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

An Evaluative Head in Romance: The Palermitan Verbal Affix -vu

In this work, we analyse the behaviour of a verbal suffix in Palermitan – the Romance language spoken in Palermo, Sicily – expressing an evaluative meaning with sentential scope. The suffix is (V)vu, where V is the thematic vowel of the verb. The Palermitan dialect we study here is the one spoken in the urban area, approximately by 250,000 speakers. This suffix is interesting for several reasons. In the first place, an evaluative morpheme with clausal scope is quite unusual in Romance and to our knowledge it has not been reported outside Sicily. In the second place, as we will illustrate later, its use is spreading among the youngest generation, showing that the dialect is productive and the form under scrutiny is adopted by larger segments of the population, even if it does not exist in Italian

Solidification and separation in saline water

Motivated by the formation of brine channels, this paper is devoted to a continuum model for salt separation and phase transition in saline water. The mass density and the concentrations of salt and ice are the pertinent variables describing saline water. Hence the balance of mass is considered for the single constituents (salt, water, ice). To keep the model as simple as possible, the balance of momentum and energy are considered for the mixture as a whole. However, due to the internal structure of the mixture, an extra-energy flux is allowed to occur in addition to the heat flux. Also, the mixture is allowed to be viscous. The constitutive equations involve the dependence on the temperature, the mass density of the mixture, the salt concentration and the ice concentration, in addition to the stretching tensor, and the gradient of temperature and concentrations. The balance of mass for the single constituents eventually result in the evolution equations for the concentrations. A whole set of constitutive equations compatible with thermodynamics are established. A free energy function is given which allows for capturing the main feature which occurs during the freezing of the salted water. That is, the salt entrapment in small regions (brine channels) where the cryoscopic effect forbid complete ice formation

Well-posedness for solid-liquid phase transitions with a forth-order nonlinearity

A phase-field system which describes the evolution of both the absolute temperature $\theta$ and the phase variable $f$ during first-order transitions in thermal insulators is considered. A thermodynamic approach is developed by regarding the order parameter as a phase field and its evolution equation as a balance law. By virtue of the special form of the internal energy, a third-order nonlinearity $G_2^\prime(f)$ appears into the energy balance in place of the (customary constant) latent-heat. As a consequence, the bounds $0\le f\le1$ hold true whenever $\theta$ is positive valued. In addition, a nonlinear Fourier law with conductivity proportional to temperature is assumed. Well-posedness for the resulting initial and boundary value problem are then established in a suitable setting

On a doubly nonlinear phase-field model for first-order transitions with memory

Solid-liquid transitions in thermal insulators and weakly conducting media are modeled through a phase-field system with memory. The evolution of the phase variable $\varphi$ is ruled by a balance law which takes the form of a Ginzburg-Landau equation. A thermodynamic approach is developed starting from a special form of the internal energy and a nonlinear hereditary heat conduction flow of Coleman-Gurtin type. After some approximation of the energy balance, the absolute temperature $\theta$ obeys a doubly nonlinear "heat equation" where a third-order nonlinearity in $\varphi$ appears in place of the (customarily constant) latent-heat. The related initial and boundary value problem is then formulated in a suitable setting and its well--posedness and stability is proved

A three-dimensional phase transition model in ferromagnetism: existence and uniqueness

We scrutinize both from the physical and the analytical viewpoint the equations ruling the paramagnetic-ferromagnetic phase transition in a rigid three dimensional body. Starting from the order structure balance, we propose a non-isothermal phase-field model which is thermodynamically consistent and accounts for variations in space and time of all fields (the temperature $\theta$, the magnetic field vector H and the magnetization vector M). In particular, we are able to establish a well-posedness result for the resulting coupled system

Recovering the unsigned photospheric magnetic field from Ca II K observations

We reassess the relationship between the photospheric magnetic field strength and the Ca II K intensity for a variety of surface features as a function of the position on the disc and the solar activity level. This relationship can be used to recover the unsigned photospheric magnetic field from images recorded in the core of Ca II K line. We have analysed 131 pairs of high-quality, full-disc, near-co-temporal observations from SDO/HMI and Rome/PSPT spanning half a solar cycle. To analytically describe the observationally-determined relation, we considered three different functions: a power law with an offset, a logarithmic function, and a power law function of the logarithm of the magnetic flux density. We used the obtained relations to reconstruct maps of the line-of-sight component of the unsigned magnetic field (unsigned magnetograms) from Ca II K observations, which were then compared to the original magnetograms. We find that both power-law functions represent the data well, while the logarithmic function is good only for quiet periods. We see no significant variation over the solar cycle or over the disc in the derived fit parameters, independently of the function used. We find that errors in the independent variable, usually not accounted for, introduce attenuation bias. To address this, we binned the data with respect to the magnetic field strength and Ca II K contrast separately and derived the relation for the bisector of the two binned curves. The reconstructed unsigned magnetograms show good agreement with the original ones. RMS differences are less than 90 G. The results were unaffected by the stray-light correction of the SDO/HMI and Rome/PSPT data. Our results imply that Ca~II~K observations, accurately processed and calibrated, can be used to reconstruct unsigned magnetograms by using the relations derived in our study.Comment: 18 pages, 22 figures, accepted in A&