Vincenzo Coscia

ON THE MATHEMATICAL THEORY OF VEHICULAR TRAFFIC FLOW I: FLUID DYNAMIC AND KINETIC MODELLING

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.

FIRST-ORDER MACROSCOPIC MODELLING OF HUMAN CROWD DYNAMICS

This paper deals with the mathematical modelling of crowd dynamics within the framework of continuum mechanics. The method uses the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients. The closures take into account movement in more than one space dimension, presence of obstacles, pedestrian strategies, and modelling of panic conditions. Numerical simulations of the initial-boundary value problems visualize the ability of the models to predict several interesting phenomena related to the complex system under consideration.

Existence, uniqueness and stability of regular steady motions of a second-grade fluid

Abstract In this paper we study the well-posedness of the steady motions problem for a second-grade fluid in a bounded domain, with adherence conditions at the boundary. We prove the existence and uniqueness of steady classical solutions for any value of the normal stress moduli α1 and α2, thus showing that the thermodynamical restrictions are not needed for the mathematical problem being well-set. Moreover, we find that such steady motions are exponentially non-linearly stable, provided α1 > 0.

Flap size/flow rate relationship in perforator flaps and its importance in DIEAP flap drainage ☆

The vascular architecture within a perforator flap is different from a conventional muscle or myocutaneous flap. The purpose of this paper is to understand the correlation between flow rate and flap size in perforator flaps. With extrapolation of these data, we have provided an indirect analysis of the venous drainage and its correlation with flap size. A prospective study was planned. Twenty-five patients were enrolled in this study: six patients were operated on using an anterolateral thigh (ALT) flap and 19 using a deep inferior epigastric artery perforator (DIEAP) flap. One month postoperatively, echo-colour-Doppler measurements were performed on pedicle and perforator arteries to calculate blood flow rate in the flaps. A correlation between weight and flow rate was analysed. Spearman rho statistic was calculated. A linear regression model was made from patient data of flow rate/flap weight and predicted values of flow per flap weight were calculated. Then, flow rate values of veins of various diameters were estimated using Hagen-Poiseuilles formula. Our data show that flow rate measured postoperatively on flap arteries is significantly correlated with flap weight [rho(23 d.f.)=0.725, P<0.01 (two-tailed)]. Moreover, we have calculated the minimum size of veins able to drain flaps of increasing weights with different patterns, i.e. our data show that veins of 1.30, 1.50 and 1.75 mm diameter could safely drain flaps of, respectively, 300, 500 and 900 g in weight. This can be useful preoperatively to estimate the risk of flap congestion and in planning additional drainage.

On the mathematical theory of living systems II: The interplay between mathematics and system biology

This paper aims at showing how the so-called mathematical kinetic theory for active particles can be properly developed to propose a new system biology approach. The investigation begins with an analysis of complexity in biological systems, continues with reviewing a general methodology to reduce complexity and furnishes the mathematical tools to describe the time evolution of such systems by capturing all their features.

On the coupling of steady and adaptive velocity grids in vehicular traffic modelling

This work deals with the derivation and the analysis of a new mathematical model for vehicular traffic along a one-way road obtained by the coupling of a uniform and an adaptive discretization of the velocity variable in the framework of the kinetic theory. Interactions are modelled by stochastic games where the output of interactions depends on the local density and is not linearly additive.

Finite volume and WENO scheme in one-dimensional vascular system modelling

A finite volume and WENO model is developed, validated and applied to a one-dimensional numerical investigation of the vascular system. Suitable balance laws are spatially integrated using a finite volume WENO approach. Time integration is performed using a five steps, fourth order accurate Runge-Kutta strong stability preserving scheme. A detailed investigation of different sets of reflecting and non-reflecting boundary conditions is performed to achieve a correct representation of limited portions of the vascular system. Moreover, special attention is devoted to the numerical discretisation of the source term. Finally, as an application of the whole model, the effect of an abdominal aorta aneurysm on the blood flow is evaluated.

Existence and uniqueness of classical solutions for a class of complexity-2 fluids

Abstract In this paper we investigate the existence of classical solutions for a general system of evolution equations and for a corresponding system of steady equations arising in the study of the equations of motion of incompressible non-Newtonian fluids. Once we have made suitable assumptions on the differential operators involved, we show existence and uniqueness, for all times, for the abstract evolutionary problem. Using the general framework, we carry on our study to a particular class of complexity-2 fluids, and show that this set of equations admits a unique, global (in time), classical solution for sufficiently small data. Finally, we establish similar results for the corresponding set of steady equations.

On the mathematical theory of living systems, I: Complexity analysis and representation

Abstract This paper is the first one of a sequel devoted to the challenging goal of developing a mathematical theory for living systems. We consider systems constituted of a number of living entities, called active particles, which have the ability to express specific strategies and interact with other entities. The author proposes a personal path, starting from the identification of a number of common features of living systems that can be viewed as sources of complexity, focusing specifically on the representation of systems based also on a strategy to reduce their complexity. The overall system is decomposed into functional subsystems whose representation is delivered by a probability distribution over the microscopic state of the active particles belonging to such system. Looking ahead, this paper indicates some guidelines to derive mathematical structures, where interactions involving active particles are nonlinearly additive.

Nonlinear energy stability in a compressible atmosphere

Abstract We provide sufficient conditions for nonlinear exponential stability of the compressible Benard problem. In particular, by using a generalized energy analysis we prove stability whenever the Rayleigh number does not exceed a computable critical number Rc . The value of Rc is given for finite amplitude depth and for thin layers as well, and such values are compared with those already computed in the linear theory. In the limit of depth which goes to zero a necessary and sufficient condition for nonlinear stability of the Benard problem is proved. The principle of exchange of stabilities is not required to hold.