Showing papers by "Chun Liu published in 2016"

Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension Three

Abstract: In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, while the second example is constructed for any generic initial data \({(u_0,d_0)}\) that has sufficiently small energy, and \({d_0}\) has a nontrivial topology.

The derivation and approximation of coarse-grained dynamics from Langevin dynamics

Abstract: We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain. These approximations offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient.

A Generalized Poisson--Nernst--Planck--Navier--Stokes Model on the Fluid with the Crowded Charged Particles: Derivation and Its Well-Posedness

Abstract: We derive a hydrodynamic model of the compressible conductive fluid by using an energetic variational approach, which could be called a generalized Poisson--Nernst--Planck--Navier--Stokes system. This system characterizes the micro-macro interactions of the charged fluid and the mutual friction between the crowded charged particles. In particular, it reveals the cross-diffusion phenomenon which does not happen in the fluid with the dilute charged particles. The cross-diffusion is tricky; however, we develop a general method to show that the system is globally asymptotically stable under small perturbations around a constant equilibrium state. Under some conditions, we also obtain the optimal decay rates of the solution and its derivatives of any order. Our method will apply equally well to a class of cross-diffusion systems if their linearized diffusion matrices are diagonally dominant.

Boundary layer solutions of charge conserving Poisson–Boltzmann equations: One-dimensional case

Abstract: For multispecies ions, we study boundary layer solutions of charge conserving Poisson-Boltzmann (CCPB) equations [50] (with a small parameter \k{o}) 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 \k{o} 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.

From generalized Langevin equations to Brownian dynamics and embedded Brownian dynamics

Abstract: We present the reduction of generalized Langevin equations to a coordinate-only stochastic model, which in its exact form involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.

Existence of weak solutions to an evolutionary model for magnetoelasticity

Abstract: We prove existence of weak solutions to an evolutionary model derived for magnetoelastic materials. The model is phrased in Eulerian coordinates and consists in particular of (i) a Navier-Stokes equation that involves magnetic and elastic terms in the stress tensor obtained by a variational approach, of (ii) a regularized transport equation for the deformation gradient and of (iii) the Landau-Lifshitz-Gilbert equation for the dynamics of the magnetization. The proof is built on a Galerkin method and a fixed-point argument. It is based on ideas from F.-H. Lin and the third author for systems modeling the flow of liquid crystals as well as on methods by G. Carbou and P. Fabrie for solutions of the Landau-Lifshitz equation.

The Derivation and Approximation of Coarse-grained Dynamics from Langevin Dynamics

Abstract: We present a derivation of a coarse-grained model from the Langevin dynamics The focus is placed on the memory kernel function and the fluctuation-dissipation theorem Also presented is an hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain These approximations offer increasing accuracy More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step

From Generalized Langevin Equations to Brownian Dynamics and Embedded Brownian Dynamics

Abstract: We present the reduction of generalized Langevin equations to a coordinate-only stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.

The Poisson Boltzmann equation and the charge separation phenomenon at the silica-water interface: A holistic approach

Abstract: The Poisson Boltzmann equation is known for its success in describing the Debye layer that arises from the charge separation phenomenon at the silica/water interface. However, by treating only the mobile ionic charges in the liquid, the Poisson Boltzmann equation accounts for only half of the electrical double layer, with the other half, the surface charge layer, being beyond its computational domain. In this work, we take a holistic approach to the charge separation phenomenon at the silica/water interface by treating, within a single computational domain, the electrical double layer that comprises both the mobile ions in the liquid and the surface charge density. The Poisson Nernst Planck equations are used as the rigorous basis for our methodology. The holistic approach has the advantage of being able to predict surface charge variations that arise either from the addition of salt and acid to the liquid, or from the decrease of the liquid channel width to below twice the Debye length. As the electrical double layer must be overall neutral, we use this constraint to derive both the form of the static limit of the Poisson Nernst Planck equations, as well as a global chemical potential that replaces the classical zeta potential as the boundary value for the PB equation, which can be re-derived from our formalism. We present several predictions of our theory that are beyond the framework of the PB equation alone, e.g., the surface capacitance and the so-called pK and pL values, the isoelectronic point at which the surface charge layer is neutralized, and the appearance of a Donnan potential that arises from the formation of an electrical double layer at the inlet regions of a nano-channel connected to the bulk reservoir. All theory predictions are shown to be in good agreement with the experimental observations.

The Poisson–Boltzmann equation and the charge separation phenomenon at the silica-water interface: A holistic approach

Abstract: The Poisson-Boltzmann (PB) equation is well known for its success in describing the Debye layer that arises from the charge separation phenomenon at the silica-water interface. However, by treating only the mobile ionic charges in the liquid, the PB equation essentially accounts for only half of the electrical double layer, with the other half—the surface charge layer—being beyond the PB equation’s computational domain. In this work, we take a holistic approach to the charge separation phenomenon at the silica-water interface by treating, within a single computational domain, the electrical double layer that comprises both the mobile ions in the liquid and the surface charge density. The Poisson-Nernst-Planck (PNP) equations are used as the rigorous basis for our methodology. This holistic approach has the inherent advantage of being able to predict surface charge variations that arise either from the addition of salt and acid to the liquid, or from the decrease of the liquid channel width to below twice the Debye length. The latter is usually known as the charge regulation phenomenon. We enumerate the “difficulty” of the holistic approach that leads to the introduction of a surface potential trap as the single physical input to drive the charge separation within the computational domain. As the electrical double layer must be overall neutral, we use this constraint to derive both the form of the static limit of the PNP equations, as well as a global chemical potential μ that is