Connection between a degenerate particle flow model and a free boundary problem

TL;DR

This paper explores the connection between a degenerate particle flow model and a free boundary problem, establishing existence, uniqueness, and convergence properties of solutions, and illustrating segregation phenomena through numerical experiments.

Contribution

It introduces a novel link between a strongly degenerate parabolic equation and a free boundary problem, including proofs of global existence and convergence behaviors.

Findings

Unique global bounded weak solutions exist and converge to steady states.

Convergence is exponential if average density exceeds critical density.

Segregation phenomena occur when initial density is below critical density.

Abstract

In this paper a strongly degenerate parabolic equation derived from a density dependent particle flow model is studied. Furthermore, a free boundary problem and its connection to the strongly degenerate parabolic equation is investigated. First, it is shown that the strongly degenerate parabolic equation has a unique global bounded weak solution that converges towards a steady state for large time horizons. Two scenarios might occur: When the average density $\rho_{\infty}$ is larger than a certain critical density $\rho_{cr}$, the steady state coincides with $\rho_{\infty}$ and the convergence rate is exponential in the $L^2$ norm; while in the opposite case $\rho_{\infty}<\rho_{cr}$, the steady state is unknown and the convergence is algebraic in a negative Sobolev seminorm. Further investigations show that for radially symmetric and decreasing initial data, the solution of the strongly degenerate parabolic equation can be constructed by using the solution of a corresponding free boundary problem. Moreover, the global existence of weak solutions to the latter problem is proved. Finally, numerical experiments in two space dimensions are presented, which show that segregation phenomena can appear when the initial average density is smaller than the critical density.