American Institute of Mathematical Sciences

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.

A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.

A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary $Γ$-limit. As examples we treat (ⅰ) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ⅱ) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers [7] and [9] from the viewpoint of well-posedness, long-time behavior and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in [9] by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates.

Starting from three-dimensional variational models with energies subject to a general type of PDE constraint, we use Γ-convergence methods to derive reduced limit models for thin strings by letting the diameter of the cross section tend to zero. A combination of dimension reduction with homogenization techniques allows for addressing the case of thin strings with fine heterogeneities in the form of periodically oscillating structures. Finally, applications of the results in the classical gradient case, corresponding to nonlinear elasticity with Cosserat vectors, as well as in micromagnetics are discussed.

Consider wave equations of the form

\begin{document}$\begin{align*}u''(t)+ A^2u(t)=0\end{align*}$ \end{document}

with $A$ an injective selfadjoint operator on a complex Hilbert space \begin{document}$\mathcal{H}$\end{document}. The kinetic, potential, and total energies of a solution $u$ are

\begin{document}$\begin{align*}K(t)= \| u'(t)\|^2, P(t)= \|Au(t)\|^2, E(t) = K(t)+P(t).\end{align*}$ \end{document}

Finite energy solutions are those mild solutions for which \begin{document}$E(t)$\end{document} is finite. For such solutions \begin{document}$E(t)= E(0)$\end{document}, that is, energy is conserved, and asymptotic equipartition of energy

\begin{document}$\begin{align*}\lim_{t \longrightarrow ± ∞}K(t) = \lim_{t \longrightarrow ± ∞}P(t) = \frac{E(0)}{2}\end{align*}$ \end{document}

holds for all finite energy mild solutions iff \begin{document}$e^{itA}\longrightarrow 0$\end{document} in the weak operator topology. In this paper we present the first extension of this result to the case where \begin{document}$A$\end{document} is time dependent.

The identification of optimal structures in reaction-diffusion models is of great importance in numerous physicochemical systems. We propose here a simple method to monitor the number of interphases formed after long simulated times by using a boundary flux condition as a control parameter. We consider as an illustration a 1-D Allen-Cahn equation with Neumann boundary conditions. Numerical examples are provided and perspectives for the application of this approach to electrochemical systems are discussed.

We study by Γ-convergence the stochastic homogenization of discrete energies on a class of random lattices as the lattice spacing vanishes. We consider general bounded spin systems at the bulk scaling and prove a homogenization result for stationary lattices. In the ergodic case we obtain a deterministic limit.

In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large.

We investigate carbon-nanotubes under the perspective ofgeometry optimization. Nanotube geometries are assumed to correspondto atomic configurations whichlocally minimize Tersoff-type interactionenergies. In the specific cases of so-called zigzag and armchairtopologies, candidate optimal configurations are analytically identifiedand their local minimality is numerically checked. Inparticular, these optimal configurations do not correspond neither tothe classical Rolled-up model [5] nor to themore recent polyhedral model [3]. Eventually, theelastic response of the structure under uniaxial testing is numericallyinvestigated and the validity of the Cauchy-Born rule is confirmed.