A microscopic mechanism for the porous medium equation
The porous medium equation (PME) \[ {\partial \over \partial t} u(t,\xi ) = K\cdot \Delta u^{\alpha }(t,\xi ) \] is considered, where \(u(t,\xi )\) is a scalar function on \({\mathcal R}_+\times {\mathcal R}^d\), \(\Delta \) is the Laplace operator, and \(K>0\) and \(\alpha >1\) are constants. It is shown that, providing \(u(t,\xi )\) is a (weak) solution of PME, \(u(t,\theta ) d\theta \) agrees with the limit of a sequence of empirical measures \(\mu _t^N(d\theta )\), where each \(\mu _t^N({\roman d}\theta )\) is associated with the Markov process \(x^N(t)\) on the \({\mathcal R}_+^{Z_N^d}\). The generator of the Markov process is given by \[ L_N f(x) = N^2 \sum _{k,l \in Z_N^d, |k-l |=1} \int _0^{x_k} u^{\alpha -2} [f(x^{u,k,l })-f(x)] du \] where \(x_i^{u,k,l } = x_i\) for \( i\neq k,l \), \(x_i^{u,k,l } = x_k-u \) for \( i=k\), \(x_i^{u,k,l } = x_l +u \) for \( i=l \), and \(Z_N^d\) is the \(d\)-dimensional periodic lattice of size \(N^d\). This hydrodynamic scaling limit provides rigorous justification for regarding the stochastic model as a microscopic analogue of the PME.
scientific article; zbMATH DE number 1134811
scientific article