A Sequence of Integro-Differential Equations Approximating a Viscous Porous Medium Equation

Abstract

We consider a sequence of particular integro-differential equations, whose solutions ${\rho }_N$ converge as $N\to \infty$ to the solution $\rho$ of a viscous porous medium equation. First, it is demonstrated that under suitable regularity conditions the functions ${\rho }_N$ are smooth uniformly in $N\in \mathbb{N}$. Furthermore, an asymptotic expansion for ${\rho }_N$ as $N\to \infty$ is provided, which precisely describes the convergence to $\rho$. The results of this paper are needed in particular for the numerical simulation of a viscous porous medium equation by a particle method.