Showing papers in "Mathematical Models and Methods in Applied Sciences in 2002"
TL;DR: It was thought that the most important characteristics of soft tissues were their complex mechanical properties: they often exhibit nonlinear, anisotropic, nearly incoherent, and often incoherent properties as discussed by the authors.
Abstract: Not long ago it was thought that the most important characteristics of the mechanics of soft tissues were their complex mechanical properties: they often exhibit nonlinear, anisotropic, nearly inco...
663 citations
TL;DR: A critical review of some approaches devised to describe tumor growth can be found in this paper, where the authors start from the observation that the phenomenological description of a tumor spheroid suggests to model it as a growing and deformable porous material.
Abstract: Mass balance equations typically adopted to describe tumor growth are to be closed by introducing a suitable velocity field. The first part of this paper is devoted to a critical review of some approaches devised to this aim in the relevant literature. In the second part we start from the observation that the phenomenological description of a tumor spheroid suggests to model it as a growing and deformable porous material. The concept of volume fraction and the essentials of the mechanics of multicomponent continua are then introduced and applied to the problem at hand. The system of equations regulating such a system is stated and its validity is then discussed at the light of numerical simulations.
301 citations
TL;DR: This review reports the existing literature on traffic flow modelling in the framework of a critical overview which aims to indicate research perspectives and mainly refers to modelling by fluid dynamic and kinetic equations.
Abstract: This review reports the existing literature on traffic flow modelling in the framework of a critical overview which aims to indicate research perspectives The contents mainly refer to modelling by fluid dynamic and kinetic equations and are arranged in three parts The first part refers to methodological aspects of mathematical modelling and to the interpretation of experimental results The second part is devoted to modelling and deals both with methodological aspects and with the description of some specific models The third part reports about an overview on applications and research perspectives
187 citations
TL;DR: In this article, the effect of a third-order fluid on the peristaltic transport is analyzed in a circular cylindrical tube, such as some organs in the living body.
Abstract: The effect of a third-order fluid on the peristaltic transport is analysed in a circular cylindrical tube, such as some organs in the living body. The third-order flow of an incompressible fluid in a circular cylindrical tube, on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The wavelength of the peristaltic waves is assumed to be large compared to the tube average radius, whereas the amplitude of the wave need not be small compared to the average radius. Both analytic (perturbation) and numerical solutions are given. For the perturbation solution, a systematic approach based on an asymptotic expansion of the solution in terms of a small Deborah number is used and solutions up to the first order are presented in closed forms. The numerical solution, valid for any Deborah number, represents a new approach to peristaltic flows, and its features illuminate the physical behaviour much more than the analytical research on this problem. Comparison is made between the analytic (perturbation) and numerical results. Furthermore, the obtained results could also have applications to a range of peristaltic flows for a variety of non-Newtonian fluids such as aqueous solutions of high-molecular weight polyethylene oxide and polyacrylamide.
142 citations
TL;DR: In this article, hyperbolic models (in 1-D and n-D) and a transport model for chemosensitive movement are discussed and compared to the classical diffusion based Patlak-Keller-Segel model.
Abstract: Chemosensitive movement describes the active orientation of individuals on chemical signals. In cases of cellular slime molds or flagellated bacteria, chemosensitive movement leads to aggregation and pattern formation. The classical mathematical model to describe chemosensitive movement is the diffusion based Patlak–Keller–Segel model. It suffers from the drawback of infinite propagation speeds. The relevant model parameters (motility and chemosensitivity) are related to population statistics. Hyperbolic models respect finite propagation speeds and the relevant model parameters (turning rate, distribution of new chosen velocities) are based on the individual movement patterns of the species at hand. In this paper hyperbolic models (in 1-D) and a transport model (in n-D) for chemosensitive movement are discussed and compared to the classical model. For the hyperbolic and transport models the following topics are reviewed: parabolic limit (which in some cases leads to the Patlak–Keller–Segel model), local and global existence, asymptotic behavior and moment closure. The moment closure approach leads to models based on Cattaneo's law of heat conduction (telegraph equation).
121 citations
TL;DR: In this paper, the authors deal with the development of suitable general mathematical structures including a large variety of Boltzmann type models and develop a critical analysis towards research perspectives both on modeling and analytic problems.
Abstract: This paper deals with the development of suitable general mathematical structures including a large variety of Boltzmann type models. The contents are organized in three parts. The first part is devoted to modeling the above general framework. The second part to the development of specific models of interest in applied sciences. The third part develops a critical analysis towards research perspectives both on modeling and analytic problems.
117 citations
TL;DR: In this paper, a change of variables that turns the critical nonlinear Schrodinger equation into the critical nonsmrodinger equations with isotropic harmonic potential, in any space dimension, is presented.
Abstract: We use a change of variables that turns the critical nonlinear Schrodinger equation into the critical nonlinear Schrodinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on L2, and bijective on some time intervals. Using the known results for the critical nonlinear Schrodinger equation, this provides information for the properties of Bose–Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon.
82 citations
TL;DR: A discontinuous Galerkin approximation for the Stokes problem is proposed and a priori error estimates generalizing the abstract theory of mixed methods are derived.
Abstract: We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability.
79 citations
TL;DR: In this article, the authors considered a variant of transport equations for age-structured populations, which allows one to introduce more stochasticity in the birth process and in the aging phenomena.
Abstract: We consider models for population structured by maturation/maturation speed proposed by Rotenberg. It is a variant of transport equations for age-structured populations which presents particularly interesting mathematical difficulties. It allows one to introduce more stochasticity in the birth process and in the aging phenomena. We present a new method for studying the time asymptotics which is also illustrated on the simpler McKendrick–Von Foerster model. The nonlinear variants of these models are shown to exhibit either nonlinear stability or periodic solutions depending on the datum.
73 citations
TL;DR: In this article, it was shown that the free boundary problem for an incompressible Euler equation with surface tension is uniquely solvable, locally in time, in a class of functions of finite smoothness.
Abstract: We prove that a free boundary problem for an incompressible Euler equation with surface tension is uniquely solvable, locally in time, in a class of functions of finite smoothness. Moreover, it is shown that the solution of this problem converges to the solution of the problem without surface tension as the coefficient of the surface tension tends to zero.
71 citations
TL;DR: In this article, a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo is studied, where in each step, in a sense, local minimizers which are sufficiently close to the approximate solution obtained in the previous step are considered.
Abstract: We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.
TL;DR: In this paper, the authors characterized the functionals which are Mosco-limits in the L2(Ω) topology of some sequence of functionals of the kind where Ω is a bounded domain of ℝN (N ≥ 3).
Abstract: We characterize the functionals which are Mosco-limits, in the L2(Ω) topology, of some sequence of functionals of the kind where Ω is a bounded domain of ℝN (N ≥ 3). It is known that this family of functionals is included in the closed set of Dirichlet forms. Here, we prove that the set of Dirichlet forms is actually the closure of the set of diffusion functionals. A crucial step is the explicit construction of a composite material whose effective energy contains a very simple nonlocal interaction.
TL;DR: The existence of weak solutions locally in time to the quantum hydrodynamic equations in bounded domains is shown in this article, based on a formulation of the problem as a nonlinear Schrodinger-Poisson system and using semigroup theory and fixed-point techniques.
Abstract: The existence of weak solutions locally in time to the quantum hydrodynamic equations in bounded domains is shown. These Madelung-type equations consist of the Euler equations, including the quantum Bohm potential term, for the particle density and the particle current density and are coupled to the Poisson equation for the electrostatic potential. This model has been used in the modeling of quantum semiconductors and superfluids. The proof of the existence result is based on a formulation of the problem as a nonlinear Schrodinger–Poisson system and uses semigroup theory and fixed-point techniques.
TL;DR: In this article, the authors considered a two-phase model described by a pressureless gas system with unilateral constraint and proved weak stability and the existence of weak solutions by passing to the limit in the sticky-blocks dynamics.
Abstract: We consider a two-phase model described by a pressureless gas system with unilateral constraint. We prove weak stability and the existence of weak solutions by passing to the limit in the sticky-blocks dynamics. We obtain the maximum principle on the velocity, the Oleinik entropy condition and local entropy inequalities. Initial data are taken in a very weak sense since the solution can jump initially in time.
TL;DR: In this paper, the convergence of the fully discretized scheme (finite element in space, finite difference in time, plus Monte Carlo realizations) towards the coupled solution of a partial differential equation/stochastic differential equation system was studied.
Abstract: We present in this paper the numerical analysis of a simple micro–macro simulation of a polymeric fluid flow, namely the shear flow for the Hookean dumbbells model. Although restricted to this academic case (which is however used in practice as a test problem for new numerical strategies to be applied to more sophisticated cases), our study can be considered as a first step towards that of more complicated models. Our main result states the convergence of the fully discretized scheme (finite element in space, finite difference in time, plus Monte Carlo realizations) towards the coupled solution of a partial differential equation/stochastic differential equation system.
TL;DR: In this paper, the design problem for semiconductor devices via an optimal control approach for the standard drift-diffusion model was studied via an optimization approach for a symmetric n-p-diode and the solvability of the minimization problem was proved.
Abstract: The design problem for semiconductor devices is studied via an optimal control approach for the standard drift–diffusion model. The solvability of the minimization problem is proved. The first-order optimality system is derived and the existence of Lagrange-multipliers is established. Further, estimates on the sensitivities are given. Numerical results concerning a symmetric n–p-diode are presented.
TL;DR: A model and theoretical basis of planning applying the Boltzmann-transport equation in dose calculation and MLC delivery technique is given and the existence of solutions and the optimal treatment planning are considered.
Abstract: In the external radiation therapy the source of radiation is from outside. The healthy tissue and some organs, called critical organs which are quite intolerable for radiation, are always irradiated, too. Therefore, the careful treatment plan has to be constructed to ensure high and homogeneous dose in the tumor, but on the other hand to spare the normal tissue and critical organs possibly well. In the radiation therapy treatment planning one tries to optimize the dose distribution in the way that the above aim is satisfied. The dose distributions can be generated with different techniques. The most recent of them is the so-called multileaf collimator (MLC) delivery technique. Calculation of the dose distribution demands some dose calculation model. The paper gives a model and theoretical basis of planning applying the Boltzmann-transport equation in dose calculation and MLC delivery technique. The existence of solutions and the optimal treatment planning are considered. A preliminary artificial computer simulation is included.
TL;DR: In this article, the inelastic Maxwell gas is studied as an introductory mean field model that has the major advantage of being exactly resoluble in the case of scalar velocities, showing an asymptotic velocity distribution with power law tails.
Abstract: In the present paper we review some recent progresses in the study of the dynamics of cooling granular gases, obtained using idealized models to address different issues of their kinetics. The inelastic Maxwell gas is studied as an introductory mean field model that has the major advantage of being exactly resoluble in the case of scalar velocities, showing an asymptotic velocity distribution with power law tails |v|-4. More realistic models can be obtained placing the same process on a spatial lattice. Two regimes are observed: an uncorrelated transient followed by a dynamical stage characterized by correlations in the velocity field in the form of shocks and vortices. The lattice models, in one and two dimensions, account for different numerical measurements: some of them agree with the already known results, while others have never been efficiently measured and shed light on the deviation from homogeneity. In particular in the velocity-correlated regime the computation of structure factors gives indication of a dynamics similar to that of a diffusion process on large scales with a more complex behavior at shorter scales.
TL;DR: In this paper, a mathematical description of human feelings (such as hostility, or indifference or love) toward other human beings is presented, which is carried out in a somewhat simplified scheme, by describing the psychological state of each individual by only one state variable, conceived as a kind of "measure" of its feelings.
Abstract: The paper deals with a mathematical description of human feelings (such as hostility, or indifference or love) toward other human beings. The discussion is carried out in a somewhat simplified scheme, by describing the psychological state of each individual by only one state variable, conceived as a kind of "measure" of its feelings. This is, however, to be meant as simply a first step towards a more comprehensive description of human psychology, taking into account character, tastes and possibly past experiences of each individual involved in a social relation. For the simplified description here developed, we propose a system of nonlinear integro-differential stochastic equations, aiming at giving at least a probabilistic forecast of the evolution of reciprocal feeling of two relating individuals. Thus, the solution of the system will be a couple of probability density functions, for which we present a short preliminary discussion of existence and uniqueness as well as of stability and the possible occurrence of strong instability effects. A number of numerical simulations are presented to test the significance of the model and some ways to extend the scheme, by introducing a number of additional external as well as internal parameters that can reasonably be assumed to influence the reciprocal feelings, are suggested and discussed, with special attention to the concept of stochastic "constitutive laws" for the feelings.
TL;DR: In this paper, the authors studied the direct and inverse approximation theorems of the p-version of the finite element method in the framework of the weighted Besov spaces, and proved the optimal rate of convergence for elliptic boundary value problems on polygonal domains.
Abstract: This is the second of a series devoted to the direct and inverse approximation theorems of the p-version of the finite element method in the framework of the weighted Besov spaces. In this paper, we combine the approximability of singular solutions in the Jacobi-weighted Besov spaces, which were analyzed in the previous paper,4 with the technique of partition of unity in order to prove the optimal rate of convergence of the p-version of the finite element method for elliptic boundary value problems on polygonal domains.
TL;DR: In this article, the steady translational fall of a homogeneous body of revolution around an axis a, with fore-and-aft symmetry, in a second-order liquid at nonzero Reynolds (Re) and Weissenberg (We) numbers was studied.
Abstract: We study the steady translational fall of a homogeneous body of revolution around an axis a, with fore-and-aft symmetry, in a second-order liquid at nonzero Reynolds (Re) and Weissenberg (We) numbers. We show that, at first order in these parameters, only two orientations are allowed, namely, those with a either parallel or perpendicular to the direction of the gravity g. In both cases the translational velocity is parallel to g. The stability of the orientations can be described in terms of a critical value Ec for the elasticity number E = We/Re, where Ec depends only on the geometric properties of the body, such as size or shape, and on the quantity (Ψ1 + Ψ2)/Ψ1, where Ψ1 and Ψ2 are the first and second normal stress coefficients. These results are then applied to the case when the body is a prolate spheroid. Our analysis shows, in particular, that there is no tilt-angle phenomenon at first order in Re and We.
TL;DR: In this article, a Cauchy-Dirichlet quasilinear parabolic problem with a gradient lower order term was studied, and it was shown that if p ≥ 1, γ ≥ ½ and p < 2 γ + 2, then there exists a global weak solution for all initial data in L 1 (Ω).
Abstract: In this paper we deal with a Cauchy–Dirichlet quasilinear parabolic problem containing a gradient lower order term; namely, ut - Δu + |u|2 γ-2u |∇u|2 = |u|p-2u. We prove that if p ≥ 1, γ ≥ ½ and p < 2 γ + 2, then there exists a global weak solution for all initial data in L1 (Ω). We also see that there exists a non-negative solution if the initial datum is non-negative.
TL;DR: A review of some results and research perspectives for the general class of bilinear systems of Boltzmann-like integro-differential equations (generalized kinetic models) describing the dynamics of individuals undergoing kinetic (stochastic) interactions is presented in this paper.
Abstract: In this paper a review of some results and research perspectives for the general class of bilinear systems of Boltzmann-like integro-differential equations (generalized kinetic models) describing the dynamics of individuals undergoing kinetic (stochastic) interactions is presented. Some macroscopic limits (the diffusive limit and the hydrodynamic limit) are discussed.
TL;DR: In this article, a class of infinite systems of linear ODEs was considered and it was shown that these systems are topologically chaotic. But the assumption on the "birth" coefficients, the "death" coefficients and the averages of the life-spans was not made.
Abstract: In this paper we consider a class of infinite systems of linear ODEs. Each system corresponds to a process characterized by two components: the conservative one (birth-and-death process) and the proliferative one. A system of this type can describe the population of neoplastic cells divided into subpopulations characterized by different levels of cellular resistance to antineoplastic drugs. Under suitable assumptions on the "birth" (amplification) coefficients, the "death" (deamplification) coefficients and the averages of the life-spans we prove that this class of models is topologically chaotic.
TL;DR: In this article, the Rational Large Eddy Simulation (RLEDD) model was used to prove the existence and uniqueness of a class of strong solutions for the Euler and Navier-Stokes equations.
Abstract: In this paper we consider the Rational Large Eddy Simulation model recently introduced by Galdi and Layton We briefly present this model, which (in principle) is similar to others commonly used, and we prove the existence and uniqueness of a class of strong solutions Contrary to the gradient model, the main feature of this model is that it allows a better control of the kinetic energy Consequently, to prove existence of strong solutions, we do not need subgrid-scale regularization operators, as proposed by Smagorinsky We also introduce some breakdown criteria that are related to the Euler and Navier–Stokes equations
TL;DR: In this paper, a connection between the asymptotic behavior and real interpolation theory is established, and a detailed study of the cases when neither the bending energy nor the membrane energy dominate is provided.
Abstract: The shell problem and its asymptotic are investigated. A connection between the asymptotic behavior and real interpolation theory is established. Thus, a detailed study of the cases when neither the bending energy nor the membrane energy dominate is provided. An application to a cylindrical shell is also detailed. Although only the Koiter shells have been considered, the same procedure can be used for other models, such as Naghdi's one, for example.
TL;DR: In this paper, numerical approximations for positive solutions of nonlinear heat equations with a nonlinear boundary condition are studied in terms of the nonlinearities when solutions of a semidiscretization in space exist globally in time and when they blow up in finite time.
Abstract: In this paper we study numerical approximations for positive solutions of a nonlinear heat equation with a nonlinear boundary condition. We describe in terms of the nonlinearities when solutions of a semidiscretization in space exist globally in time and when they blow up in finite time. We also find the blow-up rates and the blow-up sets. In particular we prove that regional blow-up is not reproduced by the numerical scheme. However, in the appropriate variables we can reproduce the correct blow-up set when the mesh parameter goes to zero.
TL;DR: In this article, the governing nonlinear equation for the unsteady flow of an incompressible fourth grade fluid is modelled and the fluid is also subjected to a magnetic field.
Abstract: The governing nonlinear equation for the unsteady flow of an incompressible fourth grade fluid is modelled. The fluid is also subjected to a magnetic field. In addition, we investigate steady flow between parallel plates with one plate at rest and the other moving parallel to it at constant speed with a suction velocity normal to the plates. Boundary conditions play a significant role. We construct the numerical solution to the sixth order nonlinear differential equation. It is found that the velocity increases with the increase in the material parameters of the fourth grade terms of the fluid.
TL;DR: In this article, the authors prove the convergence of the solutions for the incompressible homogeneous magnetohydrodynamics (MHD) system to the solutions to ideal MHD one in the inviscid and non-resistive limit.
Abstract: We prove the convergence of the solutions for the incompressible homogeneous magnetohydrodynamics (MHD) system to the solutions to ideal MHD one in the inviscid and non-resistive limit, detailing the explicit convergence rates. For this study we consider a fluid occupying the whole space ℝ3 and we assume that the viscosity effects in this fluid can be described by two different operators: the usual Laplacian operator affected by the inverse of the Reynolds number or by a viscosity operator introduced by S. I. Braginskii in 1965.
TL;DR: In this paper, a model consisting of a fluid described by the incompressible Navier-Stokes equations interacting with a solid under large deformations is presented, and a linearized problem is formulated to compute the derivative of the state variable with respect to a given boundary parameter.
Abstract: This paper deals with problems arising in the sensitivity analysis for fluid-structure interaction systems. Our model consists of a fluid described by the incompressible Navier–Stokes equations interacting with a solid under large deformations. We obtain a linearized problem which allow us to compute the derivative of the state variable with respect to a given boundary parameter. We use a particular definition of the first-order correction for the perturbed state and consider a weak arbitrary Euler–Lagrange formulation for the coupled system.