论文标题
退化粒子流模型与自由边界问题之间的联系
Connection between a degenerate particle flow model and a free boundary problem
论文作者
论文摘要
在本文中，研究了从密度依赖粒子流模型得出的强烈退化抛物线方程。此外，研究了一个自由边界问题及其与强烈退化的抛物线方程的联系。首先，结果表明，强烈的退化抛物线方程具有独特的全局界弱解，该解决方案在大时光范围内会收敛到稳态。可能会发生两种情况：当平均密度$ρ_ {\ infty} $大于某个关键密度$ρ_{Cr} $时，稳态与$ρ_{\ infty} $相吻合，并且收敛速率在$ l^2 $ narmy中是指数的；在相反的情况下，$ρ_ {\ infty} <ρ_{cr} $，稳态是未知的，并且收敛性为代数为负Sobolev seminorm。进一步的研究表明，对于径向对称和减少初始数据，可以通过使用相应的自由边界问题的解决方案来构建强烈退化的抛物线方程的解决方案。此外，证明了全球解决后一种问题的薄弱解决方案的存在。最后，提出了两个空间维度的数值实验，这表明当初始平均密度小于临界密度时，隔离现象会出现。
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 $ρ_{\infty}$ is larger than a certain critical density $ρ_{cr}$, the steady state coincides with $ρ_{\infty}$ and the convergence rate is exponential in the $L^2$ norm; while in the opposite case $ρ_{\infty}<ρ_{cr}$, 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.
