Global attractors for gradient flows in metric spaces

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope of Ambrosio et al. (2005) [5], we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach (De Giorgi, 1993; Ambrosio, 1995 [3], [15]).For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced in Ball (1997) [7].The notions of generalized and energy solutions are quite flexible, and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces, to Wasserstein spaces of probability measures. We present applications of our abstract results, by proving the existence of the global attractor for the energy solutions, both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.     

Magnetic control of magnetic shape-memory single crystals

We present a phenomenological model for the magneto-mechanical evolution of shape-memory alloy single crystals. The existence of solutions for given magnetic field is commented and optimal control results are established.     

Young-Measure Quasi-Static Damage Evolution

An existence result for the quasi-static evolution of incomplete damage in elastic materials is presented. The absence of gradient terms in the damage variable causes a critical lack of compactness. Therefore, the analysis is developed in the framework of Young measures, where a notion of solution is defined, presenting some improvements with respect to previous contributions. The main new feature in the proof of the existence result regards a delicate construction of the joint-recovery sequence.     

$$N^{3/4}$$ N 3 / 4 Law in the Cubic Lattice

Journal of Statistical Physics - We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape...     

Study of a system for the isothermal separation of components in a binary alloy with change of phase

An isothermal model describing the separation of the componentsof a binary metallic alloy is considered. A process of phasetransition is also assumed to occur in the solder; hence, thestate of the material is described by two order parameters,i.e. the concentration c of the first component and the phasefield . A physical derivation is provided starting from energybalance considerations. The resulting system of PDEs consistsof a rather regular second-order parabolic equation for coupledwith a fourth-order relation of Cahn–Hilliard type forc with constraint and solution-dependent mobility. Global existenceof solutions is proved and several regularity properties arediscussed under more restrictive assumptions on the physicalparameters. Continuous dependence on data is shown in a specialcase. An asymptotic analysis of the model is also performed,yielding at the limit step a coupling of the original phasefield equation with a Stefan-like system for c.     

On the nature of thermal anomalies in aqueous systems

New experimental data put in evidence a time dependence of the “thermal anomalies” in aqueous heterogeneous systems and suggest a reexamination of their nature.     

Evaluation of the reproducibility standard deviation in the pesticide multi-residue methods on olive oil from past proficiency tests

Accreditation and Quality Assurance - The data reported in the frame of proficiency testing (PT) exercises organized from 2007 to 2016 have been statistically re-evaluated using Algorithm A of the...     

Sharp $$N^{3/4}$$ Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice

We investigate the edge-isoperimetric problem (EIP) for sets of n points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of n. In view of this isoperimetric characterizations, EIP minimizers \(M_n\) are seen to be given by hexagonal configurations with some extra points at their boundary. By a careful computation of the cardinality of these extra points, minimizers \(M_n\) are estimated to deviate from such hexagonal configurations by at most \(K_t\, n^{3/4}+\mathrm{o}(n^{3/4})\) points. The constant \(K_t\) is explicitly determined and shown to be sharp.     

A variational principle for gradient flows in metric spaces

We present a novel variational approach to gradient-flow evolution in metric spaces. In particular, we advance a functional defined on entire trajectories, whose minimizers converge to curves of maximal slope for geodesically convex energies. The crucial step of the argument is the reformulation of the variational approach in terms of a dynamic programming principle, and the use of the corresponding Hamilton–Jacobi equation. The result is applicable to a large class of nonlinear evolution PDEs including nonlinear drift-diffusion, Fokker–Planck, and heat flows on metric-measure spaces.