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Point defects in tight binding models for insulators Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2021-01-06 Christoph Ortner; Jack Thomas
We consider atomistic geometry relaxation in the context of linear tight binding models for point defects. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. We also formulate the thermodynamic limit model at zero Fermi-temperature, extending the results of [H. Chen, J. Lu and C. Ortner, Thermodynamic
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DG approach to large bending plate deformations with isometry constraint Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2021-01-06 Andrea Bonito; Ricardo H. Nochetto; Dimitrios Ntogkas
We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove Γ-convergence of the
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Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-11-12 Rafael Bailo; José A. Carrillo; Hideki Murakawa; Markus Schmidtchen
We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient flow structure, arXiv:1811.11502]. Crucially, this scheme keeps the dissipation property
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Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot’s consolidation and multiple-network poroelasticity models Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-10 Qingguo Hong; Johannes Kraus; Maria Lymbery; Fadi Philo
This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each
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Atomistic origins of continuum dislocation dynamics Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-17 Thomas Hudson; Patrick van Meurs; Mark Peletier
This paper focuses on the connections between four stochastic and deterministic models for the motion of straight screw dislocations. Starting from a description of screw dislocation motion as interacting random walks on a lattice, we prove explicit estimates of the distance between solutions of this model, an SDE system for the dislocation positions, and two deterministic mean-field models describing
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Global boundedness and asymptotic behavior in a quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-14 Guoqiang Ren; Bin Liu
In this work, we consider the quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source. We present the global existence of classical solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature
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The one-phase fractional Stefan problem Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-28 Félix del Teso; Jørgen Endal; Juan Luis Vázquez
We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in ℝN. In terms of the enthalpy h(x,t), the evolution equation reads ∂th+(−Δ)sΦ(h)=0, while the temperature is defined as u:=Φ(h):=max{h−L,0} for some constant L>0 called the latent heat, and (−Δ)s stands for the fractional Laplacian with exponent s∈(0,1). We prove
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Modeling glioma invasion with anisotropy- and hypoxia-triggered motility enhancement: From subcellular dynamics to macroscopic PDEs with multiple taxis Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-28 Gregor Corbin; Axel Klar; Christina Surulescu; Christian Engwer; Michael Wenske; Juanjo Nieto; Juan Soler
We deduce a model for glioma invasion that accounts for the dynamics of brain tissue being actively degraded by tumor cells via excessive acidity production, but also according to the local orientation of tissue fibers. Our approach has a multiscale character: we start with a microscopic description of single cell dynamics including biochemical and/or biophysical effects of the tumor microenvironment
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A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-22 Stefano Almi; Sandro Belz; Stefano Micheletti; Simona Perotto
In this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the Γ-limit of the functional as the
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A ternary Cahn–Hilliard–Navier–Stokes model for two-phase flow with precipitation and dissolution Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-17 Christian Rohde; Lars von Wolff
We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The
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An adaptive edge element approximation of a quasilinear H(curl)-elliptic problem Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-10 Yifeng Xu; Irwin Yousept; Jun Zou
An adaptive edge element method is designed to approximate a quasilinear H(curl)-elliptic problem in magnetism, based on a residual-type a posteriori error estimator and general marking strategies. The error estimator is shown to be both reliable and efficient, and its resulting sequence of adaptively generated solutions converges strongly to the exact solution of the original quasilinear system. Numerical
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Convergence of knowledge in a stochastic cultural evolution model with population structure, social learning and credibility biases Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-07 Sylvain Billiard; Maxime Derex; Ludovic Maisonneuve; Thomas Rey
Understanding how knowledge emerges and propagates within groups is crucial to explain the evolution of human populations. In this work, we introduce a mathematically oriented model that draws on individual-based approaches, inhomogeneous Markov chains and learning algorithms, such as those introduced in [F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc. 39
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Linear instability of Z-pinch in plasma: Viscous case Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-12-05 Dongfen Bian; Yan Guo; Ian Tice
The z-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the z-direction. We develop a scaled variational framework to study its stability in the presence of viscosity effect, and demonstrate that any such z-pinch is always unstable. We also establish the existence
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On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-10-19 Xiaoli Li; Jie Shen
We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently
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Instability of the abstract Rayleigh–Taylor problem and applications Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-10-23 Fei Jiang; Song Jiang; Weicheng Zhan
Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of L1-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady)
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Density-induced consensus protocol Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-10-23 Piotr Minakowski; Piotr B. Mucha; Jan Peszek
The paper introduces a model of collective behavior where agents receive information only from sufficiently dense crowds in their immediate vicinity. The system is an asymmetric, density-induced version of the Cucker–Smale model with short-range interactions. We prove the basic mathematical properties of the system and concentrate on the presentation of interesting behaviors of the solutions. The results
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Convergence of a first-order consensus-based global optimization algorithm Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-19 Seung-Yeal Ha; Shi Jin; Doheon Kim
Global optimization of a non-convex objective function often appears in large-scale machine learning and artificial intelligence applications. Recently, consensus-based optimization (CBO) methods have been introduced as one of the gradient-free optimization methods. In this paper, we provide a convergence analysis for the first-order CBO method in [J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based
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Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-10-23 Zhilei Liang; Dehua Wang
A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate
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Condition number bounds for IETI-DP methods that are explicit in h and p Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-10-23 Rainer Schneckenleitner; Stefan Takacs
We study the convergence behavior of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods for solving large-scale algebraic systems arising from multi-patch Isogeometric Analysis. We focus on the Poisson problem on two-dimensional computational domains. We provide a convergence analysis that covers several choices of the primal degrees of freedom: the vertex values, the edge averages
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Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-14 Nancy Rodríguez; Michael Winkler
We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci.18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium
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Element-splitting-invariant local-length-scale calculation in B-Spline meshes for complex geometries Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-11-04 Yuki Ueda; Yuto Otoguro; Kenji Takizawa; Tayfun E. Tezduyar
Variational multiscale methods and their precursors, stabilized methods, which are sometimes supplemented with discontinuity-capturing (DC) methods, have been playing their core-method role in flow computations increasingly with isogeometric discretization. The stabilization and DC parameters embedded in most of these methods play a significant role. The parameters almost always involve some local-length-scale
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A hydrodynamic model for synchronization phenomena Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-11 Young-Pil Choi; Jaeseung Lee
We present a new hydrodynamic model for synchronization phenomena which is a type of pressureless Euler system with nonlocal interaction forces. This system can be formally derived from the Kuramoto model with inertia, which is a classical model of interacting phase oscillators widely used to investigate synchronization phenomena, through a kinetic description under the mono-kinetic closure assumption
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Social climbing and Amoroso distribution Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-10-06 Giacomo Dimarco; Giuseppe Toscani
We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable
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Recent results and challenges in behavioral systems Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-19 N. Bellomo; F. Brezzi; J. Soler
This editorial paper is devoted to present the papers published in a special issue focused on modeling, qualitative analysis and simulation of the collective dynamics of living, self-propelled particles. A critical analysis of the overall contents of the issue is proposed, thus leading to a forward look to research perspectives.
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Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-26 Michael R. Lindstrom; Andrea L. Bertozzi
In this paper, we develop a continuum model for the movement of agents on a lattice, taking into account location desirability, local and far-range migration, and localized entry and exit rates. Specifically, our motivation is to qualitatively describe the homeless population in Los Angeles. The model takes the form of a fully nonlinear, nonlocal, non-degenerate parabolic partial differential equation
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Coupling kinetic theory approaches for pedestrian dynamics and disease contagion in a confined environment Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-03 Daewa Kim; Annalisa Quaini
The goal of this work is to study an infectious disease spreading in a medium size population occupying a confined environment. For this purpose, we consider a kinetic theory approach to model crowd dynamics in bounded domains and couple it to a kinetic equation to model contagion. The interactions of a person with other pedestrians and the environment are modeled by using tools of game theory. The
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The evolution of communication mechanisms in self-organised ecological aggregations: Impact on pattern formation Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-28 R. Eftimie
Collective behaviours in animal communities are the result of inter-individual communication. However, communication signals are not fixed; they evolve to ensure more effective interactions between the emitter and receiver of these signals. In this study, we use a mathematical approach and investigate the effect of changes in communication signals (at both receiver and emitter levels) on the aggregation
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Nematic alignment of self-propelled particles: From particle to macroscopic dynamics Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-26 Pierre Degond; Sara Merino-Aceituno
Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution
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On the critical exponent of the one-dimensional Cucker–Smale model on a general graph Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-07 Seung-Yeal Ha; Zhuchun Li; Xiongtao Zhang
We study a critical exponent of the flocking behavior to the one-dimensional 1D Cucker–Smale (C–S) model with a regular inverse power law communication on a general network with a spanning tree. For this, we propose a new nonlinear functional which can control the velocity diameter and decays exponentially fast as time goes on. As an application of the time-evolution of the nonlinear functional, we
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Preconditioning the EFIE on screens Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-05 Ralf Hiptmair; Carolina Urzúa-Torres
We consider the electric field integral equation (EFIE) modeling the scattering of time-harmonic electromagnetic waves at a perfectly conducting screen. When discretizing the EFIE by means of low-order Galerkin boundary methods (BEM), one obtains linear systems that are ill-conditioned on fine meshes and for low wave numbers k. This makes iterative solvers perform poorly and entails the use of preconditioning
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The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-19 Linfeng Mei; Juncheng Wei
Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D’Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci107 (2008) 1249–1267; M. B. Short, A
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Coarse-graining via EDP-convergence for linear fast-slow reaction systems Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-09-12 Alexander Mielke; Artur Stephan
We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass
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Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-26 Daniele A. Di Pietro; Jérôme Droniou; Francesca Rapetti
In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving the exactness, we show that the usual three-dimensional sequence of trimmed Finite Element
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A phase-field-based graded-material topology optimization with stress constraint Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-26 Ferdinando Auricchio; Elena Bonetti; Massimo Carraturo; Dietmar Hömberg; Alessandro Reali; Elisabetta Rocca
In this paper, a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraints and potentially multiple materials or multiscales is analyzed. First-order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for
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Global existence of weak solutions to the incompressible Vlasov–Navier–Stokes system coupled to convection–diffusion equations Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-07-20 Laurent Boudin; David Michel; Ayman Moussa
We study the existence of global weak solutions in a three-dimensional time-dependent bounded domain for the incompressible Vlasov–Navier–Stokes system which is coupled with two convection–diffusion equations describing the air temperature and its water vapor mass fraction. This newly introduced model describes respiratory aerosols in the human aiways when one takes into account the hygroscopic effects
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On weak solutions to a dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-21 Young-Sam Kwon; Antonin Novotny; C. H. Arthur Cheng
In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential
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Polynomial preserving virtual elements with curved edges Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-26 L. Beirão da Veiga; F. Brezzi; L. D. Marini; A. Russo
In this paper, we tackle the problem of constructing conforming Virtual Element spaces on polygons with curved edges. Unlike previous VEM approaches for curvilinear elements, the present construction ensures that the local VEM spaces contain all the polynomials of a given degree, thus providing the full satisfaction of the patch test. Moreover, unlike standard isoparametric FEM, this approach allows
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A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-19 Nicola Bellomo; Richard Bingham; Mark A. J. Chaplain; Giovanni Dosi; Guido Forni; Damian A. Knopoff; John Lowengrub; Reidun Twarock; Maria Enrica Virgillito
This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS–CoV–2 Severe Acute Respiratory Syndrome, corona virus n.2. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further
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Fluid models with phase transition for kinetic equations in swarming Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-08-07 Mihaï Bostan; José Antonio Carrillo
We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs
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Some aspects of the inertial spin model for flocks and related kinetic equations Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-07-25 D. Benedetto; P. Buttà; E. Caglioti
In this paper, we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the
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Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-07-10 Pierluigi Colli; Hector Gomez; Guillermo Lorenzo; Gabriela Marinoschi; Alessandro Reali; Elisabetta Rocca
Chemotherapy is a common treatment for advanced prostate cancer. The standard approach relies on cytotoxic drugs, which aim at inhibiting proliferation and promoting cell death. Advanced prostatic tumors are known to rely on angiogenesis, i.e. the growth of local microvasculature via chemical signaling produced by the tumor. Thus, several clinical studies have been investigating antiangiogenic therapy
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Global bounded solution of the higher-dimensional forager–exploiter model with/without growth sources Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-10 Jianping Wang; Mingxin Wang
This paper concerns with the global existence and boundedness of classical solution of the higher-dimensional forager–exploiter model with homogeneous Neumann boundary condition and nonnegative initial data. For cases where there are no forager and exploiter growth sources, it will be shown that if either the initial data and the production rate of nutrient are small or the taxis effects are small
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Asymptotic dynamics on a chemotaxis-Navier–Stokes system with nonlinear diffusion and inhomogeneous boundary conditions Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-07 Chunyan Wu; Zhaoyin Xiang
The diffusion of cells in a viscous incompressible fluid (e.g. water) may be viewed like movement in a porous medium and there is a bidirectorial oxygen exchange between water and their surrounding air in thin fluid layers near the air–water contact surface. This leads to the following chemotaxis-Navier–Stokes system with nonlinear diffusion: nt+u⋅∇n=Δnm−∇⋅(n∇c),x∈Ω, t>0,ct+u⋅∇c=Δc−nc,x∈Ω, t>0,ut+(u⋅∇)u=Δu+∇P+n∇ϕ
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Brownian fluctuations of flame fronts with small random advection Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-10 Christopher Henderson; Panagiotis E. Souganidis
We study the effect of small random advection in two models in turbulent combustion. Assuming that the velocity field decorrelates sufficiently fast, we (i) identify the order of the fluctuations of the front with respect to the size of the advection; and (ii) characterize them by the solution of a Hamilton–Jacobi equation forced by white noise. In the simplest case, the result yields, for both models
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Construction of boundary conditions for hyperbolic relaxation approximations I: The linearized Suliciu model Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-07 Yizhou Zhou; Wen-An Yong
Starting with this paper, we intend to develop a program aiming at construction of boundary conditions (BCs) for hyperbolic relaxation systems. Physically, such BCs are not always available. The construction is based on the assumption that the relaxation systems and well-posed BCs for the corresponding equilibrium systems are given. This paper focuses on the linearized Suliciu model. We obtain strictly
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From particles to firms: on the kinetic theory of climbing up evolutionary landscapes Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-10 Nicola Bellomo; Giovanni Dosi; Damián A. Knopoff; Maria Enrica Virgillito
This paper constitutes the first attempt to bridge the evolutionary theory in economics and the theory of active particles in mathematics. It seeks to present a kinetic model for an evolutionary formalization of economic dynamics. The new derived mathematical representation intends to formalize the processes of learning and selection as the two fundamental drivers of evolutionary environments [G. Dosi
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Global radial renormalized solution to a producer–scrounger model with singular sensitivities Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-19 Xinru Cao
This paper is concerned with the parabolic system ut=Δu−∇⋅uw∇w,(x,t)∈Ω×(0,T),vt=Δv−∇⋅vu∇u,(x,t)∈Ω×(0,T),wt=Δw−(u+v)w−λw+g(x,t),(x,t)∈Ω×(0,T),(0.1) in a bounded ball Ω=BR(0)⊂ℝn (n≥2) with R>0. Where λ≥0 and 0≤g∈C1(Ω¯×(0,∞)). It is shown that for arbitrarily radially symmetric initial data (u0,v0,w0), which are nonnegative and suitably regular, the corresponding Neumann initial-boundary problem admits
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A general cell–fluid Navier–Stokes model with inclusion of chemotaxis Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-13 Yangyang Qiao; Steinar Evje
The main purpose of this work is to explore a general cell–fluid model which is based on a mixture theory formulation that accounts for the interplay between oxytactically (chemotaxis toward gradient in oxygen) moving bacteria cells in water and the buoyance forces caused by the difference in density between cells and fluid. The model involves two mass balance and two general momentum balance equations
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Multiple asymptotics of kinetic equations with internal states Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-10 Benoit Perthame; Weiran Sun; Min Tang; Shugo Yasuda
The run and tumble process is well established in order to describe the movement of bacteria in response to a chemical stimulus. However, the relation between the tumbling rate and the internal state of bacteria is poorly understood. This study aims at deriving macroscopic models as limits of the mesoscopic kinetic equation in different regimes. In particular, we are interested in the roles of the
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Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-10 Yulan Wang
We consider the coupled chemotaxis–Navier–Stokes system with logistic source term nt+u⋅∇n=Δn−∇⋅(n∇c)+rn−μnα,ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0 in a bounded, smooth domain Ω⊂ℝ3, where Φ∈W2,∞(Ω) and where r≥0, μ>0 and 1<α<2 are given parameters. Although the degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the initial-value problem
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Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-06-07 Tobias Black
In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the
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Doubly nonlinear stochastic evolution equations Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-05-07 Luca Scarpa; Ulisse Stefanelli
Nonlinear diffusion problems featuring stochastic effects may be described by stochastic partial differential equations of the form dα(u)−div(β1(∇u))dt+β0(u)dt∋f(u)dt+G(u)dW. We present an existence theory for such equations under general monotonicity assumptions on the nonlinearities. In particular, α, β0, and β1 are allowed to be multivalued, as required by the modelization of solid–liquid phase
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Asymptotic limit of a spatially-extended mean-field FitzHugh–Nagumo model Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-05-04 Joachim Crevat
We consider a spatially extended mean-field model of a FitzHugh–Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its asymptotic limit in the regime of strong local interactions converges toward a system of reaction–diffusion equations taking account for the average quantities of the network. Our approach is based on a modulated energy argument, to compare
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A kinetic theory approach for modelling tumour and macrophages heterogeneity and plasticity during cancer progression Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-05-04 R. Eftimie; L. Gibelli
The heterogeneity and plasticity of macrophages have become a topic of great interest, due to their role in various diseases ranging from cancer to bacterial infections. While initial experimental studies assumed an extreme polarisation situation, with the (anti-tumour) M1 and (pro-tumour) M2 macrophages representing the two extreme cell phenotypes, more recent studies showed a continuum of macrophages
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Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-04-23 Gabriel Barrenechea; Erik Burman; Johnny Guzmán
We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using H(div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the L2-norm of order O(hk+12). We also prove error estimates
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Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-04-16 Robert Altmann; Patrick Henning; Daniel Peterseim
This paper analyzes spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps among the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs
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An approach to congestion analysis in crowd dynamics models Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-04-16 Liang Li; Hong Liu; Yanbin Han
This paper presents a novel approach to quantitatively analyzing pedestrian congestion in evacuation management based on the Hughes and social force models. An accurate analysis of crowds plays an important role in illustrating their dynamics. However, the majority of the existing approaches to analyzing pedestrian congestion are qualitative. Few methods focus on the quantification of the interactions
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Equilibria of an aggregation model with linear diffusion in domains with boundaries Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-04-09 Daniel Messenger; Razvan C. Fetecau
We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends
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Critical thresholds in one-dimensional damped Euler–Poisson systems Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-03-25 Manas Bhatnagar; Hailiang Liu
This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler–Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J.50 (2001) 109–157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical
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Stochastic persistency of nematic alignment state for the Justh–Krishnaprasad model with additive white noises Math. Models Methods Appl. Sci. (IF 3.044) Pub Date : 2020-03-25 Seung-Yeal Ha; Dongnam Ko; Woojoo Shim; Hui Yu
We present a stochastic Justh–Krishnaprasad flocking model describing interactions among individuals in a planar domain with their positions and heading angles. The deterministic counterpart of the proposed model describes the formation of nematic alignment in an ensemble of planar particles moving with a unit speed. When the noise is turned off, we show that the nematic alignment state, in which all