Finite speed of propagation in Degenerate Einstein Brownian Motion Model

We consider a generalization of Einstein model of Brownian motion when the key parameter of the time interval of free jump degenerates via solution and its gradient. This phenomena manifest in two scenarios: a) Flow of the fluid which is highly dispersing like a non-dense gas and b)Flow of fluid far away from the source of flow, when velocity of the flow is not-comparably smaller than gradient of the pressure. First, we will show that both types of flows can be modeled by using the Einstein paradigm. We will investigate the question: what feature will exhibit particle flow if the time interval of free jump is inverse proportional to density of the fluids and its gradient. It was shown that in this scenario, the flow exhibits localization property, namely: if at some moment of time $t_0$ in the region gradient of pressure or pressure itself is equal zero then for some $T$ during time interval $[t_{0}, t_0+T]$ is no flow in the region. This directly links to Barenblatt's finite speed of propagation property for the degenerate equation. Method of proof is very different to Barenblatt's method and based on application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique.