An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

Abstract

We consider the degenerate Einstein's Brownian motion model for the case when the time interval ($\tau$) of particle Jumps before collision (free jumps) reciprocal to the number of particles per unit volume $u(x,t)>0$ at the point of observation $x$ at time $t$. The parameter $0<\tau \le C<\infty$, controls characteristic of the fluid "almost decreases" to $0$ when $u\to \infty$. This degeneration leads to the localisation of the spread of particle propagation in the media. In our report we will present a structural condition of the time interval of free jumps - $\tau$ and the frequency of these free jumps $\varphi$ as functions of $u$ which guarantees the finite speed of propagation of $u$.