Journal ArticleActa Applicandae Mathematicae · December 1, 2024
Linear response theory is a fundamental framework studying the macroscopic response of a physical system to an external perturbation. This paper focuses on the rigorous mathematical justification of linear response theory for Langevin dynamics. We give som ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · July 1, 2024
We formulate a class of mean field games on a finite state space with variational principles resembling those in continuous-state mean field games. We construct a controlled continuity equation featuring a nonlinear activation function on graphs induced by ...
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ConferenceMed Phys · May 2024
BACKGROUND: Delta radiomics is a high-throughput computational technique used to describe quantitative changes in serial, time-series imaging by considering the relative change in radiomic features of images extracted at two distinct time points. Recent wo ...
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Journal ArticleJournal of Scientific Computing · May 1, 2024
The dynamical equation of the boundary vorticity has been obtained, which shows that the viscosity at a solid wall is doubled as if the fluid became more viscous at the boundary. For certain viscous flows the boundary vorticity can be determined via the dy ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · April 1, 2024
We study the existence of weak solutions to the p-Navier-Stokes equations with a symmetric p-Laplacian on bounded domains. We construct a particular Schauder basis in W01, p(Ω) with divergence free constraint and prove existence of weak solutions using the ...
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Journal ArticleSIAM Journal on Numerical Analysis · January 1, 2024
We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we pr ...
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Journal ArticleStochastics and Dynamics · January 1, 2024
We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-conca ...
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Journal ArticleChaos (Woodbury, N.Y.) · October 2023
We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are close ...
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Journal ArticleMed Phys · June 2023
BACKGROUND: Due to intrinsic differences in data formatting, data structure, and underlying semantic information, the integration of imaging data with clinical data can be non-trivial. Optimal integration requires robust data fusion, that is, the process o ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · March 1, 2023
In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we const ...
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Journal ArticleMultiscale Modeling and Simulation · March 1, 2023
We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the tr ...
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Journal ArticleChaos (Woodbury, N.Y.) · February 2023
A new efficient ensemble prediction strategy is developed for a multiscale turbulent model framework with emphasis on the nonlinear interactions between large and small-scale variables. The high computational cost in running large ensemble simulations of h ...
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Journal ArticleApplied and Computational Harmonic Analysis · January 1, 2023
We study the Langevin dynamics of a physical system with manifold structure M⊂Rp based on collected sample points {xi}i=1n⊂M that probe the unknown manifold M. Through the diffusion map, we first learn the reaction coordinates {yi}i=1n⊂N corresponding to { ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2023
The Poisson-Nernst-Planck-Bikermann (PNPB) model, in which the ions and water molecules are treated as different species with non-uniform sizes and valences with interstitial voids, can describe the steric and correlation effects in ionic solution neglecte ...
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Journal ArticleCommunications in Computational Physics · January 1, 2023
Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifold, us ...
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Journal ArticleJournal of Statistical Physics · November 1, 2022
Most biochemical reactions in living cells are open systems interacting with environment through chemostats to exchange both energy and materials. At a mesoscopic scale, the number of each species in those biochemical reactions can be modeled by a random t ...
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Journal ArticleNumerische Mathematik · November 1, 2022
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the mac ...
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Journal ArticleCommunications in Computational Physics · July 1, 2022
We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a tru ...
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Journal ArticleKinetic and Related Models · June 1, 2022
This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion me ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · January 1, 2022
This paper investigates the global existence of weak solutions for the incompressible p-Navier-Stokes equations in Rd (2 ≤ d ≤ p). The pNavier-Stokes equations are obtained by adding viscosity term to the p-Euler equations. The diffusion added is represent ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2022
In this work, the primary goal is to establish a rigorous connection between the Fokker-Planck equation of neural networks and its microscopic model: the diffusion-jump stochastic process that captures the mean-field behavior of collections of neurons in t ...
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Journal ArticleSIAM Journal on Scientific Computing · January 1, 2022
The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After i ...
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Journal ArticleCommunications in Computational Physics · January 1, 2022
Although interest in numerical approximations of the water wave equation grows in recent years, the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In practice of water wave simulations, the tradeoff b ...
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Journal ArticlePhysica D: Nonlinear Phenomena · December 15, 2021
We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lin ...
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Journal ArticleMathematical Neuroscience and Applications · November 30, 2021
In the mean field integrate-and-fire model, the dynamics of a typical neuron
within a large network is modeled as a diffusion-jump stochastic process whose
jump takes place once the voltage reaches a threshold. In this work, the main
goal is to est ...
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Journal ArticleNonlinearity · November 1, 2021
We focus on the existence and rigidity problems of the vectorial Peierls- Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-l ...
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Journal ArticleMed Phys · July 2021
PURPOSE: This study investigated the prognostic potential of intra-treatment PET radiomics data in patients undergoing definitive (chemo) radiation therapy for oropharyngeal cancer (OPC) on a prospective clinical trial. We hypothesized that the radiomic ex ...
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Journal ArticleJournal of Computational Physics · May 15, 2021
In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unc ...
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Journal ArticleInterfaces and Free Boundaries · January 1, 2021
We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. U ...
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Journal ArticleInverse Problems & Imaging · 2021
<p style='text-indent:20px;'>We propose an equilibrium-driven deformation algorithm (EDDA) to simulate the inbetweening transformations starting from an initial image to an equilibrium image, which covers images varying from a g ...
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Journal ArticleSIAM-ASA Journal on Uncertainty Quantification · January 1, 2021
The generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but t ...
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Journal ArticleSIAM Journal on Applied Mathematics · January 1, 2021
At the continuous level, we consider two types of tumor growth models: the cell density model, based on the fluid mechanical construction, is more favorable for scientific interpretation and numerical simulations, and the free boundary model, as the incomp ...
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Journal ArticleCommunications in Computational Physics · January 1, 2021
In this paper, we consider the field model for complex ionic fluids with an energy variational structure, and analyze the well-posedness to this model with regularized kernels. Furthermore, we deduce the estimate of the maximal density function to quantify ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2021
In this paper we study a cross-diffusion system whose coefficient matrix is non-symmetric and degenerate. The system arises in the study of tissue growth with autophagy. The existence of a weak solution is established. We also investigate the limiting beha ...
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Journal ArticlePure and Applied Analysis · January 1, 2021
We analytically and numerically study a fourth-order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long-time dynamics for the PDE model. The PDE, originally derived by Katsevic ...
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Journal ArticleJournal of Statistical Physics · December 1, 2020
We rigorously justify the mean-field limit of an N-particle system subject to Brownian motions and interacting through the Newtonian potential in R3. Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized ...
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Journal ArticleNonlinearity · August 1, 2020
We study a 4th order degenerate parabolic PDE model in one-dimension with a 2nd order correction modeling the evolution of a crystal surface under the influence of both thermal fluctuations and evaporation/deposition effects. First, we provide a non-rigoro ...
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Journal ArticleQuarterly of Applied Mathematics · May 21, 2020
We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex ...
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Journal ArticleSIAM Journal on Numerical Analysis · March 16, 2020
We consider in this work the convergence of the random batch method proposed in our previous work [Jin et al., J. Comput. Phys., 400(2020), 108877] for interacting particles to the case of disparate species and weights. We show that the strong error is of ...
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Journal ArticleCommunications in Applied Mathematics and Computational Science · January 1, 2020
We propose in this work RBM-SVGD, a stochastic version of the Stein variational gradient descent (SVGD) method for efficiently sampling from a given probability measure, which is thus useful for Bayesian inference. The method is to apply the random batch m ...
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Journal ArticleJournal of Computational Physics · January 1, 2020
We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N2) per time step to O(N), for a system wit ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2020
Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential equations into the ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2020
We translate a coagulation-fragmentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the nonlinear evolution equation with a Markov jump process of a type introduced in class ...
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Journal ArticleMathematics of Computation · January 1, 2020
We revisit some standard schemes, including upwind schemes and some B-schemes, for linear conservation laws from the viewpoint of jump processes, allowing the study of them using probabilistic tools. For Fokker-Planck equations on R, in the case of weak co ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · January 1, 2020
In this paper, we revisit the mathematical validation of the Peierls–Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocat ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2020
. In this note, we study the Nernst-Planck-Navier-Stokes system for the transport and diffusion of ions in electrolyte solutions. The key feature is to establish three energy-dissipation equalities. As their direct consequence, we obtain global existence f ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2020
We study a class of fourth-order nonlinear parabolic equations which include the thinfilm equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type [Fourmula pres ...
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Journal ArticleJournal of Mathematical Physics · November 1, 2019
In this paper, we consider the mean field limit of Brownian particles with Coulomb repulsion in 3D space using compactness. Using a symmetrization technique, we are able to control the singularity and prove that the limit measure almost surely is a weak so ...
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Journal ArticleChinese Annals of Mathematics. Series B · November 1, 2019
Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ide ...
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Journal ArticleNonlinearity · October 4, 2019
We study local-time well-posedness and breakdown for solutions of regularized Saint-Venant equations (regularized classical shallow water equations) recently introduced by Clamond and Dutykh. The system is linearly non-dispersive, and smooth solutions cons ...
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Journal ArticleCalculus of Variations and Partial Differential Equations · October 1, 2019
As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as ...
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Journal ArticleSci Rep · August 8, 2019
Contemporary medical imaging is becoming increasingly more quantitative. The emerging field of radiomics is a leading example. By translating unstructured data (i.e., images) into structured data (i.e., imaging features), radiomics can potentially characte ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · July 1, 2019
In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law vi ...
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Journal ArticleIEEE Transactions on Geoscience and Remote Sensing · July 1, 2019
In this paper, full mechanisms of dissipation and dispersion in poro-viscoelastic media are accurately simulated in time domain. Specifically, four Q values are first proposed to depict a poro-viscoelastic medium: two for the attenuation of the bulk and sh ...
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Journal ArticleJournal of Statistical Physics · June 30, 2019
In a recent study of certain merging-splitting models of animal-group size (Degond et al. in J Nonlinear Sci 27(2):379–424, 2017), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresp ...
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Journal ArticlePhysica D: Nonlinear Phenomena · June 1, 2019
In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 di ...
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Journal ArticlePhysica D: Nonlinear Phenomena · March 1, 2019
In this paper, we study traveling wave solutions and peakon weak solutions of the modified Camassa–Holm (mCH) equation with dispersive term 2kux for k∈R. We study traveling wave solutions through a Hamiltonian system obtained from the mCH equation by using ...
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Journal ArticlePhys Med Biol · January 8, 2019
The purpose of this work was to investigate the potential relationship between radiomic features extracted from pre-treatment x-ray CT images and clinical outcomes following stereotactic body radiation therapy (SBRT) for non-small-cell lung cancer (NSCLC). ...
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Journal ArticleMathematical Models and Methods in Applied Sciences · January 1, 2019
In this paper, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation and Brownian diffusion. Specifically, we construct an estimator with partial observed data to approximate the diffusion parameter , and th ...
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Journal ArticleProceedings. Mathematical, physical, and engineering sciences · January 2019
A compact Green's function for general dispersive anisotropic poroelastic media in a full-frequency regime is presented for the first time. First, starting in a frequency domain, the anisotropic dispersion is exactly incorporated into the constitutive rela ...
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Journal ArticleESAIM - Control, Optimisation and Calculus of Variations · January 1, 2019
In this work, we study a fourth order exponential equation, ut = Δe-Δu derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong so ...
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Journal ArticleInterfaces and Free Boundaries · January 1, 2019
In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE ht D eh in the whole space Rdx . We assume the initial data is of medium size in the Wiener algebra A.Rd /; we ...
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Journal ArticleQuarterly of Applied Mathematics · January 1, 2019
This paper introduces a novel data clustering algorithm based on Langevin dynamics, where the associated potential is constructed directly from the data. To introduce a self-consistent potential, we adopt the potential model from the established Quantum Cl ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2019
We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's b ...
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ConferenceSpringer Proceedings in Mathematics and Statistics · January 1, 2019
We give a complete study of the asymptotic behavior of a simple model of alignment of unit vectors, both at the level of particles, which corresponds to a system of coupled differential equations, and at the continuum level, under the form of an aggregatio ...
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Journal ArticleQuarterly of Applied Mathematics · January 1, 2019
We consider a sequence of identical independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the ∞-Wasserstein distance between the empiric ...
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Journal ArticleSIAM Journal on Numerical Analysis · January 1, 2019
We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the correspondin ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · December 1, 2018
We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, ut = -u2(u3 + au)hhhh, gives the ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · October 1, 2018
In this paper, we study 1D autonomous fractional ODEs D c γu = f(u); 0 < γ < 1, where u : [0;∞) → R is the unknown function and D c is the generalized Caputo derivative introduced by Li and Liu ( arXiv:1612.05103). Based on the existence and uniqueness the ...
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Journal ArticleApplied Mathematics Letters · September 1, 2018
In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work Feng et al. [15]. Here we use the definition of Caputo derivative proposed in Li and Liu (2017) [5,7] based on a convolu ...
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Journal ArticleJournal of Differential Equations · August 5, 2018
This paper investigates Cauchy problems for nonlinear fractional time–space generalized Keller–Segel equation Dtβ0cρ+(−△)[Formula presented]ρ+∇⋅(ρB(ρ))=0, where Caputo derivative Dtβ0cρ models memory effects in time, fractional Laplacian (−△)[Formula prese ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · August 1, 2018
In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data m0, we show that classical solutions to this equation blow up in finite time Tmax. Before Tmax, existence and uniqueness of classical solution ...
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Journal ArticleJournal of Computational Physics · July 1, 2018
We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear. In particular, th ...
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Journal ArticleInverse Problems · May 29, 2018
We study the inverse problem of radiative transfer equation (RTE) using stochastic gradient descent method (SGD) in this paper. Mathematically, optical tomography amounts to recovering the optical parameters in RTE using the incoming-outgoing pair of light ...
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Journal ArticleJournal of Differential Equations · April 15, 2018
In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biologica ...
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Journal ArticleJournal of Differential Equations · April 5, 2018
We propose in this work new systems of equations which we call p-Euler equations and p-Navier–Stokes equations. p-Euler equations are derived as the Euler–Lagrange equations for the action represented by the Benamou–Brenier characterization of Wasserstein- ...
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Journal ArticleCalculus of Variations and Partial Differential Equations · April 1, 2018
In this work we consider (Formula presented.) which is derived from a thin film equation for epitaxial growth on vicinal surface. We formulate the problem as the gradient flow of a suitably-defined convex functional in a non-reflexive space. Then by restri ...
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Journal ArticleQuarterly of Applied Mathematics · January 1, 2018
We study in this work convolution groups generated by completely monotone sequences related to the ubiquitous time-delay memory effect in physics and engineering. In the first part, we give an accurate description of the convolution inverse of a completely ...
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Journal ArticleMathematics of Computation · January 1, 2018
We introduce a sub-cell shock capturing method for scalar conservation laws built upon the Jin-Xin relaxation framework. Here, sub-cell shock capturing is achieved using the original defect measure correction technique. The proposed method exactly restores ...
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Journal ArticleMathematics of Computation · January 1, 2018
We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is stable as long as the initial co ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2018
In this paper, we present a dispersive regularization approach to construct a global N-peakon weak solution to the modified Camassa–Holm equation (mCH) in one dimension. In particular, we perform a double mollification for the system of ODEs describing tra ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2018
We propose a generalized definition of Caputo derivatives from t = 0 of order \gamma \in (0, 1) using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2018
We study the Markov semigroups for two important algorithms from machine learning: stochastic gradient descent (SGD) and online principal component analysis (PCA). We investigate the effects of small jumps on the properties of the semigroups. Properties in ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2018
The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin-Lions lemma for the existence of weak s ...
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Journal ArticleJournal of Statistical Physics · October 1, 2017
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the ‘fluctuation-dissipation theorem’, the differential equations driven by ...
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Journal ArticleJournal of Scientific Computing · July 1, 2017
In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are sh ...
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Journal ArticlePhysica D: Nonlinear Phenomena · July 1, 2017
Motivated by a model proposed by Peng et al. (2014) for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDEs for the film thickn ...
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Journal ArticleJournal of Nonlinear Science · June 1, 2017
This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton–Cabrera–Frank-type model following the Xiang’s work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We prove that as the lattice parameter goe ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · June 1, 2017
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case ht + x(hn xxxh) + x(hn+2xh) = 0; where n 1. There exists a critical mass Mc = 2 p 6 3 found by Witelski et al. (2004 E ...
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Journal ArticleJournal of Nonlinear Science · April 1, 2017
We study coagulation–fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · March 1, 2017
This paper investigates the existence of a uniform in time L∞ bounded weak entropy solution for the quasilinear parabolic-parabolic KellerSegel model with the supercritical diffusion exponent 0 < m < 2 - 2/d in the multi-dimensional space ℝd under the cond ...
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Journal ArticleNonlinearity · January 1, 2017
In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions d ≥ 1 J(h): = (∫ℝd|h|dx)a-1 ∫ℝd |∇h|2 dx/(∫ℝd |h|m+1 dx)a+1/m+1 ≥ β0, where m > 0 for d = 1 ...
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Journal ArticleMathematical Models and Methods in Applied Sciences · January 1, 2017
We investigate the long-Time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker-Planck-Type equation under the assumption that interactions between individuals produce little consensus of o ...
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Journal ArticleMathematics of Computation · January 1, 2017
In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension d ≥ 2) and establish a rigorous convergence analysis. ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2017
We study in this work a continuum model derived from a one-dimensional attachmentdetachment-limited type step flow on a vicinal surface, ut = -u2(u3)hhhh, where u, considered as a function of step height h, is the step slope of the surface. We formulate a ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2017
In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2017
Mean-field games are games with a continuum of players that incorporate the timedimension through a control-theoretic approach. Recently, simpler approaches relying on the Best-Reply Strategy have been proposed. They assume that the agents navigate their s ...
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Journal ArticleMathematics of Computation · January 1, 2017
We establish an optimal error estimate for a random particle blob method for the Keller-Segel equation in ℝd (d ≥ 2). With a blob size ε = hκ (1/2 < κ < 1), we prove a rate h| ln h| of convergence in ℓhp (p > d/1-κ) norm up to a probability 1-hC| ln h|, wh ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2017
In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation ∂u/∂t = Δ(Δu)-3. This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solutio ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2017
We establish an error estimate of a discrete-in-time random particle blob method for the Keller{Segel (KS) equation in ℝd (d≥2). With a blob size ε=N-1/d(d+1) log(N), we prove the convergence rate between the solution to the KS equation and the empirical m ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · December 1, 2016
In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is CΔt ...
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Journal ArticleApplied Mathematics Letters · November 1, 2016
This note studies the Monge-Ampère Keller-Segel equation in a periodic domain Td(d≥2), a fully nonlinear modification of the Keller-Segel equation where the Monge-Ampère equation det(I+2v)=u+1 substitutes for the usual Poisson equation Δv=u. The existence ...
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Journal ArticleActa Applicandae Mathematicae · April 1, 2016
In this note we establish the uniform (Formula presented.) -bound for the weak solutions to a degenerate Keller-Segel equation with the diffusion exponent (Formula presented.) under a sharp condition on the initial data for the global existence. As a conse ...
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Journal ArticlePhysics of Fluids · April 1, 2016
In biological transport mechanisms such as insect respiration and renal filtration, fluid travels along a leaky channel allowing material exchange with systems exterior to the channel. The channels in these systems may undergo peristaltic pumping which is ...
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Journal ArticleTransactions of the American Mathematical Society · January 1, 2016
The class of generating functions for completely monotone sequences (moments of finite positive measures on [0, 1]) has an elegant characterization as the class of Pick functions analytic and positive on (−∞, 1). We establish this and another such characte ...
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ConferenceSpringer Proceedings in Mathematics and Statistics · January 1, 2016
In this paperwedevelop a newmartingale method to showthe convergence of the regularized empirical measure of many particle systems in probability under a Sobolev norm to the corresponding mean field PDE. Our method works well for the simple case of Fokker ...
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Journal ArticleKinetic and Related Models · January 1, 2016
This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term -ν(-Δ) α/2 ρ (1 < α < 2). Firstly, the global existence of weak solutions is proved for the initial density ρ0 ∈ L1∩L d/α (ℝd) (d ≥ 2) with [norm of matrix]ρ0[n ...
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Journal ArticleKinetic and Related Models · January 1, 2016
This paper investigates the existence of a uniform in time L∞ bounded weak solution for the p-Laplacian Keller-Segel system with the supercritical diffusion exponent 1 < p < 3d/d+1 in the multi-dimensional space ℝd under the condition that the L d(3-p)/p n ...
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Journal ArticleMultiscale Modeling and Simulation · January 1, 2016
From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an "s-d" potential term, including the Schrödinger-Landau-Lifshitz- Maxwell system, the Pauli-Landau-Lifshitz system, and the Schrödinger-Landa ...
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Journal ArticleSIAM Journal on Mathematical Analysis · January 1, 2016
In this paper we study the existence assertion of the initial boundary value problem for the equation @u/@t = Δe-Δu. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in th ...
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Journal ArticleArchive for Rational Mechanics and Analysis · April 1, 2015
We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensi ...
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Journal ArticleJournal of Scientific Computing · March 14, 2015
In this paper, we apply a simple finite element numerical scheme, proposed in an earlier work (Liu in Math Comput 70(234):579–593, 2000), to perform a high resolution numerical simulation of incompressible flow over an irregular domain and analyze its boun ...
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Journal ArticlePhysical Review E - Statistical, Nonlinear, and Soft Matter Physics · March 4, 2015
The Burton-Cabrera-Frank (BCF) model for the flow of line defects (steps) on crystal surfaces has offered useful insights into nanostructure evolution. This model has rested on phenomenological grounds. Our goal is to show via scaling arguments the emergen ...
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Journal ArticleSIAM Journal on Applied Mathematics · January 1, 2015
We derive an exact solution for Stokes flow in a channel with permeable walls. At the channel walls, the normal component of the fluid velocity is described by Darcy's law, and the tangential component of the fluid velocity is described by the no slip cond ...
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Journal ArticleApplied Numerical Mathematics · January 1, 2015
The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa-Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These s ...
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Journal ArticlePhilosophical transactions. Series A, Mathematical, physical, and engineering sciences · November 2014
We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed in Degond et al. (2014 J. Stat. Phys. 154, 751-780 (doi:10.1007/s10955-013-0888-4)). The model considers a system of rational agents int ...
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Journal ArticleArchive for Rational Mechanics and Analysis · October 17, 2014
Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very sm ...
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Journal ArticleJournal of Scientific Computing · September 1, 2014
A spectral collocation scheme for the three-dimensional incompressible (u,p) formulation of the Navier–Stokes equations, in domains Ω with a non-periodic boundary condition, is described. The key feature is the high order approximation, by means of a local ...
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Journal ArticleInternational Journal for Numerical Methods in Fluids · May 20, 2014
We are concerned with a coupled system describing the interaction between suspended particles and a dense fluid. The particles are modeled by a kinetic equation of Vlasov-Fokker-Planck type, and the fluid is described by the incompressible Navier-Stokes sy ...
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Journal ArticleDiscrete and Continuous Dynamical Systems- Series A · May 1, 2014
In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equa ...
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Journal ArticleKinetic and Related Models · March 1, 2014
This paper establishes the hyper-contractivity in L∞(ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent m > 1-2/d. The results show that for the super- critical and critical case 1-2/d < m ≤ 2 ...
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Journal ArticleJournal of Statistical Physics · January 1, 2014
We present and analyze a model for the evolution of the wealth distribution within a heterogeneous economic environment. The model considers a system of rational agents interacting in a game theoretical framework, through fairly general assumptions on the ...
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Journal ArticleJournal of Nonlinear Science · January 1, 2014
We introduce a new mean field kinetic model for systems of rational agents interacting in a game-theoretical framework. This model is inspired from noncooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2014
We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto, and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With a no- ...
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Journal ArticleMultiscale Modeling and Simulation · January 1, 2014
We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quant ...
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Journal ArticleActa Applicandae Mathematicae · January 1, 2014
Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinskiǐ lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a result of Dubinskiǐ (in ...
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Journal ArticleCommunications in Mathematical Physics · November 1, 2013
This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 - 2/d, the L p space for both dynamic and steady solutions are detected with (Formula presented.). Firstly, the global existence of the weak ...
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Journal ArticleSIAM Journal on Mathematical Analysis · October 4, 2013
We studied coupled systems of the Fokker-Planck equation and the Navier-Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in wei ...
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Journal ArticleJournal of Computational Physics · August 1, 2013
We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme i ...
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Journal ArticleJournal of Differential Equations · April 1, 2013
We prove the existence of the global weak entropy solution to the Doi-Saintillan-Shelley model for active and passive rod-like particle suspensions, which couples a Fokker-Planck equation with the incompressible Navier-Stokes or Stokes equation, under the ...
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Journal ArticleSIAM Journal on Scientific Computing · January 1, 2013
We propose a simple finite difference scheme for Navier-Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact H ...
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Journal ArticlePhysical Review A - Atomic, Molecular, and Optical Physics · November 14, 2011
We develop an approach, by calculating the autocorrelation function of spins, to derive the magnetic field gradient-induced transverse (T2) relaxation of spins undergoing restricted diffusion. This approach is an extension to the method adopted by McGregor ...
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Journal ArticleCommun. Comput. Phy. · 2011
We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differ- ential operator. As a result, the Fast Fourier Transform can be applied to solve the Pois ...
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Journal ArticleSIAM J. Numer. Anal. · 2010
We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31 (2008), pp. 334–368] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is unifor ...
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Journal ArticleJournal of Computational Physics · January 1, 2010
How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier-Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the press ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2010
We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving ...
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Journal ArticleChinese Annals of Mathematics. Series B · December 1, 2009
The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which p ...
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Journal ArticleSIAM Journal on Numerical Analysis · November 10, 2008
In a previous work [P. Crispel, P. Degond, and M.-H. Vignal, J. Comput. Phys., 223 (2007), pp. 208-234], a new numerical discretization of the Euler-Poisson system was proposed. This scheme is "asymptotic preserving" in the quasineutral limit (i.e., when t ...
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Journal ArticleMathematics of Computation · July 1, 2008
The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method wit ...
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Journal ArticleCommunications in Computational Physics · January 1, 2008
A fourth-order finite difference method is proposed and studied for the primitive equations (PEs) of large-scale atmospheric and oceanic flow based on mean vorticity formulation. Since the vertical average of the horizontal velocity field is divergence-fre ...
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Journal ArticleSIAM Journal on Scientific Computing · January 1, 2008
The sequential regularization method is a reformulation of the unsteady Navier-Stokes equations from the viewpoint of constrained dynamical systems or the approximate Helmholtz-Hodge projection. In this paper we study the long time behavior of the sequenti ...
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Journal ArticleJournal of agricultural and food chemistry · December 2007
The bran fraction of wheat grain is known to contain significant quantities of bioactive components. This study evaluated the potential of solid-state yeast fermentation to improve the health beneficial properties of wheat bran, including extractable antio ...
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Journal ArticleMilan Journal of Mathematics · December 1, 2007
The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concr ...
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Journal ArticleCommunications on Pure and Applied Mathematics · October 1, 2007
For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velo ...
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Journal ArticleBulletin of the Institute of Mathematics Academia Sinica (New Series) · 2007
We present novel algorithms for compressible flows that are
efficient for all Mach numbers. The approach is based on several
ingredients: semi-implicit schemes, the gauge decomposition of the
velocity field and a second order formulation of the density
equ ...
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Journal ArticleSIAM Journal on Numerical Analysis · December 1, 2006
We give an error estimate for the energy and helicity preserving scheme (EHPS) in second order finite difference setting on axisymmetric incompressible flows with swirling velocity. This is accomplished by a weighted energy estimate, along with careful and ...
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Journal ArticleMultiscale Modeling and Simulation · September 1, 2006
This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid mod ...
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Journal ArticleJournal of agricultural and food chemistry · July 2006
Recent consumer interest in controlling and preventing chronic diseases through improved diet has promoted research on the bioactive components of agricultural products. Wheat is an important agricultural and dietary commodity worldwide with known antioxid ...
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Journal ArticleJournal of Computational Physics · October 10, 2004
We propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry. By introducing a generalized vorticity-stream formulation, the divergence free constraints are automatically satisfied. In addition, wi ...
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Journal ArticlePhysica D: Nonlinear Phenomena · October 1, 2004
We study the detailed process of bifurcation in the flow's topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory ...
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Journal ArticleJournal of Nonlinear Science · October 1, 2004
We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. ...
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Journal ArticleJournal of Computational Physics · September 1, 2004
We present numerical schemes for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, co ...
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Journal ArticleNumerische Mathematik · May 1, 2004
The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discr ...
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Journal ArticleQuarterly of Applied Mathematics · January 1, 2004
In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by t ...
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Journal ArticleCommunications on Pure and Applied Analysis · January 1, 2004
In this paper we study the effects of small viscosity term and the far-field boundary conditions for systems of convection-diffusion equations in the zero viscosity limit. The far-field boundary conditions are classified and the corresponding solution stru ...
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Journal ArticleComputers and Fluids · January 1, 2004
Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order ...
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Journal ArticleEuropean Journal of Applied Mathematics · December 1, 2003
Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich-Schwoebel effect - the kinetic asymmetry in attachment and detachment of adatoms to and from terrace bou ...
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Journal ArticleJournal of Scientific Computing · December 1, 2003
A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependen ...
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Journal ArticleJournal of Scientific Computing · April 1, 2003
A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equat ...
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Journal ArticleCommunications in Mathematical Sciences · January 1, 2003
Gauge transformation is a well-known concept in physics and has been used as a computational tool also. In fluid dynamics, Buttke was the first to use it as a computational tool to design vortex methods [1], following earlier work of Oseledets and others [ ...
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Journal ArticleMathematical Modelling and Numerical Analysis · January 1, 2003
We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equati ...
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Journal ArticleDiscrete and Continuous Dynamical Systems - Series B · January 1, 2003
A class of upwind flux splitting methods in the Euler equations of compressible flow is considered in this paper. Using the property that Euler flux F(U) is a homogeneous function of degree one in U, we reformulate the splitting fluxes with F+ = A+U, F- = ...
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Journal ArticleJournal of Computational Physics · July 20, 2002
In this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible Navier-Stokes equations in the velocity-pressure formulation. Their use is especially well suited for the co ...
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Journal ArticleNumerische Mathematik · May 1, 2002
In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted ...
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Journal ArticleMathematics of Computation · January 1, 2002
In E & Liu (SIAM J Numer. Anal., 1995), we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous (we call this the semi-discrete case). In this paper, we address similar questions fo ...
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Journal ArticleJournal of Computational Physics · November 20, 2001
We propose a simple and efficient finite-difference method for the incompressible MHD equation. The numerical method combines the advantage of the MAC scheme for the Navier-Stokes equation and Yee's scheme for the Maxwell equation. In particular, the semi- ...
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Journal ArticleMathematics of Computation · April 1, 2001
A very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation. This method relies on recasting the traditional finite element method in the spirit of the high orde ...
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Journal ArticleMathematics of Computation · April 1, 2001
We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial ve ...
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Journal ArticleInternational Journal for Numerical Methods in Fluids · December 1, 2000
A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E. Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation ...
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Journal ArticleJournal of Computational Physics · May 20, 2000
In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin ...
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Journal ArticleMathematics of Computation · January 1, 2000
A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a and a gauge variable φ, u = a + ∇φ, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation an ...
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Journal ArticleCommunications on Pure and Applied Mathematics · January 1, 2000
We prove the convergence of a discontinuous Galerkin method approximating the 2-D incompressible Euler equations with discontinuous initial vorticity: ω0 ∈ L2(Ω). Furthermore, when ω0 ∈ L∞(Ω), the whole sequence is shown to be strongly convergent. This is ...
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Journal ArticlePhysics of Fluids · January 1, 1999
We investigate, by numerical simulation, the shear layer instability associated with the outer layer of a spiral vortex formed behind an impulsively started thin ellipse. The unstable free shear layer undergoes a secondary instability. We connect this inst ...
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Journal ArticleMathematics of Computation · January 1, 1999
Solutions of conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numer ...
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Journal ArticleJournal of Computational Physics · August 10, 1998
The Euler equation of compressible flows is solved by the finite volume method, where high order accuracy is achieved by the reconstruction of each component of upwind fluxes of a flux splitting using the biased averaging procedure. Compared to the solutio ...
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Journal ArticleJournal of Computational Physics · November 15, 1997
Simple, efficient, and accurate finite difference methods are introduced for 3D unsteady viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids. Two different types of methods are discussed. They differ in the impl ...
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Journal ArticleJournal of Computational Physics · January 1, 1997
We consider finite difference schemes based on the impulse density variable. We show that the original velocity - impulse density formulation of Oseledets is marginally ill-posed for the inviscid flow, and this has the consequence that some ordinarily stab ...
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Journal ArticleMathematics of Computation · January 1, 1997
We are concerned with the convergence of Lax-Weridroff type schemes with high resolution to the entropy solutions fo: conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-t ...
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Journal ArticleJournal of Computational Physics · March 15, 1996
This paper discusses three basic issues related to the design of finite difference schemes for unsteady viscous incompressible flows using vorticity formulations: the boundary condition for vorticity, an efficient time-stepping procedure, and the relation ...
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Journal ArticleTransport Theory and Statistical Physics · January 1, 1996
In this paper, we investigate the boundary layer behavior of solutions to the one dimensional Broadwell model of the nonlinear Boltzmann equation for small mean free path. We consider the analogue of Maxwell's diffusive and the reflexive boundary condition ...
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Journal ArticlePhysica D: Nonlinear Phenomena · January 1, 1996
We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of that equation develops a singularity. Modulation equations are derived that describe period-two oscillation ...
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Journal ArticleSIAM Journal on Numerical Analysis · January 1, 1996
This is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by (1) the numerical b ...
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Journal ArticleJournal of Computational Physics · January 1, 1996
A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced. The scheme is based on the vorticity-stream function formulation. It is essentially compact and has the nice features of a comp ...
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Journal ArticleJournal of Computational Physics · January 1, 1996
We begin a systematical study on the effect of numerical viscosities. In this paper we investigate the behavior of shock-capturing methods for slowly moving shocks. It is known that for slowly moving shocks even a first-order scheme, such as the Godunov or ...
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Journal ArticleProceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences · January 1, 1994
The development of shocks in nonlinear hyperbolic conservation laws may be regularized through either diffusion or relaxation. However, we have observed surprisingly that for some physical problems, when both of the smoothing factors diffusion and relaxati ...
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Journal ArticleArchive for Rational Mechanics and Analysis · September 1, 1993
In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are n ...
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