Authors:S. Lanthaler, S. Mishra, C. Parés-Pulido Pages: 223 - 292 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 31, Issue 02, Page 223-292, February 2021. We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-02-16T08:00:00Z DOI: 10.1142/S0218202521500068 Issue No:Vol. 31, No. 02 (2021)

Authors:Laurent Bétermin, Markus Faulhuber, Hans Knüpfer Pages: 293 - 325 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 31, Issue 02, Page 293-325, February 2021. The goal of this paper is to investigate the optimality of the [math]-dimensional rock-salt structure, i.e. the cubic lattice [math] of volume [math] with an alternation of charges [math] at lattice points, among periodic distributions of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the [math]-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials. Numerically, we investigate several particular cases in dimensions [math], [math] and [math]. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For [math], we observe a phase transition of the type “triangular-rhombic-square-rectangular” for the minimizer’s shape as the density decreases. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-02-24T08:00:00Z DOI: 10.1142/S021820252150007X Issue No:Vol. 31, No. 02 (2021)

Authors:José A. Carrillo, Young-Pil Choi, Jinwook Jung Pages: 327 - 408 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 31, Issue 02, Page 327-408, February 2021. In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker–Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulomb interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. The existence of weak and strong solutions to the kinetic and fluid equations is also obtained. We emphasize that the existence of global weak solution with the needed free energy dissipation for the kinetic model is established. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-02-24T08:00:00Z DOI: 10.1142/S0218202521500081 Issue No:Vol. 31, No. 02 (2021)

Authors:Dongfen Bian, Yan Guo, Ian Tice Pages: 409 - 472 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 31, Issue 02, Page 409-472, February 2021. The [math]-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 [math] direction. We develop a variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that such a [math]-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-02-26T08:00:00Z DOI: 10.1142/S0218202521500093 Issue No:Vol. 31, No. 02 (2021)

Authors:Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi Pages: 1 - 39 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we propose a novel space-dependent multiscale model for the spread of infectious diseases in a two-dimensional spatial context on realistic geographical scenarios. The model couples a system of kinetic transport equations describing a population of commuters moving on a large scale (extra-urban) with a system of diffusion equations characterizing the non-commuting population acting over a small scale (urban). The modeling approach permits to avoid unrealistic effects of traditional diffusion models in epidemiology, like infinite propagation speed on large scales and mass migration dynamics. A construction based on the transport formalism of kinetic theory allows to give a clear model interpretation to the interactions between infected and susceptible in compartmental space-dependent models. In addition, in a suitable scaling limit, our approach permits to couple the two populations through a consistent diffusion model acting at the urban scale. A discretization of the system based on finite volumes on unstructured grids, combined with an asymptotic preserving method in time, shows that the model is able to describe correctly the main features of the spatial expansion of an epidemic. An application to the initial spread of COVID-19 is finally presented. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S0218202521400017

Authors:Seung-Yeal Ha, Shi Jin, Doheon Kim, Dongnam Ko Pages: 1 - 37 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We present a uniform-in-time (and in particle numbers as well) error estimate for the random batch method (RBM) [S. Jin, L. Li and J.-G. Liu, Random batch methods (RBM) for interacting particle systems, J. Comput. Phys. 400 (2020) 108877] to the Cucker–Smale (CS) model. The uniform-in-time error estimates of the RBM have been obtained for various interacting particle systems, when corresponding flow generates a contraction semigroup. In this paper, we derive a uniform-in-time error estimate for RBM-approximation to the CS model in which the corresponding flow does not generate contractive semigroup. To derive uniform error estimate, we use asymptotic flocking estimate of the RBM-approximated CS model which yields the decay of relative velocities to zero, at least in the order of [math], while velocities of the original system decay exponentially. Here, [math] is the decay rate of the communication weight with respect to the distance between particles in the CS model. We also provide several numerical simulations to confirm the analytical results. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S0218202521400029

Authors:Jeongho Kim, David Poyato, Juan Soler Pages: 1 - 73 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker–Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes’ law. While the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S0218202521400042

Authors:Nicolás Barnafi, Gabriel N. Gatica, Daniel E. Hurtado, Willian Miranda, Ricardo Ruiz-Baier Pages: 1 - 42 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This work introduces and analyzes new primal and dual-mixed finite element methods for deformable image registration, in which the regularizer has a nontrivial kernel, and constructed under minimal assumptions of the registration model: Lipschitz continuity of the similarity measure and ellipticity of the regularizer on the orthogonal complement of its kernel. The aforementioned singularity of the regularizer suggests to modify the original model by incorporating the additional degrees of freedom arising from its kernel, thus granting ellipticity of the former on the whole solution space. In this way, we are able to prove well-posedness of the resulting extended primal and dual-mixed continuous formulations, as well as of the associated Galerkin schemes. A priori error estimates and corresponding rates of convergence are also established for both discrete methods. Finally, we provide numerical examples confronting our formulations with the standard ones, which prove our finite element methods to be particularly more efficient on the registration of translations and rotations, in addition for the dual-mixed approach to be much more suitable for the quasi-incompressible case, and all the above without losing the flexibility to solve problems arising from more realistic scenarios such as the image registration of the human brain. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S021820252150024X

Authors:Riccardo Durastanti, Lorenzo Giacomelli, Giuseppe Tomassetti Pages: 1 - 36 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-26T07:00:00Z DOI: 10.1142/S0218202521500160

Authors:Benedetto Piccoli, Francesco Rossi Pages: 1 - 40 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations (ODEs) with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing strength at the boundary. Various works in the literature analyzed properties of solutions, such as barycenter invariance and clustering. On the other side, the problem of giving a precise definition of solution, from an analytical point of view, was often overlooked. However, a rich literature proposing different concepts of solution to discontinuous differential equations is available. Using several concepts of solution, we show how existence is granted under general assumptions, while uniqueness may fail even in dimension one, but holds for almost every initial conditions. Consequently, various properties of solutions depend on the useddefinition and initial conditions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-22T07:00:00Z DOI: 10.1142/S0218202521400054

Authors:Alejandro Allendes, Enrique Otárola, Abner J. Salgado Pages: 1 - 39 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to [math], where [math] is a weight in the Muckenhoupt class [math] that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and [math], show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability. We also explore efficiency estimates. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-22T07:00:00Z DOI: 10.1142/S0218202521500196

Authors:Daewa Kim, Kaylie O’Connell, William Ott, Annalisa Quaini Pages: 1 - 26 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we present a computational modeling approach for the dynamics of human crowds, where the spreading of an emotion (specifically fear) has an influence on the pedestrians’ behavior. Our approach is based on the methods of the kinetic theory of active particles. The model allows us to weight between two competing behaviors depending on fear level: the search for less congested areas and the tendency to follow the stream unconsciously (herding). The fear level of each pedestrian influences their walking speed and is influenced by the fear levels of their neighbors. Numerically, we solve our pedestrian model with emotional contagion using an operator splitting scheme. We simulate evacuation scenarios involving two groups of interacting pedestrians to assess how domain geometry and the details of fear propagation impact evacuation dynamics. Further, we reproduce the evacuation dynamics of an experimental study involving distressed ants. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-17T07:00:00Z DOI: 10.1142/S0218202521400030

Authors:Andrea Cangiani, Emmanuil H. Georgoulis, Oliver J. Sutton Pages: 1 - 41 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the [math] and [math] norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-17T07:00:00Z DOI: 10.1142/S0218202521500172

Authors:Franco Flandoli, Eleonora La Fauci, Martina Riva Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. A Markov chain individual-based model for virus diffusion is investigated. Both the virus growth within an individual and the complexity of the contagion within a population are taken into account. A careful work of parameter choice is performed. The model captures very well the statistical variability of quantities like the incubation period, the serial interval and the time series of infected people in Tuscany towns. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-17T07:00:00Z DOI: 10.1142/S0218202521500226

Authors:Xiaofeng Yang Pages: 1 - 35 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we establish a new hydrodynamically coupled phase-field model for three immiscible fluid components system. The model consists of the Navier–Stokes equations and three coupled nonlinear Allen–Cahn type equations, to which we add nonlocal type Lagrange multipliers to conserve the volume of each phase accurately. To solve the model, a linear and energy stable time-marching method is constructed by combining the stabilized-Invariant Energy Quadratization (S-IEQ) approach and the projection method. The well-posedness of the scheme and its unconditional energy stability are rigorously proved. Several numerical simulations in 2D and 3D are carried out, including spinodal decomposition, dynamical deformations of a liquid lens and rising liquid drops, to validate the model and demonstrate the efficiency and energy stability of the proposed scheme, numerically. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-13T07:00:00Z DOI: 10.1142/S0218202521500184

Authors:Xiaoming Fu, Quentin Griette, Pierre Magal Pages: 1 - 45 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this work, we describe a hyperbolic model with cell–cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-13T07:00:00Z DOI: 10.1142/S0218202521500214

Authors:Parveena Shamim Abdul Salam, Wolfgang Bock, Axel Klar, Sudarshan Tiwari Pages: 1 - 19 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Modeling and simulation of disease spreading in pedestrian crowds have recently become a topic of increasing relevance. In this paper, we consider the influence of the crowd motion in a complex dynamical environment on the course of infection of the pedestrians. To model the pedestrian dynamics, we consider a kinetic equation for multi-group pedestrian flow based on a social force model coupled with an Eikonal equation. This model is coupled with a non-local SEIS contagion model for disease spread, where besides the description of local contacts, the influence of contact times has also been modeled. Hydrodynamic approximations of the coupled system are derived. Finally, simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigated, including uni- and bi-directional flow in a passage with and without obstacles. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521400066

Authors:Qingwu Gao, Jun Zhuang, Ting Wu, Houcai Shen Pages: 1 - 30 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Coronavirus Disease 2019 (COVID-19) is a zoonotic illness which has spread rapidly and widely since December, 2019, and is identified as a global pandemic by the World Health Organization. The pandemic to date has been characterized by ongoing cluster community transmission. Quarantine intervention to prevent and control the transmission are expected to have a substantial impact on delaying the growth and mitigating the size of the epidemic. To our best knowledge, our study is among the initial efforts to analyze the interplay between transmission dynamics and quarantine intervention of the COVID-19 outbreak in a cluster community. In the paper, we propose a novel Transmission-Quarantine epidemiological model by nonlinear ordinary differential equations system. With the use of detailed epidemiologic data from the Cruise ship “Diamond Princess”, we design a Transmission-Quarantine work-flow to determine the optimal case-specific parameters, and validate the proposed model by comparing the simulated curve with the real data. First, we apply a general SEIR-type epidemic model to study the transmission dynamics of COVID-19 without quarantine intervention, and present the analytic and simulation results for the epidemiological parameters such as the basic reproduction number, the maximal scale of infectious cases, the instant number of recovered cases, the popularity level and the final scope of the epidemic of COVID-19. Second, we adopt the proposed Transmission-Quarantine interplay model to predict the varying trend of COVID-19 with quarantine intervention, and compare the transmission dynamics with and without quarantine to illustrate the effectiveness of the quarantine measure, which indicates that with quarantine intervention, the number of infectious cases in 7 days decrease by about 60%, compared with the scenario of no intervention. Finally, we conduct sensitivity analysis to simulate the impacts of different parameters and different quarantine measures, and identify the optimal quarantine strategy that will be used by the decision makers to achieve the maximal protection of population with the minimal interruption of economic and social development. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521500147

Authors:Guoqiang Ren, Bin Liu Pages: 1 - 38 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this work, we consider the two-species chemotaxis system with Lotka–Volterra competitive kinetics in a bounded domain with smooth boundary. We construct weak solutions and prove that they become smooth after some waiting time. In addition, the asymptotic behavior of the solutions is studied. Our results generalize some well-known results in the literature. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521500238

Authors:Luis Almeida, Pierre-Alexandre Bliman, Grégoire Nadin, Benoît Perthame, Nicolas Vauchelet Pages: 1 - 31 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We formulate a general SEIR epidemic model in a heterogeneous population characterized by some trait in a discrete or continuous subset of a space [math]. The incubation and recovery rates governing the evolution of each homogeneous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines the probability for an individual of a given trait to be infected by an individual with another trait. Our goal is to derive and study the final size equation fulfilled by the limit distribution of the population. We show that this limit exists and satisfies the final size equation. The main contribution of this work is to prove the uniqueness of this solution among the distributions smaller than the initial condition. We also establish that the dominant eigenvalue of the next-generation operator (whose initial value is equal to the basic reproduction number) decreases along every trajectory until a limit smaller than 1. The results are shown to remain valid in the presence of a diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrices (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another). Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521500251

Authors:Martin Jesenko, Bernd Schmidt Pages: 1 - 32 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-03-22T07:00:00Z DOI: 10.1142/S0218202521500202

Authors:Eduard Feireisl, Mária Lukáčová–Medvi’ová, Bangwei She, Yue Wang Pages: 1 - 40 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We develop a method to compute effectively the Young measures associated to sequences of numerical solutions of the compressible Euler system. Our approach is based on the concept of [math]-convergence adapted to sequences of parameterized measures. The convergence is strong in space and time (a.e. pointwise or in certain [math] spaces) whereas the measures converge narrowly or in the Wasserstein distance to the corresponding limit. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-03-13T08:00:00Z DOI: 10.1142/S0218202521500123

Authors:Peter Y. H. Pang, Yifu Wang, Jingxue Yin Pages: 1 - 42 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This paper is concerned with a spatially two-dimensional version of a chemotaxis system with logistic cell proliferation and death, for a singular tactic response of standard logarithmic type, and with interaction with a surrounding incompressible fluid through transport and buoyancy. Systems of this form are of significant relevance to the understanding of chemotaxis-fluid interaction, but the rigorous knowledge of their qualitative properties is yet far from complete. In this direction, using the conditional energy functional method, the present work provides some interesting contributions by establishing results on global boundedness, and especially on large time stabilization toward homogeneous equilibria, under mild assumptions on the initial data and appropriate conditions on the strength of the damping death effects. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-03-06T08:00:00Z DOI: 10.1142/S0218202521500135

Authors:Hiromichi Itou, Victor A. Kovtunenko, Kumbakonam R. Rajagopal Pages: 1 - 26 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-02-26T08:00:00Z DOI: 10.1142/S0218202521500159

Authors:Qianyun Miao, Changhui Tan, Liutang Xue Pages: 1 - 52 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We study one-dimensional Eulerian dynamics with nonlocal alignment interactions, featuring strong short-range alignment, and long-range misalignment. Compared with the well-studied Euler-alignment system, the presence of the misalignment brings different behaviors of the solutions, including the possible creation of vacuum at infinite time, which destabilizes the solutions. We show that with a strongly singular short-range alignment interaction, the solution is globally regular, despite the effect of misalignment. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-02-16T08:00:00Z DOI: 10.1142/S021820252150010X