arXiv subjects

Weishi Liu

Publications and source records attributed to Weishi Liu.

13 recordsLinked to original sources

arXiv2026

Bifurcation and stability of stationary shear flows of Ericksen-Leslie model for nematic liquid crystals

In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of the solutions of an algebraic equation for a cusp case. This one-to-one correspondence is established essentially based on the treatment in the work of Jiao, et. al. [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}] for a different case, and the relation gives directly parameter ranges for existence of multiple stationary solutions; in particular, multiple stationary solutions are created through countably many saddle-node bifurcations for the algebraic equation at critical shear speeds. The main result of the paper is on the stability of stationary solutions associated to the bifurcations; more precisely, (i) for each critical shear speed, there is a unique stationary solution and, for smaller shear speed, the stationary solution disappears but, for larger shear speed, two stationary solutions nearby bifurcate; (ii) more importantly, under a generic condition, there is a simple zero eigenvalue for the linearization of the shear flow at the critical stationary solution and, for larger shear speed, the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution.

math.DS↗

arXiv2024

"Hidden" mechanisms for Gouy-Chapman layer and other critical features via Poisson-Boltzmann equations

In this work a dynamical system approach is taken to systematically investigate the one-dimensional classical Poisson-Boltzmann (PB) equation with various boundary conditions. This framework, particularly, the phase space portrait, has a unique advantage of a geometric view of the dynamical systems, which allows one to reveal and examine critical features of the PB models. More specifically, we are able to reveal the mechanism of Gouy-Chapman layer: the presence of an {\em equilibrium} for the PB equation, including equilibrium-at-infinity for Gouy-Chapman's original setup as the limiting case. Several other critical, somehow counterintuitive, features revealed in this work are the saturation phenomenon of surface charge density, the uniform boundedness of electric pressure (given length) and of length (given electric pressure) in surface charge, and the critical length for a reversal of electric force direction. All have a common mechanism: the presence of an equilibrium for the PB equation. We believe that the critical features presented from the classical PB models persist for modified PB systems up to a certain degree. On the other hand, any qualitative change in these features as the sophistication of the model is increasing is an indication of new phenomena with new mechanisms.

math.CA↗

arXiv2023

Flux ratios for effects of permanent charges on ionic flows with three ion species: Case study (II)

In this paper, we study effects of permanent charges on ion flows through membrane channels via a quasi-one-dimensional classical Poisson-Nernst-Planck system. This system includes three ion species, two cations with different valences and one anion, and permanent charges with a simple structure, zeros at the two end regions and a constant over the middle region. For small permanent charges, our main goal is to analyze the effects of permanent charges on ionic flows, interacting with the boundary conditions and channel structure. Continuing from a previous work, we investigate the problem for a new case toward a more comprehensive understanding about effects of permanent charges on ionic fluxes.

cond-mat.soft↗

arXiv2023

The Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals: The general Case

In this work, we study the Cauchy problem of Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of two partial differential equations: One is a quasi-linear wave equation for the director field representing the crystallization of the nematics, and the other is a parabolic PDE for the velocity field characterizing the liquidity of the material. We extend the work in [Chen, et. al. {\em Arch. Ration. Mech. Anal.} {\bf 236} (2020), 839-891] for a special case to the general physical setup. The Cauchy problem is shown to have global solutions beyond singularity formation. Among a number of progresses made in this paper, a particular contribution is a systematic treatment of a parabolic PDE with only Hölder continuous diffusion coefficient and rough (worse than Hölder) nonhomogeneous terms.

math.AP↗

arXiv2022

Local Hard-Sphere Poisson-Nernst-Planck Models for Ionic Channels with Permanent Charges

The main goal of this work is to examine the qualitative effect of ion sizes via a steady-state boundary value problem. We study a one-dimensional version of a Poisson-Nernst-Planck system with a local hard-sphere potential model for ionic flow through a membrane channel with fixed boundary ion concentrations and electric potentials. A complete set of integrals for the inner system is illustrated that delivers information for boundary and internal layers. In addition, a group of simultaneous equations appears in the construction of singular orbits. The research aims to set up a simple formation defined by the profile of permanent charges with two mobile ion species, one positively charged, cation, and one negatively charged, anion. A local hard-sphere potential that depends pointwise on ion concentrations is included in the model to estimate ion-size impacts on the ionic flow. The analysis is built on the geometric singular perturbation theory, particularly on specific structures of this concrete model. For 1:1 ionic mixtures, we first conduct precise mathematical analysis and derive a matching system of nonlinear algebraic equations. We then extend the results by directing on a critical case where the current is zero. Treating the ion sizes as small parameters, we derive an approximation of zero-current fluxes for general values of permanent charge. We then focus on small values of permanent charge to obtain more concrete outputs. We will also examine ion size effects on the flow rate of matter for the zero-current case.

math-ph↗

arXiv2020

Reversal potential and reversal permanent charge with unequal diffusion coefficients via classical Poisson-Nernst-Planck models

In this paper, based on geometric singular perturbation analysis of a quasi-one dimensional Poisson-Nernst-Planck model for ionic flows, we study the problem of zero current condition for ionic flows through membrane channels with a simple profile of permanent charges. For ionic mixtures of multiple ion species, under equal diffusion constant condition, Eisenberg, et al [{\em Nonlinearity {\bf 28} (2015), 103-128}] derived a system of two equations for determining the reversal potential and an equation for the reversal permanent charge. The equal diffusion constant condition is significantly degenerate from physical points of view. For unequal diffusion coefficients, the analysis becomes extremely challenging. This work will focus only on two ion species, one positively charged (cation) and one negatively charged (anion), with two arbitrary diffusion coefficients. Dependence of reversal potential on channel geometry and diffusion coefficients has been investigated experimentally, numerically, and analytically in simple setups, in many works. In this paper, we identify two governing equations for the zero current, which enable one to mathematically analyze how the reversal potential depends on the channel structure and diffusion coefficients. In particular, we are able to show, with a number of concrete results, that the possible different diffusion constants indeed make significant differences. The inclusion of channel structures is also far beyond the situation where the Goldman-Hodgkin-Katz (GHK) equation might be applicable. A comparison of our result with the GHK equation is provided. The dual problem of reversal permanent charges is briefly discussed too.

math.DS↗

arXiv2020

Permanent charge effects on ionic flow: a numerical study of flux ratios and their bifurcation

Ionic flow carries electrical signals for cells to communicate with each other. The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flow properties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planck model for small and relatively large permanent charges. The analytical studies have led to the introduction of flux ratios that reflect permanent charge effects and have a universal property. The studies also show that the flux ratios have different behaviors for small and large permanent charges. However, the existing analytical techniques can reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios to bridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. In particular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numerical techniques. An adaptive moving mesh finite element method is used in the numerical studies.

math.NA↗

arXiv2019

Effects of diffusion coefficients on reversal potentials in ionic channels

In this work, the dependence of reversal potentials and zero-current fluxes on diffusion coefficients are examined for ionic flows through membrane channels. The study is conducted for the setup of a simple structure defined by the profile of permanent charges with two mobile ion species, one positively charged (cation) and one negatively charged (anion). Numerical observations are obtained from analytical results established using geometric singular perturbation analysis of classical Poisson-Nernst-Planck models. For 1:1 ionic mixtures with arbitrary diffusion coefficients, Mofidi and Liu [arXiv:1909.01192] conducted a rigorous mathematical analysis and derived an equation for reversal potentials that, in its particular case, can be compared to Goldman-Hodgkin-Katz equation. We summarize and extend these results with numerical observations for biological relevant situations. The numerical investigations on profiles of the electrochemical potentials, ion concentrations, and electrical potential across ion channels are also presented for the zero-current case. Moreover, the behavior of current and fluxes with respect to voltages and permanent charges are investigated. In the opinion of the authors, many results in the paper are not intuitive, and it is difficult, if not impossible, to see all cases without investigations of this type.

cond-mat.soft↗

arXiv2019

Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model

In this paper, we study the Cauchy problem of the Poiseuille flow of full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities - blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen-Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are Hölder continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation - the flux density remains uniformly bounded while its two components approach infinity at formations of cusp singularities.

math.AP↗

arXiv2017

An effect of large permanent charge: Decreasing flux to zero with increasing transmembrane potential to infinity

In this work, we examine effects of large permanent charges on ionic flow through ion channels based on a quasi-one dimensional Poisson-Nernst-Planck model. It turns out large positive permanent charges inhibit the flux of cation as expected, but strikingly, as the transmembrane electrochemical potential for anion increases in a particular way, the flux of anion decreases. The latter phenomenon was observed experimentally but the cause seemed to be unclear. The mechanisms for these phenomena are examined with the help of the profiles of the ionic concentrations, electric fields and electrochemical potentials. The underlying reasons for the near zero flux of cation and for the decreasing flux of anion are shown to be different over different regions of the permanent charge. Our model is oversimplified. More structural detail and more correlations between ions can and should be included. But the basic finding seems striking and important and deserving of further investigation.

q-bio.QM↗

arXiv2017

Relative dielectric constants and selectivity ratios in open ionic channels

We investigate the effects of the relative dielectric coefficient on ionic flows in open ion channels, using mathematical analysis of reasonably general Poisson-Nernst-Planck type models that can include the finite sizes of ions. The value of the relative dielectric coefficient is of course a crucial parameter for ionic behavior in general. Using the powerful theory of singularly perturbed problems in applied mathematics, we show that some properties of open channels are quite insensitive to variation in the relative dielectric coefficient, thereby explaining such effects seen unexpectedly in simulations. The ratio between the total number of one ion species and that of another ion species, and the ratio between the flux of one ion species and that of another ion species do not depend significantly on the relative dielectric coefficient.

q-bio.BM↗

arXiv2016

Flux ratios and channel structures

We investigate Ussing's unidirectional fluxes and flux ratios of charged tracers motivated particularly by the insightful proposal of Hodgkin and Keynes on a relation between flux ratios and channel structure. Our study is based on analysis of quasi-one-dimensional Poisson-Nernst-Planck type models for ionic flows through membrane channels. This class of models includes the Poisson equation that determines the electrical potential from the charges present and is in that sense consistent. Ussing's flux ratios generally depend on all physical parameters involved in ionic flows, particularly, on bulk conditions and channel structures. Certain setups of ion channel experiments result in flux ratios that are universal in the sense that their values depend on bulk conditions but not on channel structures; other setups lead to flux ratios that are specific in the sense that their values depend on channel structures too. Universal flux ratios could serve some purposes better than specific flux ratios in some circumstances and worse in other circumstances. We focus on two treatments of tracer flux measurements that serve as estimators of important properties of ion channels. The first estimator determines the flux of the main ion species from measurements of the flux of its tracer. Our analysis suggests a better experimental design so that the flux ratio of the tracer flux and the main ion flux is universal. The second treatment of tracer fluxes concerns ratios of fluxes and experimental setups that try to determine some properties of channel structure. We analyze the two widely used experimental designs of estimating flux ratios and show that the most widely used method depends on the spatial distribution of permanent charge so this flux ratio is specific and thus allows estimation of (some of) the properties of that permanent charge, even with ideal ionic solutions. ...

q-bio.OT↗

arXiv2009

Poisson-Nernst-Planck Systems for Narrow Tubular-like Membrane Channels

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.

math.AP↗