arXiv subjects

Tai-Chia Lin

Publications and source records attributed to Tai-Chia Lin.

16 recordsLinked to original sources

arXiv2025

Asymptotic Analysis of Boundary Layer Solutions to Poisson-Boltzmann Type Equations in General Bounded Smooth Domains

We study the boundary layer solution to singular perturbation problems involving Poisson-Boltzmann (PB) type equations with a small parameter $\epsilon$ in general bounded smooth domains (including multiply connected domains) under the Robin boundary condition. The PB type equations include the classical PB, modified PB and charge-conserving PB (CCPB) equations, which are mathematical models for the electric potential and ion distributions. The CCPB equations present particular analytical challenges due to their nonlocal nonlinearity introduced through integral terms enforcing charge conservation. Using the principal coordinate system, exponential-type estimates and the moving plane agruments, we rigorously prove asymptotic expansions of boundary layer solutions throughout the whole domain. The solution domain is partitioned into three characteristic regions based on the distance from the boundary: Region I, where the distance from the boundary is at most $T\sqrt\epsilon$, Region II, where the distance ranges between $T\sqrt\epsilon$ and $\epsilon^\beta$, and Region III, where the distance is at least $\epsilon^\beta$, for given parameters $T>0$ and $0<\beta<1/2$. In Region I, we derive second-order asymptotic formulas explicitly incorporating the effects of boundary mean curvature, while exponential decay estimates are established for Regions II and III. Furthermore, we obtain asymptotic expansions for key physical quantities, including the electric potential, electric field, total ionic charge density and total ionic charge, revealing how domain geometry regulates electrostatic interactions.

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arXiv2023

PB-steric equations: A general model of Poisson-Boltzmann equations

When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson-Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter $\Lambda$, which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As $\Lambda=0$, the PB-steric equation becomes the conventional PB equation, but as $\Lambda>0$, the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as $\Lambda$ goes to infinity. Our theoretical results show that the PB-steric equations (for $0\leq\Lambda\leq\infty$) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations.

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arXiv2022

On the asymptotic limit of steady state Poisson--Nernst--Planck equations with steric effects

When ions are crowded, the effect of steric repulsion between ions becomes significant and the conventional Poisson--Boltzmann (PB) equation (without steric effect) should be modified. For this purpose, we study the asymptotic limit of steady state Poisson--Nernst--Planck equations with steric effects (PNP-steric equations). By the assumptions of steric effects, we transform steady state PNP-steric equations into a PB equation with steric effects (PB-steric equation) which has a parameter $\Lambda$ and positive constants $\lambda_i$'s (depend on the radii of ions and solvent molecules). The nonlinear term of PB-steric equation is mainly determined by a Lambert type function which represents the concentration of solvent molecules. As $\Lambda=0$, the PB-steric equation becomes the conventional PB equation but as $\Lambda>0$, a large $\Lambda$ makes the steric repulsion (between ions and solvent molecules) stronger. This motivates us to find the asymptotic limit of PB-steric equation as $\Lambda$ goes to infinity. Under the Robin (or Neumann) boundary condition, we prove theoretically and numerically that the PB-steric equation has a unique solution $\phi_\Lambda$ which converges to solution $\phi^*$ of a modified PB (mPB) equation as $\Lambda$ tends to infinity. Our results show that the limiting equation of PB-steric equation (as $\Lambda$ goes to infinity) is a mPB equation which has the same form (up to scalar multiples) as those mPB equations in \cite{1942bikerman,1997borukhov,2007kilic,2009li,2009li2,2013li,2011lu}. Therefore, the PB-steric equation can be regarded as a generalized model of mPB equations.

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arXiv2021

Near- and far-field expansions for stationary solutions of Poisson--Nernst--Planck equations

This work is concerned with the stationary Poisson--Nernst--Planck equation with a large parameter which describes a huge number of ions occupying an electrolytic region. Firstly, we focus on the model with a single specie of positive charges in one-dimensional bounded domains due to the assumption that these ions are transported in the same direction along a tubular-like mircodomain. We show that the solution asymptotically blows up in a thin region attached to the boundary, and establish the refined "near-field" and "far-field" expansions for the solutions with respect to the parameter. Moreover, we obtain the boundary concentration phenomenon of the net charge density, which mathematically confirms the physical description that the non-neutral phenomenon occurs near the charged surface. In addition, we revisit a nonlocal Poisson--Boltzmann model for monovalent binary ions and establish a novel comparison for these two models.

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arXiv2020

Global Existence of the Non-isothermal Poisson-Nernst-Planck-Fourier System

In this paper, we consider a non-isothermal electrokinetic model, which is derived from the Energetic Variational Approach. The charge transport is described through the Poisson-Nernst-Planck equations with variable temperature, and the heat flux satisfies the Fourier's law. This Poisson-Nernst-Planck-Fourier model satisfies both the first law and second law of thermodynamics as well as the Onsager's reciprocal relations, thus it is thermodynamic-consistent. Finally, we prove the global well-posedness for this model under the smallness assumption of the initial data by the energy method.

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arXiv2019

Semi-classical analysis with new Galilean transformations for a Gross--Pitaevskii system with non-zero conditions at infinity

Recently, a rich variety of the micro-phenomena of the superfluid passing an obstacle has been observed in the binary mixture of rotating Bose--Einstein condensates (BECs). Among such phenomena, the interaction of dark--bright solitons is one of the most important issues. In this work we investigate the semi-classical limit for a coupled system of Gross--Pitaevskii (GP) equations with rotating fields and trap potentials in a two-dimensional exterior domain, where the superfluid is non-vanishing at infinity. We establish a new Galilean type transformation and follow the argument of the modulated energy functional (a Lyapunov type functional) in \cite{ll08,lz05} to control the propagation of mass densities and linear momenta of the solution via a compressible Euler equation with Coriolis force in a semi-classical regime. Moreover, the effect of the rotating field on the superfluid in the region far away from the obstacle is precisely described.

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arXiv2016

The virial theorem and ground state energy estimate of nonlinear Schr\"odinger equations in $\mathbb{R}^2$ with square root and saturable nonlinearities in nonlinear optics

The virial theorem is a nice property for the linear Schrodinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrodinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems no way of getting any eigenvalue estimate. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimate of nonlinear Schrodinger (NLS) equations in R2 with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimate can be used to obtain the 2nd order term (which is of order $ln\Gamma$) of the lower bound of the ground state energy as the coefficient $\Gamma$ of the nonlinear term tends to infinity.

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arXiv2015

Exponential decay estimates for the stability of boundary layer solutions to Poisson-Nernst-Planck systems: one spatial dimension case

With a small parameter $ε$, Poisson-Nernst-Planck (PNP) systems over a finite one-dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, which profiles form boundary layers near boundary points and become at in the interior domain as $ε$ approaches zero. For the stability of 1D boundary layer solutions to (time-dependent) PNP systems, we estimate the solution of the perturbed problem with global electroneutrality. We prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially (in time) with exponent independent of $ε$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound. The main difficulty is that the gradients of 1D boundary layer solutions at boundary points may blow up as $ε$ tends to zero. The main idea of our argument is to transform the perturbed problem into another parabolic system with a new and useful energy law for the proof of the exponential decay estimate.

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arXiv2015

Boundary Layer Solutions of Charge Conserving Poisson-Boltzmann Equations: One-Dimensional Case

For multispecies ions, we study boundary layer solutions of charge conserving Poisson-Boltzmann (CCPB) equations [50] (with a small parameter ǫ) over a finite one-dimensional (1D) spatial domain, subjected to Robin type boundary conditions with variable coefficients. Hereafter, 1D boundary layer solutions mean that as ǫ approaches zero, the profiles of solutions form boundary layers near boundary points and become flat in the interior domain. These solutions are related to electric double layers with many applications in biology and physics. We rigorously prove the asymptotic behaviors of 1D boundary layer solutions at interior and boundary points. The asymptotic limits of the solution values(electric potentials) at interior and boundary points with a potential gap (related to zeta potential) are uniquely determined by explicit nonlinear formulas (cannot be found in classical Poisson-Boltzmann equations) which are solvable by numerical computations.

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arXiv2014

Transport of Charged Particles: Entropy Production and Maximum Dissipation Principle

In order to describe the dynamics of crowded ions (charged particles), we use an energetic variation approach to derive a modified Poisson-Nernst-Planck (PNP) system which includes an extra dissipation due to the effective velocity differences between ion species. Such a system is more complicated than the original PNP system but with the same equilibrium states. Using Schauder's fixed-point theorem, we develop a local existence theorem of classical solutions for the modified PNP system. Different dynamics (but same equilibrium states) between the original and modified PNP systems can be represented by numerical simulations using finite element method techniques.

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arXiv2014

Multiple solutions of steady-state Poisson-Nernst-Planck equations with steric effects

Experiments measuring currents through single protein channels show unstable currents. Channels switch between 'open' or 'closed' states in a spontaneous stochastic process called gating. Currents are either (nearly) zero or at a definite level, characteristic of each type of protein, independent of time, once the channel is open. The steady state Poisson-Nernst-Planck equations with steric effects (PNP-steric equations) describe steady current through the open channel quite well, in a wide variety of conditions. Here we study the existence of multiple solutions of steady state PNP-steric equations to see if they themselves, without modification or augmentation, can describe two levels of current. We prove that there are two steady state solutions of PNP-steric equations for (a) three types of ion species (two types of cations and one type of anion) with a positive constant permanent charge, and (b) four types of ion species (two types of cations and their counter-ions) with a constant permanent charge but no sign condition. The excess currents (due to steric effects) associated with these two steady state solutions are derived and expressed as two distinct formulas. Our results indicate that PNP-steric equations may become a useful model to study spontaneous gating of ion channels. Spontaneous gating is thought to involve small structural changes in the channel protein that perhaps produce large changes in the profiles of free energy that determine ion flow. Gating is known to be modulated by external structures. Both can be included in future extensions of our present analysis.

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arXiv2013

Diffusion limit of kinetic equations for multiple species charged particles

In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson-Nernst-Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov-Poisson-Fokker-Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.

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arXiv2012

Ground state of nonlinear Schrodinger systems with saturable nonlinearity

We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.

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arXiv2010

Orbital stability of bound states of nonlinear Schrodinger equations with linear and nonlinear optical lattices

We study the orbital stability and instability of single-spike bound states of semiclassical nonlinear Schrodinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develop mathematical theorems which are most general on the orbital stability problem.

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arXiv2007

Half-Skyrmions and Spike-Vortex Solutions of Two-Component Nonlinear Schrodinger Systems

Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of nonlinear Schrodinger equations (NLSE) describing a binary mixture of Bose-Einstein condensates. Besides, half-skyrmions characterized by half-integer topological charges can also be found in the nonlinear sigma model which is a model of the Bose-Einstein condensate of the Schwinger bosons. Here we prove rigorously the existence of half-skyrmions which may come from a new type of soliton solutions called spike-vortex solutions of two-component systems of NLSE on the entire plane. These spike-vortex solutions having spikes in one component and a vortex in the other component may form half-skyrmions. By Liapunov-Schmidt reduction process, we may find spike-vortex solutions of two-component systems of NLSE.

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arXiv2006

Symbiotic Bright Solitary Wave Solutions of Coupled Nonlinear Schrodinger Equations

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two and three dimensional domains, we prove the existence of least energy solutions and study the location and configuration of symbiotic bright solitons. We use Nehari's manifold to construct least energy solutions and derive their asymptotic behaviors by some techniques of singular perturbation problems.

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