窄管状膜通道的Poisson-Nernst-Planck系统
Poisson-Nernst-Planck Systems for Narrow Tubular-like Membrane Channels
摘要：
本文研究了两种离子通过狭窄的管状膜通道的Poisson-Nernst-Planck（PNP）系统的全局动力学。当三维管状膜通道的横截面半径趋于零时，导出了一维的极限PNP系统。这个一维限制系统不同于以前研究的一维PNP系统，因为它编码的三维膜通道的定义几何形状。为了证明这个限制过程，我们证明了三维PNP系统的全局吸引子是上半连续的限制PNP系统。然后，我们研究的一维限制PNP系统的动力学。在大德拜数条件下，利用几何奇异扰动理论对一维极限PNP系统的定态进行了分析。对于一种特殊的情形，构造了一个熵型李雅普诺夫泛函，证明了系统的全局渐近稳定性。
英文摘要：
We study global dynamics of the Poisson-Nernst-Planck (PNP) system for flows of two types of ions through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previous studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous to that of the limiting PNP system. We then examine the dynamics of the one-dimensional limiting PNP system. For large Debye number, the steady-state of the one-dimensional limiting PNP system is completed analyzed using the geometric singular perturbation theory. For a special case, an entropy-type Lyapunov functional is constructed to show the global, asymptotic stability of the steady-state.
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