arXiv subjects
Publications and source records attributed to Yanxiang Zhao.
In this paper we study the global well-posedness of the Allen-Cahn Ohta-Nakazawa model with two fixed nonlinear volume constraints. Utilizing the gradient flow structure of its free energy, we prove the existence and uniqueness of the solution by following De Giorgi's minimizing movement scheme in a novel way.
We study some maximum principle preserving and energy stable schemes for the Allen-Cahn-Ohta-Kawasaki model with fixed volume constraint. With the inclusion of a nonlinear term in the Ohta-Kawasaki free energy functional, we show that the Allen-Cahn-Ohta-Kawasaki dynamics is maximum principle preserving. We further design some first order energy stable numerical schemes which inherit the maximum principle preservation in both semi-discrete and fully-discrete levels. Furthermore, we apply the maximum principle preserving schemes to a general framework for binary systems with long-range interactions. We also present some numerical results to support our theoretical findings.
We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+Δu +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0, x \in \mathbb{R}^d$. First, we consider the $L^2$-critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate but also the log-log correction (via asymptotic analysis and functional fitting). In this setting we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled $Q$, a ground state solution of the elliptic equation $-ΔQ+Q- \left(|x|^{-(d-2)} \ast |Q|^p \right) |Q|^{p-2} Q =0$. We also consider the $L^2$-supercritical case in dimensions d=3,4. We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS $L^2$-supercritical regime, the profile equation exhibits branches of non-oscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ODE is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the $Q_{1,0}$ is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level $10^{-5}$, and thus, numerically observable (unlike the $L^2$-critical case). In summary, we find that the results are similar to the behavior of stable blowup dynamics in the corresponding NLS settings. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in stable blow-up.
We study stable blow-up dynamics in the $L^2$-supercritical nonlinear Schrödinger equation in various dimensions. We first investigate the profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4] on the existence and local uniqueness of solutions of the cubic profile equation to other $L^2$-supercritical nonlinearities and dimensions $d \geq 2$. We then numerically observe the multi-bump structure of such solutions, and in particular, exhibit the $Q_{1,0}$ solution, a candidate for the stable blow-up profile. Next, using the dynamic rescaling method, we investigate stable blow-up solutions in the $L^2$-supercritical NLS and confirm the square root rate of the blow-up as well as the convergence of blow-up profiles to the $Q_{1,0}$ profile.
We study stable blow-up dynamics in the $L^2$-critical nonlinear Schrödinger equation in high dimensions. First, we show that in dimensions $d=4$ to $d=12$ generic blow-up behavior confirms the "log-log" regime in our numerical simulations, including the log-log rate and the convergence of the blow-up profiles to the rescaled ground state; this matches the description of the stable blow-up regime in the dimension $d =2$ (for the 2d cubic NLS equation). Next, we address the question of rigorous justification of the "log-log" dynamics in higher dimensions ($d \geq5)$, at least for the initial data with the mass slightly larger than the mass of the ground state, for which the spectral conjecture has yet to be proved, see [34] and [10]. We give a numerically-assisted proof of the spectral property for the dimensions from $d=5$ to $d=12$, and a modification of it in dimensions $2 \leq d \leq 12$. This, combined with previous results of Merle-Raphaël, proves the "log-log" stable blow-up regime in dimensions $d \leq 10$ and radially stable for $d \leq 12$.
Adhesive cell-substrate interactions are crucial for cell motility and are responsible for the necessary traction that propels cells. These interactions can also change the shape of the cell, analogous to liquid droplet wetting on adhesive substrates. To address how these shape changes affect cell migration and cell speed we model motility using deformable, 2D cross-sections of cells in which adhesion and frictional forces between cell and substrate can be varied separately. Our simulations show that increasing the adhesion results in increased spreading of cells and larger cell speeds. We propose an analytical model which shows that the cell speed is inversely proportional to an effective height of the cell and that increasing this height results in increased internal shear stress. The numerical and analytical results are confirmed in experiments on motile eukaryotic cells.
In this paper, we propose a first order energy stable linear semi-implicit method for solving the Allen-Cahn-Ohta-Kawasaki equation. By introducing a new nonlinear term in the Ohta-Kawasaki free energy functional, all the system forces in the dynamics are localized near the interfaces which results in the desire hyperbolic tangent profile. In our numerical method, the time discretization is done by some stabilization technique in which some extra nonlocal but linear term is introduced and treated explicitly together with other linear terms, while other nonlinear and nonlocal terms are treated implicitly. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are proved for this method in both semi-discretization and full discretization levels. Numerical experiments indicate the force localization and desire hyperbolic tangent profile due to the new nonlinear term. We test the first order temporal convergence rate of the proposed scheme. We also present hexagonal bubble assembly as one type of equilibrium for the Ohta-Kawasaki model. Additionally, the two-third law between the number of bubbles and the strength of long-range interaction is verified which agrees with the theoretical studies.
We construct a new phase-field model for the solvation of charged molecules with a variational implicit solvent. Our phase-field free-energy functional includes the surface energy, solute-solvent van der Waals dispersion energy, and electrostatic interaction energy that is described by the Coulomb-field approximation, all coupled together self-consistently through a phase field. By introducing a new phase-field term in the description of the solute-solvent van der Waals and electrostatic interactions, we can keep the phase-field values closer to those describing the solute and solvent regions, respectively, making it more accurate in the free-energy estimate. We first prove that our phase-field functionals $Γ$-converge to the corresponding sharp-interface limit. We then develop and implement an efficient and stable numerical method to solve the resulting gradient-flow equation to obtain equilibrium conformations and their associated free energies of the underlying charged molecular system. Our numerical method combines a linear splitting scheme, spectral discretization, and exponential time differencing Runge-Kutta approximations. Applications to the solvation of single ions and a two-plate system demonstrate that our new phase-field implementation improves the previous ones by achieving the localization of the system forces near the solute-solvent interface and maintaining more robustly the desirable hyperbolic tangent profile for even larger interfacial width. This work provides a scheme to resolve the possible unphysical feature of negative values in the phase-field function found in the previous phase-field modeling (cf. H. Sun, et al. J. Chem. Phys., 2015) of charged molecules with the Poisson--Boltzmann equation for the electrostatic interaction.
A nonlocal diffuse interface model is used to study bubble assemblies in ternary systems. As model parameters vary, a large number of morphological phases appear as stable stationary states. One open question related to the polarity direction of double bubble assemblies is answered numerically. Moreover, the average size of bubbles in a single bubble assembly depends on the sum of the minority constituent areas and the long range interaction coefficients. One identifies the ranges for area fractions and the long range interaction coefficients for double bubble assemblies.
We study a minimal model of a crawling eukaryotic cell with a chemical polarity controlled by a reaction-diffusion mechanism describing Rho GTPase dynamics. The size, shape, and speed of the cell emerge from the combination of the chemical polarity, which controls the locations where actin polymerization occurs, and the physical properties of the cell, including its membrane tension. We find in our model both highly persistent trajectories, in which the cell crawls in a straight line, and turning trajectories, where the cell transitions from crawling in a line to crawling in a circle. We discuss the controlling variables for this turning instability, and argue that turning arises from a coupling between the reaction-diffusion mechanism and the shape of the cell. This emphasizes the surprising features that can arise from simple links between cell mechanics and biochemistry. Our results suggest that similar instabilities may be present in a broad class of biochemical descriptions of cell polarity.
The simulation of quantum transport in a realistic, many-particle system is a nontrivial problem with no quantitatively satisfactory solution. While real-time propagation has the potential to overcome the shortcomings of conventional transport methods, this approach is prone to finite size effects that are associated with modeling an open system on a closed spatial domain. Using a master equation framework, we exploit an equivalence between the superoperators coupling an open system to external particle reservoirs and non-Hermitian terms defined at the periphery of a quantum device. By taking the mean-field limit, the equation of motion for the single-particle reduced density matrix becomes equivalent to real-time time-dependent density functional theory in the presence of imaginary source and sink potentials. This method may be used to converge nonequilibrium steady states for a many-body quantum system using a previously reported constraint algorithm.
The accurate simulation of real--time quantum transport is notoriously difficult, requiring a consistent scheme to treat incoming and outgoing fluxes at the boundary of an open system. We demonstrate a method to converge non--equilibrium steady states using non--Hermitian source and sink potentials alongside the application of dynamic density constraints during wavefunction propagation. This scheme adds negligible cost to existing computational methods while exhibiting exceptional stability and numerical accuracy.
We extend a model for the morphology and dynamics of a crawling eukaryotic cell to describe cells on micropatterned substrates. This model couples cell morphology, adhesion, and cytoskeletal flow in response to active stresses induced by actin and myosin. We propose that protrusive stresses are only generated where the cell adheres, leading to the cell's effective confinement to the pattern. Consistent with experimental results, simulated cells exhibit a broad range of behaviors, including steady motion, turning, bipedal motion, and periodic migration, in which the cell crawls persistently in one direction before reversing periodically. We show that periodic motion emerges naturally from the coupling of cell polarization to cell shape by reducing the model to a simplified one-dimensional form that can be understood analytically.
In this work, we study the adhesion of multi-component vesicle membrane to both flat and curved substrates, based on the conventional Helfrich bending energy for multi-component vesicles and adhesion potentials of different forms. A phase field formulation is used to describe the different components of the vesicle. For the axisymmetric case, a number of representative equilibrium vesicle shapes are computed and some energy diagrams are presented which reveal the dependence of the calculated shapes and solution branches on various parameters including both bending moduli and spontaneous curvatures as well as the adhesion potential constants. Our computation also confirms a recent experimental observation that the adhesion effect may promote phase separation in two-component vesicle membranes.