The finite speed of propagation in the degenerate Einstein-Brownian motion model

TL;DR

This paper studies the finite speed of propagation in a generalized Einstein-Brownian motion model, showing that under certain degeneracies, the particle flow localizes and stops, linking to Barenblatt's finite speed property.

Contribution

It introduces a new analysis of the degenerate Einstein-Brownian model using PDE techniques, demonstrating localization and finite propagation speed with novel proof methods.

Findings

Flow exhibits localization when density or its gradient is zero.

No flow occurs in a region for a time interval if initial conditions are zero.

The method employs Ladyzhenskaya-De Giorgi and Vespri-Tedeev techniques.

Abstract

We considered the qualitative behavior of the generalization of Einstein's model of Brownian motion when the key parameter of the time interval of \textit{free jump} degenerates. Fluids will be characterized by the number of particles per unit volume (density of fluid) at the point of observation. Degeneration of the phenomenon manifests 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 the velocity of the flow is incomparably smaller than the gradient of the density. First, we will show that both types of flows can be modeled using the Einstein paradigm. We will investigate the question: What features will particle flow exhibit if the time interval of the\textit{ free jump} is inverse proportional to the density and its gradient? We will show that in this scenario, the flow exhibits localization property, namely: if at some moment of time $t_0$ in the region, the gradient of the density or density itself is equal to zero, then for some time $T$ during t interval $[ t_{0}, t_0+T]$ there is no flow in the region. This directly links to Barenblatt's finite speed of propagation property for the degenerate equation. The method of the proof is very different from Barenblatt's method and based on the application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique. From PDE's point of view, it assumed that solution exists in appropriate Sobolev type of space.