Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I

This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v ₀ of a front that is sufficiently small in L ² norm, and sufficiently localized that x ² v ₀(x)² dx<, yields a solution that relaxes to another front, selected by a conservation law, in the L ¹ norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L ² norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.on leave from     

Characterisation of foxing stains in eighteenth to nineteenth century drawings using non-destructive techniques

The reddish-brown, brown or yellowish stains of circular or irregular shape known as foxing spots have been fully described in conservation literature but still, this phenomenon does not find any scientific agreement since many hypotheses have been raised concerning their origin. In this work a contribution to foxing definition not only focussed on its appearance but also reported on its chemical information. For this purpose foxing stains present in drawings from two Portuguese artists dated from the eighteenth to nineteenth centuries were observed under ultra-violet light and optical microscope and analysed by three non-invasive spectroscopy techniques. The observations carried out on the stains provided information on their surface morphology. The use of energy-dispersive X-ray fluorescence revealed a variation on the elemental content between foxing and paper region. Although the results from X-ray diffraction analysis showed no signs of cellulose degradation in foxing stains, Fourier-transformed infrared analysis revealed the presence of oxide groups. Both the information on the chemical nature and surface morphology of the stains achieved in this study will contribute to increase foxing formation information and develop future protocols for conservation purposes.