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• W2398710667 abstract "We describe interacting lattice models on the torus whose special feature is that the macroscopic equation of the empirical density is a degenerate parabolic equation, namely the equation of an ideal gas flowing isothermally through a porous medium. The models come in two versions: one with continuous variables and one with particles on the sites. In the particle model a degenerate equation is obtained only if the size of the particle vanishes in the limit, otherwise the limiting equation is a nondegenerate equation that also governs the densities of certain exclusion processes with speed change. We establish basic properties of these models such as attractiveness and reversibility, and prove the hydrodynamic scaling limits for the empirical densities. 1. Introduction and results The porous medium equation ∂tu = �(u m ), m > 1, has for some time been among the most intensely studied partial differential equations. A rich theory has developed since the fundamental solutions were found in the early 1950's in Russia, but results connecting this equation with interesting stochastic dynamics are few. This equation is a degenerate parabolic equation in the sense that when written in the form ∂tu = ∇ � D(u) ∇u � , the diffusion matrix D(u) vanishes for u = 0. In this paper we describe some simple interacting lattice models whose empirical densities obey the porous medium equation ∂tu = �(u 2 ) in a hydrodynamical scaling limit. The stochastic model we study comes in two versions, one with continuous variables (the stick model) and one with discrete variables (the particle model). The stick model is a relative of the linear models discussed in Chapter IX of Liggett's monograph (L), in the sense that when an event takes place at some time t, the new configuration ηt is a linear function of the old one ηt−. But the rates are not uniform as in the linear models of (L), for an event takes place at a site x at a rate proportional to the size of the variable η(x). This model is not new. It has been studied earlier in (SU), where H. Tanaka is credited for suggesting the model. The particle model resembles the zero-range process introduced by Spitzer (S), in that particles jump from a site with an intensity determined by the" @default.
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• W2398710667 date "1996-01-01" @default.
• W2398710667 modified "2023-09-27" @default.
• W2398710667 title "STOCHASTIC DYNAMICS MACROSCOPICALLY GOVERNED BY THE POROUS MEDIUM EQUATION FOR ISOTHERMAL FLOW" @default.
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