Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases

Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials

In this paper, we investigate optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.Comment: Key words: optimal control; parabolic obstacle problems; MPECs; dynamic boundary conditions; optimality conditions. arXiv admin note: substantial text overlap with arXiv:1308.561

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.Comment: Key words: distributed optimal control, nonlinear phase field systems, first-order necessary optimality condition

Tertiary Climate Change and the Diversification of the Amazonian Gecko Genus Gonatodes (Sphaerodactylidae, Squamata)

The genus Gonatodes is a monophyletic group of small-bodied, diurnal geckos distributed across northern South America, Central America, and the Caribbean. We used fragments of three nuclear genes (RAG2, ACM4, and c-mos) and one mitochondrial gene (16S) to estimate phylogenetic relationships among Amazonian species of Gonatodes. We used Penalized Likelihood to estimate timing of diversification in the genus. Most cladogenesis occurred in the Oligocene and early Miocene and coincided with a burst of diversification in other South American animal groups including mollusks, birds, and mammals. The Oligocene and early Miocene were periods dominated by dramatic climate change and Andean orogeny and we suggest that these factors drove the burst of cladogenesis in Gonatodes geckos as well as other taxa. A common pattern in Amazonian taxa is a biogeographic split between the eastern and western Amazon basin. We observed two clades with this spatial distribution, although large differences in timing of divergence between the east–west taxon pairs indicate that these divergences were not the result of a common vicariant event

A New Genus of Miniaturized and Pug-Nosed Gecko from South America (Sphaerodactylidae: Gekkota)

Sphaerodactyl geckos comprise five genera distributed across Central and South America and the Caribbean. We estimated phylogenetic relationships among sphaerodactyl genera using both separate and combined analyses of seven nuclear genes. Relationships among genera were incongruent at different loci and phylogenies were characterized by short, in some cases zero-length, internal branches and poor phylogenetic support at most nodes. We recovered a polyphyletic Coleodactylus, with Coleodactylus amazonicus being deeply divergent from the remaining Coleodactylus species sampled. The C. amazonicus lineage possessed unique codon deletions in the genes PTPN12 and RBMX while the remaining Coleodactylus species had unique codon deletions in RAG1. Topology tests could not reject a monophyletic Coleodactylus, but we show that short internal branch lengths decreased the accuracy of topology tests because there were not enough data along these short branches to support one phylogenetic hypothesis over another. Morphological data corroborated results of the molecular phylogeny, with Coleodactylus exhibiting substantial morphological heterogeneity. We identified a suite of unique craniofacial features that differentiate C. amazonicus not only from other Coleodactylus species, but also from all other geckos. We describe this novel sphaerodactyl lineage as a new genus, Chatogekko gen. nov. We present a detailed osteology of Chatogekko, characterizing osteological correlates of miniaturization that provide a framework for future studies in sphaerodactyl systematics and biology

Coming to America: Multiple Origins of New World Geckos

Geckos in the Western Hemisphere provide an excellent model to study faunal assembly at a continental scale. We generated a time-calibrated phylogeny, including exemplars of all New World gecko genera, to produce a biogeographic scenario for the New World geckos. Patterns of New World gecko origins are consistent with almost every biogeographic scenario utilized by a terrestrial vertebrate with different New World lineages showing evidence of vicariance, dispersal via temporary land bridge, overseas dispersal, or anthropogenic introductions. We also recovered a strong relationship between clade age and species diversity, with older New World lineages having more species than more recently arrived lineages. Our data provide the first phylogenetic hypothesis for all New World geckos and highlight the intricate origins and ongoing organization of continental faunas. The phylogenetic and biogeographical hypotheses presented here provide an historical framework to further pursue research on the diversification and assembly of the New World herpetofauna