Showing papers in "Mathematical Models and Methods in Applied Sciences in 2007"
TL;DR: The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of mixture theory; the equations however not having been developed in such a manner by Fick or Darcy.
Abstract: The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of mixture theory; the equations however not having been developed in such a manner by Fick or Darcy. Relaxing the assumptions made in deriving these equations via mixture theory selectively leads to a hierarchy of mathematical models and it can be shown that popular models due to Brinkman, Biot and many others can be obtained via various approximations. It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing interesting technological applications.
265 citations
TL;DR: For the viscous and heat-conductive fluids governed by the Navier-Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity.
Abstract: For the viscous and heat-conductive fluids governed by the compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp - Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.
215 citations
TL;DR: This paper develops the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems and derivesable upper and lower bounds on the error measured in terms of a natural energy norm.
Abstract: In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.
175 citations
TL;DR: A new mathematical model of tumor spheroid growth is proposed that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages.
Abstract: Tumor spheroids grown in vitro have been widely used as models of in vivo tumor growth because they display many of the characteristics of in vivo growth, including the effects of nutrient limitations and perhaps the effect of stress on growth. In either case there are numerous biochemical and biophysical processes involved whose interactions can only be understood via a detailed mathematical model. Previous models have focused on either a continuum description or a cell-based description, but both have limitations. In this paper we propose a new mathematical model of tumor spheroid growth that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages. In this model the cell-based description is used in the region where the majority of growth and cell division occurs, at the periphery of a tumor, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of two important nutrients, oxygen and glucose, throughout the entire domain. The cell-based component of this hybrid model allows us to examine the effects of cell–cell adhesion and variable growth rates at the cellular level rather than at the continuum level. We show that the model can predict a number of cellular behaviors that have been observed experimentally.
170 citations
TL;DR: In this paper, the existence of weak solutions for a coupled system of kinetic and fluid equations was proved in a bounded domain of ℝ3 with homogeneous Dirichlet conditions on the fluid velocity field and Dirichlets or reflection boundary conditions on a kinetic distribution function.
Abstract: We establish the existence of a weak solutions for a coupled system of kinetic and fluid equations. More precisely, we consider a Vlasov–Fokker–Planck equation coupled to compressible Navier–Stokes equation via a drag force. The fluid is assumed to be barotropic with γ-pressure law (γ > 3/2). The existence of weak solutions is proved in a bounded domain of ℝ3 with homogeneous Dirichlet conditions on the fluid velocity field and Dirichlet or reflection boundary conditions on the kinetic distribution function.
152 citations
TL;DR: In this article, the authors considered the problem of finding the free boundary of a tumor in a well-defined region in space, where the boundary of this region is held together by the forces of cell-to-cell adhesion.
Abstract: In the last four decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. In this paper, we deal with tumor models in which the tumor occupies a well-defined region in space; the boundary of this region is held together by the forces of cell-to-cell adhesion. We shall refer to such tumors as "solid" tumors, although they may sometimes consist of fluid-like tissue, such as in the case of brain tumors (e.g. gliomas) and breast tumors. The most common class of solid tumors is carcinoma: a cancer originating from epithelial cells, that is, from the closely packed cells which align the internal cavities of the body. Models of solid tumors must take spatial effects into account, and are therefore described in terms of partial differential equations (PDEs). They also need to take into account the fact that the tumor region is changing in time; in fact, the tumor region, say Ω(t), and its boundary Γ(t), are unknown in advance. Thus one needs to determine both the unknown "free boundary" Γ(t) together with the solution of the PDEs in Ω(t). These types of problems are called free boundary problems. The models described in this paper are free boundary problems, and our primary interest is the spatial/geometric features of the free boundary. Some of the basic questions we shall address are: What is the shape of the free boundary? How does the free boundary behave as t → ∞? Does the tumor volume increase or shrink as t → ∞? Under what conditions does the tumor eventually become dormant? Finally, we shall explore the dependence of the free boundary on some biological parameters, and this will give rise to interesting bifurcation phenomena. The structure of the paper is as follows. In Secs. 1 and 2 we consider models in which all the cells are of one type, they are all proliferating cells. The tissue is modeled either as a porous medium (in Sec. 1) or as a fluid medium (in Sec. 2). The models are extended in Secs. 3 and 4 to include three types of cells: proliferating, quiescent, and dead. Finally, in Sec. 5 we outline a general multiphase model that includes gene mutations.
128 citations
TL;DR: This paper addresses the numerical simulation of fluid-structure interaction problems characterized by a strong added-mass effect and proposes a semi-implicit coupling scheme based on an algebraic fractional-step method for the first time.
Abstract: We address the numerical simulation of fluid-structure interaction problems characterized by a strong added-mass effect. We propose a semi-implicit coupling scheme based on an algebraic fractional-step method. The basic idea of a semi-implicit scheme consists in coupling implicitly the added-mass effect, while the other terms (dissipation, convection and geometrical nonlinearities) are treated explicitly. Thanks to this kind of explicit–implicit splitting, computational costs can be reduced (in comparison to fully implicit coupling algorithms) and the scheme remains stable for a wide range of discretization parameters. In this paper we derive this kind of splitting from the algebraic formulation of the coupled fluid-structure problem (after finite-element space discretization). From our knowledge, it is the first time that algebraic fractional step methods, used thus far only for fluid problems in computational domains with rigid boundaries, are applied to fluid-structure problems. In particular, for the specific semi-implicit method presented in this work, we adapt the Yosida scheme to the case of a coupled fluid-structure problem. This scheme relies on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the fluid-structure system. We analyze the numerical performances of this scheme on 2D fluid-structure simulations performed with a simple 1D structure model.
128 citations
TL;DR: In this paper, the authors generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term, and obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.
Abstract: The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.
120 citations
TL;DR: A discrete velocity mathematical model for vehicular traffic along a one-way road using the kinetic scale to capture the probabilistic essence of the interactions among the vehicles and offers the opportunity of a profitable analytical investigation of the relevant global features of the system.
Abstract: Following some general ideas on the discrete kinetic and stochastic game theory proposed by one of the authors in a previous work, this paper develops a discrete velocity mathematical model for vehicular traffic along a one-way road. The kinetic scale is chosen because, unlike the macroscopic one, it allows to capture the probabilistic essence of the interactions among the vehicles, and offers at the same time, unlike the microscopic one, the opportunity of a profitable analytical investigation of the relevant global features of the system. The discretization of the velocity variable, rather than being a pure mathematical technicality, plays a role in including the intrinsic granular nature of the flow of vehicles in the mathematical theory of traffic. Other important characteristics of the model concern the gain and loss terms of the kinetic equations, namely the construction of a density-dependent table of games to model velocity transitions and the introduction of a visibility length to account for non...
112 citations
TL;DR: In this paper, the authors deal with the analysis of the asymptotic limit towards the derivation of hyperbolic macroscopic equations for a class of equations modeling complex multicellular systems.
Abstract: This paper deals with the analysis of the asymptotic limit towards the derivation of hyperbolic macroscopic equations for a class of equations modeling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferating destructive events related to growth of tumor cells in competition with the immune system. The asymptotic analysis refers to the hyperbolic limit to show how the macroscopic tissue behavior can be described by linear and nonlinear hyperbolic systems which seem the most natural in this context.
107 citations
TL;DR: In this article, a stable formula for recovering a planar function from the circular Radon transform was proposed, which can be used to obtain an exact three-dimensional imaging algorithm for TACT.
Abstract: Thermoacoustic computed tomography (TACT) is an emerging hybrid imaging method for non-invasive medical diagnosis and fully three-dimensional visualization of biological probes. Within this modality electromagnetic illumination is used to induce acoustic waves inside an object of interest. In this paper, we assume that a cylindrical array of line detectors is used to record the acoustical data. This leads to the mathematical problem of inverting the circular Radon transform. The circular Radon transform arises in several other up-to-date imaging modalities, such as RADAR imaging or ultrasound tomography. In this paper we prove a novel stable formula for recovering a planar function from its circular Radon transform. We apply this formula to obtain an exact three-dimensional imaging algorithm for TACT. Numerical reconstructions from real and synthetic data demonstrate the potential and robustness of our algorithm.
TL;DR: In this article, the existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven, and the model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface.
Abstract: The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre–Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus μc = 0, one recovers the infinitesimal-displacement Reissner–Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for μc = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H1(ω, ℝ3). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.
TL;DR: Convergence results for general (successive) subspace correction methods for solving nearly singular systems of equations are discussed and parameter independent estimates under appropriate assumptions on the subspace solvers and space decompositions are provided.
Abstract: In this paper, we discuss convergence results for general (successive) subspace correction methods for solving nearly singular systems of equations. We provide parameter independent estimates under appropriate assumptions on the subspace solvers and space decompositions. The main assumption is that any component in the kernel of the singular part of the system can be decomposed into a sum of local (in each subspace) kernel components. This assumption also covers the case of "hidden" nearly singular behavior due to decreasing mesh size in the systems resulting from finite element discretizations of second order elliptic problems. To illustrate our abstract convergence framework, we analyze a multilevel method for the Neumann problem (H(grad) system), and also two-level methods for H(div) and H(curl) systems.
TL;DR: In this article, the authors address the problem of existence, approximation, and uniqueness of solutions to an abstract doubly nonlinear equation, modeling a rate-independent process with hysteretic behavior.
Abstract: In this paper, we address the problem of existence, approximation, and uniqueness of solutions to an abstract doubly nonlinear equation, modeling a rate-independent process with hysteretic behavior. Models of this kind arise in, e.g., plasticity, solid phase transformations, and several other problems in non smooth mechanics. Existence of solutions is proved via passage to the limit in a time-discretization scheme, whereas uniqueness results are obtained by means of convex analysis techniques.
TL;DR: In this article, the variational limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems is analyzed, where the lattice size tends to zero when the energy densities are of multiwell or Lennard-Jones type.
Abstract: We analyze the variational limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems as the lattice size tends to zero when the energy densities are of multiwell or Lennard–Jones type. Properly scaling the energies, we study several phenomena as the formation of boundary layers and phase transitions. We also study the presence of local patterns and of anti-phase transitions in the asymptotic behaviour of the ground states of NNN model subject to Dirichlet boundary conditions. We use this information to prove a localization of fracture result in the case of Lennard–Jones type potentials.
TL;DR: In this article, a new variant of the diamond reconstruction algorithm based on linear least squares is proposed, which unifies the treatment of internal and boundary vertices and includes information from boundaries as linear constraints of the Least Squares minimization process.
Abstract: The accuracy of the diamond scheme is experimentally investigated for anisotropic diffusion problems in two space dimensions. This finite volume formulation is cell-centered on unstructured triangulations and the numerical method approximates the cell averages of the solution by a suitable discretization of the flux balance at cell boundaries. The key ingredient that allows the method to achieve second-order accuracy is the reconstruction of vertex values from cell averages. For this purpose, we review several techniques from the literature and propose a new variant of the reconstruction algorithm that is based on linear Least Squares. Our formulation unifies the treatment of internal and boundary vertices and includes information from boundaries as linear constraints of the Least Squares minimization process. It turns out that this formulation is well-posed on those unstructured triangulations that satisfy a general regularity condition. The performance of the finite volume method with different algorithms for vertex reconstructions is examined on three benchmark problems having full Dirichlet, Dirichlet-Robin and Dirichlet–Neumann boundary conditions. Comparison of experimental results shows that an important improvement of the accuracy of the numerical solution is attained by using our Least Squares-based formulation. In particular, in the case of Dirichlet–Neumann boundary conditions and strongly anisotropic diffusions the good behavior of the method relies on the absence of locking phenomena that appear when other reconstruction techniques are used.
TL;DR: This work uses a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution.
Abstract: The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully nonlinearly coupled formulation for the study of fluid structure interactions. Many numerical methods have been introduced to reduce the difficulties related to the nonlinear coupling between the structure and the fluid evolution. However numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates of the variational formulation of the immersed boundary method. A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly. We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.
TL;DR: In this paper, the authors deal with the derivation of hybrid mathematical structures to describe the behavior of large systems of active particles by ordinary differential equations with stochastic coefficients whose evolution is modelled by equations of the mathematical kinetic theory.
Abstract: This paper deals with the derivation of hybrid mathematical structures to describe the behavior of large systems of active particles by ordinary differential equations with stochastic coefficients whose evolution is modelled by equations of the mathematical kinetic theory. A preliminary analysis shows how the above tools can be used to model complex systems of interest in applied sciences, with special attention to the immune competition.
TL;DR: In this paper, the existence theory for quasistatic initial-boundary value problems with internal variables is studied, where a system of linear partial differential equations coupled with a nonlinear system of differential equations or differential inclusions must be solved.
Abstract: We study the existence theory to quasistatic initial-boundary value problems with internal variables, which model the viscoelastic or viscoplastic behavior of solids at small strain. In these problems a system of linear partial differential equations coupled with a nonlinear system of differential equations or differential inclusions must be solved. The solution theory is based on monotonicity properties of the differential equations or differential inclusions. The differential inclusions considered in this article belong to the class of constitutive relations of monotone type with positive definite free energy and typically model solids with linear hardening behavior. Models for solids without hardening or with nonlinear hardening are considered in a companion article.
TL;DR: In this article, the authors considered the hyperbolic relaxation of the Cahn-Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain Ω of ℝ3.
Abstract: In this paper we consider the hyperbolic relaxation of the Cahn–Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain Ω of ℝ3. This equation is characterized by the presence of the additional inertial term ∊utt that accounts for the relaxation of the diffusion flux. For this equation we address the problem of the long time stability from the point of view of global attractors. The main difficulty in dealing with this system is the low regularity of its weak solutions, which prevents us from proving a uniqueness result and a proper energy identity for the solutions. We overcome this difficulty by using a density argument based on a Faedo–Galerkin approximation scheme and the recent J. M. Ball's theory of generalized semiflows. Moreover, we address the problem of the approximation of the attractor of the continuous problem with the one of Faedo–Galerkin scheme. Finally, we show that the same type of results hold also for the damped semilinear wave equation when the nonlinearity ϕ is not Lipschitz continuous and has a super critical growth.
TL;DR: The picture emerging from the modelling indicates that production of growth factors by cells considered may lead to diffusion-driven instability, which in turn may lead either to decay of both population, or to emergence of local growth foci, represented by spike-like solutions.
Abstract: The generally accepted Moolgavkar's theory of carcinogenesis assumes that all cancers are clonal, i.e. that they arise from progressive genetic deregulation in a cell pedigree originating from a single ancestral cell.18 However, recently the clonal theory has been challenged by the field theory of carcinogenesis, which admits the possibility of simultaneous changes in tissue subject to carcinogenic agents, such as tobacco smoke in lung cancer. Axelrod et al.1 formulated a more detailed framework, in which partially transformed cells depend in a mutualistic way on growth factors they produce, in this way enabling these cells to proliferate and undergo further transformations. On the other hand, the field theory assumes spatial distribution of precancerous cells and indeed there exists evidence that early-stage precancerous lesions in lung cancer progress along linear, tubular, or irregular surface structures. This seems to be the case for the atypical adenomatous hyperplasia (AAH),10 a likely precursor of adenocarcinoma of the lung. In this paper we explore the consequences of linking the model of spatial growth of precancerous cells,12 with the mutualistic hypothesis. We investigate the solutions of the model using analytical and computational techniques. The picture emerging from our modelling indicates that production of growth factors by cells considered may lead to diffusion-driven instability, which in turn may lead either to decay of both population, or to emergence of local growth foci, represented by spike-like solutions. Mutualism may, in some situations, increase the stability of solutions. One important conclusion is that models of field carcinogenesis, which include spatial effects, generally have very different behaviour compared to ODE models.
TL;DR: In this article, the authors investigated the viscous model of quantum hydrodynamics in one and higher space dimensions, and exploited the entropy dissipation method to prove the exponential decay to the thermal equilibrium state in one, two, and three dimensions, provided that the domain is a box.
Abstract: We investigate the viscous model of quantum hydrodynamics in one and higher space dimensions. Exploiting the entropy dissipation method, we prove the exponential decay to the thermal equilibrium state in one, two, and three dimensions, provided that the domain is a box. Further, we show the local in time existence of a solution in the one-dimensional case; and in the case of higher dimensions under the assumption of periodic boundary conditions. Finally, we prove the global existence in a one-dimensional setting under additional assumptions.
TL;DR: In this paper, a global in time abstract existence and uniqueness result for a general parabolic problem of reconstruction of a convolution kernel was proved for the theory of heat conduction for materials with memory.
Abstract: We prove a global in time abstract existence and uniqueness result for a general parabolic problem of reconstruction of a convolution kernel. The result is, in particular, applicable to the theory of heat conduction for materials with memory.
TL;DR: In this article, a finite element implementation of a Cosserat elasto-plastic model is presented and a rigorous numerical analysis of the introduced time-incremental algorithm is provided.
Abstract: We present a finite element implementation of a Cosserat elasto-plastic model and provide a rigorous numerical analysis of the introduced time-incremental algorithm The model allows the use of standard tools from convex analysis as known from classical Prandtl–Reuss plasticity We derive the dual stress formulation and prove that for vanishing Cosserat couple modulus µc → 0 the classical problem is approximated Our numerical results show the robustness of the approximation Notably, for positive couple modulus µc > 0 there is no need for a safe-load assumption For small µc the response is numerically indistinguishable from the classical response
TL;DR: In this article, the convergence of a global solution to an equilibrium as time goes to infinity was proved by means of a suitable Łojasiewicz-Simon type inequality for the system with homogeneous Neumann boundary conditions for both ¸ and χ.
Abstract: This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.
TL;DR: Three stochastic models of cell populations with a constant size are considered: a simple mass-action model, a spatial model and a hierarchical model which contains stem cells and daughter cells, which shows that hierarchical tissue organization lowers the risk of cancerous transformations.
Abstract: Here we review some spatial and non-spatial stochastic methods developed to study the dynamics of cancer progression. We illustrate the methodology with applications to the two most common patterns in cancer initiation and progression: loss-of-function and gain-of-function mutations. An example of a gain-of-function mutation is an activation of an oncogene; for such mutations we are interested in the process of mutant take-over. An example of a loss-of-function mutation is an inactivation of a tumor suppressor gene; for such processes we calculate the rate of production of double-hit mutants. We consider three stochastic models of cell populations with a constant size: a simple mass-action model, a spatial model and a hierarchical model which contains stem cells and daughter cells. Interestingly, the process of mutation accumulation and spread develops differently in different models. This suggests that simple mass-action models can be misleading when studying cancer dynamics. Moreover, our results also allow us to think about various types of tissue architecture and its protective role against cancer. In particular, we show that hierarchical tissue organization lowers the risk of cancerous transformations. Also, cellular motility and long-range signaling can decrease the risk of cancer in solid tissues.
TL;DR: In this paper, the authors consider time-dependent variational and quasi-variational inequalities and study under which assumptions the continuity of solutions with respect to time can be ensured, making an appropriate use of the set convergence in Mosco's sense.
Abstract: The aim of this paper is to consider time-dependent variational and quasi-variational inequalities and to study under which assumptions the continuity of solutions with respect to time can be ensured. Making an appropriate use of the set convergence in Mosco's sense, we are able to prove continuity results for strongly monotone variational and quasi-variational inequalities. The continuity results allow us to provide a discretization procedure for the calculation of solutions to the variational inequalities and, as a consequence, we can solve the time-dependent traffic network equilibrium problem.
TL;DR: Caprino, Marchioro and Pulvirenti as discussed by the authors considered a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas.
Abstract: We consider a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. In this setup, we discuss the following statement: "Let V0 be the initial velocity of the body and V∞ its asymptotic velocity, then for |V0 - V∞| small enough it results |V(t) - V∞| ≈ C t-d-2 for t large, where V(t) is the velocity of the body at time t, d the dimension of the space and C is a positive constant depending on the medium and on the shape of the body". The reason for the power law approach to the stationary state instead of the exponential one (usually assumed in viscous friction problems), is due to the long memory of the dynamical system. In a recent paper by Caprino, Marchioro and Pulvirenti,3 the case of E constant and positive, with 0 0 with V0 > V∞ and E = 0. We also approach the problem of an x-dependent external force, by choosing E of harmonic type. In this case we obtain the power-like asymptotic time behavior for the body position X(t). The investigation is done in detail for a disk orthogonal to the x-axis and then, by a sketched proof, extended to a body with a general convex shape.
TL;DR: The main result is that the local optimization ensures a very good result also for the complete network, shown by the case study of Re di Roma Square, a big traffic circle of the urban network of Rome.
Abstract: This paper focuses on the optimization of traffic flow on a road network, modeled by a fluid-dynamic approach Three cost functionals that measure average velocity, average traveling time, and total flux of cars, are considered First, such functionals are optimized for two simple networks that consist of a single junction: one with two incoming and one outgoing roads (junctions of 2 × 1 type), and the other with one incoming and two outgoing roads (junctions of 1 × 2 type) The optimization is made with respect to right of way parameters and traffic distribution coefficients, obtaining an explicit solution Then, through simulations, the traffic behavior for complex networks is studied The main result is that the local optimization ensures a very good result also for the complete network This is shown by the case study of Re di Roma Square, a big traffic circle of the urban network of Rome
TL;DR: In this paper, the authors analyzed the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type when the boundary of the domain varies very rapidly, and they showed that the limit boundary condition is given by, where γ(x) is a factor related to the oscillations of the boundary at point x.
Abstract: We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by , where γ(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary, γ is a bounded function and we show the convergence of the solutions in H1 and Cα norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.