of condensing zero range processes

Condensing zero range processes (ZRPs) are stochastic interacting particle systems that exhibit phase separation with the emergence of a condensate. Standard approaches for deriving a hydrodynamic limit of the density fail in these models, and an effective macroscopic description has not been rigorously established, yet. In this article we prove that the limiting triple $(\pi,W,\sigma)$ of the empirical density, the empirical current, and the empirical jump rate of the ZRP satisfies the continuity equation $\partial_t\pi=-{\rm{div}}W$ in the sense of distributions. Here $(\pi_t)_{t\geq 0}$ is a $w^*$-continuous curve of finite non-negative measures on the torus $\mathbb{T}^d$, $\sigma_t\in H^1(\mathbb{T}^d)$ and $W_t=-\nabla\sigma_t$ is a vector-valued measure that is absolutely continuous with respect to the Lebesgue measure, for all almost all $t\geq 0$. In order to obtain a closed equation we propose a generalization of Young measures and we prove that for symmetric ZRPs on the torus, the hydrodynamic limit of the density is a generalized Young-measure-valued weak solution $\boldsymbol{\pi}=(\boldsymbol{\pi}_t)_{t\geq 0}$ to a saturated filtration equation $\partial_t\boldsymbol{\pi}=\Delta\Phi(\boldsymbol{\pi})$. Furthermore we prove a one-sided two-blocks estimate and we give an equivalent criterion for its validity. Assuming the validity of the two-blocks estimate one obtains the equation $\partial_t\pi=\Delta\Phi(\pi^{ac})$ for the empirical density, where $\pi=\pi^{ac}+\pi^\perp$ is the Radon-Nikodym decomposition.