帕松波茲曼類方程的邊界層解

Abstract

帕松波茲曼方程式可被用於描述電解質液的離子濃度分布情形，從而在生物化學與電化學領域中有諸多重要的應用，在細胞膜上的離子通道便是一重要例子。然而傳統的帕松波茲曼方程式視離子為質點而不帶有體積，而離子通過狹窄的離子通道時便有不可忽略的體積效應。在既有的文獻中已經有許多將體積效應納入考量的模型，但這些模型均無法描述如密度泛函理論中所產生的非單調總電荷密度的現象，這就啟發我們去推導更一般的帕松波茲曼模型。這樣的模型不僅涵蓋傳統與修正後的帕松波茲曼模型，也能產生具非單調總電荷密度的方程式。

更進一步地，由於離子通道的複雜幾何結構，我們研究一般光滑區域上的帕松波茲曼類方程隨介電常數趨於零時所產生的邊界解。此處帕松波茲曼類方程包含了傳統、修正後與電中性條件下的電荷守恆帕松波茲曼模型。源於電荷守恆條件下的非局部非線性項使得離子守恆的帕松波茲曼模型的分析更加困難，其結果也有別於傳統與修正後的帕松波茲曼模型。透過主軸坐標系統、指數型估計與移動平面法，我們嚴格地證明了邊界層解的二階漸近展開式，其中包含了均曲率項。進一步地，我們也計算出許多關鍵的物理量的漸近展開式，這包括了電位、電場、總電荷密度與總電荷，而這些物理量也揭示了區域幾何的影響。

相較於前段所研究的電荷守恆帕松波茲曼模型為電中性條件，其電位保持均勻有界性。而對於非電中性條件下，電位便會隨著介電常數趨於零時而產生爆破現象。對於最極端的非電中性條件，我們研究一維上帶單粒子的電荷守恆帕松波茲曼模型。透過適當的平移假設下，我們將問題轉為邊界爆破問題。藉由解的表達式，我們研究了近場與遠場展開從而完整的刻畫了靠近邊界區域的解的所有漸近行為，也證明了總電荷密度的邊界集中性。另一方面，我們研究了雙離子的電荷守恆帕松波茲曼模型的差異。形式上來看，當雙離子模型中的其中一粒子總濃度相當小時便會趨近單離子模型，而我們給出一充分條件去證明這樣的猜測成立。

The Poisson–Boltzmann (PB) equation serves as a fundamental model for describing ionic concentration distributions in electrolyte solutions, with significant applications in both biochemical and electrochemical fields. One prominent example arises in the study of ion channels on cellular membranes. However, the classical PB equation treats ions as point particles without volume, which becomes inadequate when ions pass through narrow channels so the steric effect should not be negligible. Although several modified models incorporating steric effects have been proposed in the literature, they cannot capture the non-monotonic behavior of total ionic charge density observed in density functional theory. This motivates us to derive a more general Poisson–Boltzmann framework that not only includes the classical and modified PB models, but also includes the model that has non-monotonic total ionic charge density.

Further, motivated by the complex geometry of ion channels, we investigate the boundary layer behavior of Poisson–Boltzmann type equations as the dielectric constant tends to zero in general smooth domains. Here the PB type equations include the classical PB, modified PB, and charge-conserving PB models with the electroneutrality condition.Due to the presence of nonlocal nonlinear terms arising from the charge conservation, the analysis of charge-conserving PB models presents the analytical challenges and leads to different behaviors compared to that of other models. By employing principal coordinate systems, exponential-type estimates, and the moving plane argument, we rigorously derive second-order asymptotic expansions of boundary layer solutions, incorporating curvature-dependent terms. We also compute asymptotic expansions for several key physical quantities, including the electrostatic potential, electric field, total charge density, and total charge, thereby revealing the influence of domain geometry on the ionic distributions.

In contrast to the electroneutral regime, where the electrostatic potential remains uniformly bounded, the potential exhibits blow-up behavior under non-electroneutral conditions as the dielectric constant tends to zero. To explore this extreme case, we examine the one-dimensional charge-conserving PB model with a single species. Under an appropriate shift assumption, we reformulate the problem as a boundary blow-up problem. Thanks to the explicit solution representation, we can analyze both near-field and far-field expansions and hence provide a complete characterization of the asymptotic behavior near the boundary. In addition, we rigorously establish the boundary concentration phenomenon of the total ionic charge density. On the other hand, we also investigate the differences arising in charge-conserving PB models with two ionic species. Formally, such two-species model would approach to the single-species model when the total concentration of one ions species becomes sufficiently small. We provide a sufficient condition to justify this conjecture.