A New Approach to Equations with Memory

In this work, we present a novel approach to the mathematical analysis of equations with memory based on the notion of a state, namely, the initial configuration of the system which can be unambiguously determined by the knowledge of the future dynamics. As a model, we discuss the abstract version of an equation arising from linear viscoelasticity. It is worth mentioning that our approach goes back to the heuristic derivation of the state framework, devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state in viscoelasticity: new free energies and applications to PDEs", Arch. Ration. Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical motivations, we develop a suitable functional formulation which, as far as we know, is completely new.Comment: 39 pages, no figur

EVALUATION OF ROLLER SKATING TRAINING LOADS USING HEART RATE MONITORING. A REPORT OF 4 YEARS DATA COLLECTION

This paper reports the experience in using the heart rate (HR) monitor for roller skating in a period of 4 years. The relationship between HR and mechanical power is shown. A method for calculating the total load of the training sessions is also shown. The experience shows that HR alone can be easily used in an effective way to measure the training load for roller skating

New variational principles in quasi-static viscoelasticity

A "saddle point" (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving Fourier transform. The "minimax" property is shown to hold as a direct consequence of the thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed

Stability of abstract linear thermoelastic systems with memory

An abstract linear thermoelastic system with memory is here considered. Existence, uniqueness, and continuous dependence results are given. In presence of regular and convex memory kernels, the system is shown to be exponentially stable. An application to the Kirchhoff plate equation is given

A phase-field model for liquid-vapor transitions

2Starting from the mesoscopic description of the state equations for the vapor and liquid pure phases of a single chemical species, we propose a phase-field model ruling the liquid-vapor phase transition. Two different phases are separated by a thin layer, rather than a sharp interface, where the phase-field changes abruptly from 0 to 1. All thermodynamic quantities are allowed to vary inside the transition layer, including the mass density. The approach is based on an extra entropy flux which is proved to be non vanishing inside the transition layer, only. Unlike classical phase-field models, the kinetic equation for the phase variable is obtained as a consequence of thermodynamic restrictions and it depends only on the rescaled free enthalpy. The system turns out to be thermodynamically consistent and accounts for both temperature and pressure variations during the evaporation process. Few commonly accepted assumptions allow us to obtain the explicit expression of the Gibbs free enthalpy and the Clausius-Clapeyron formula. As a consequence, the customary form of the vapor pressure curve is recovered.AMS Subject Classification:: 74A15, 74A50, 80A17, 80A22. 82C26openopenBERTI A; C. GIORGIBerti, Alessia; Giorgi, Claudi

Phase-field modeling of transition and separation phenomena in continuum thermodynamics

In the framework of continuum thermodynamics a new approach to phase transition and separation phenomena is developed by emphasizing their nonlocal charac- ter. The phase-field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation of rate type. The second law of thermodynamics is sat- isfied by virtue of an extra entropy flux which arises from its nonlocal formulation. Such an extra flux is proved to be nonvanishing inside the transition layer, only. Different choices of the state variables distinguish transition form separation models. The former case involves the gradients of the main fields up to the second order, whereas in the latter all gradients up to the fourth order are needed and the total mass of the phase-field is conserved. In both cases, necessary and sufficient restrictions on the constitutive equations are derived from thermodynamics. On this background, some applications to scalar-valued models are developed. A simple model of the temperature-induced first-order transition is derived in connection with a state space involving second-order gradients. Dynamical models of phase separation in a binary fluid mixture are discussed and the classical nonisothermal Cahn-Hilliard system is obtained as a special case of a fourth-gradient model

A minimum principle for the quasi-static problem in linear viscoelasticity

A minimum principle is set up for the quasi-static boundary-value problem (QSP) in linear viscoelasticity. A linear homogeneous and isotropic viscoelastic solid under unidimensional displacements is considered along with the complete set of thermodynamic restrictions on the relaxation function. It is assumed that boundary conditions are of Dirichlet type and initial history data are not given. The variational formulation of QSP is set up through a convex functional based on a "weighted" $L^2$ inner product as the bilinear form and is strictly related to the thermodynamic restrictions on the relaxation function. As an aside, the same technique is proved to be applicable to analogous physical problems such as the quasi-static heat flux equation

Extremum principles in electromagnetic systems

Variational expressions and saddle-point (or "mini-max") principles for linear problems in electromagnetism are proposed. When conservative conditions are considered, well-known variational expressions for the resonant frequencies of a cavity and the propagation constant of a waveguide are revised directly in terms of electric and magnetic field vectors. In both cases the unknown constants are typefied as stationary (but not extremum) points of some energy-like functionals. On the contrary, if dissipation is involved then variational expressions achieve the extremum property. Indeed, we point out that a saddle-point characterizes the unique solution of Maxwell equations subject to impedance-like dissipative boundary conditions. In particular, we deal with the quasi-static problem and the time-harmonic case

The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels

In this paper we investigate the longtime behavior of the mathematical model of a homogeneous viscoelastic plate based on Reissner–Mindlin deformation shear assumptions. According to the approximation procedure due to Lagnese for the Kirchhoff viscoelastic plate, the resulting motion equations for the vertical displacement and the angle deflection of vertical fibers are derived in the framework of the theory of linear viscoelasticity. Assuming that in general both Lame’s functions, λ and μ, depend on time, the coupling terms between the equations of displacement and deflection depend on hereditary contributions. We associate to the model a nonlinear semigroup and show the behavior of the energy when time goes on. In particular, assuming that the kernels λ and μ decay exponentially, and not too weakly with respect to the physical properties considered in the model, then the energy decays uniformly with respect to the initial conditions; i.e., we prove the existence of an absorbing set for the semigroup associated to the model

Asymptotic behavior of a nonlinear hyperbolic heat equation with memory

n this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an integro-differential equation describing the heat flow in a rigid heat conductor with memory. This model arises matching the energy balance, in presence of a nonlinear time-dependent heat source, with a linearized heat flux law of the Gurtin-Pipkin type. Existence and uniqueness of solutions for the corresponding semilinear system (subject to initial history and Dirichlet boundary conditions) is provided. Moreover, under proper assumptions on the heat flux memory kernel and the magnitude of nonlinearity, the existence of a uniform absorbing set is achieved