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

Weak formulation for singular diffusion equation with dynamic boundary condition

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.Comment: 23 page

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

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

Population Genetic Structure and Species Delimitation of a Widespread, Neotropical Dwarf Gecko

Amazonia harbors the greatest biological diversity on Earth. One trend that spans Amazonian taxa is that most taxonomic groups either exhibit broad geographic ranges or small restricted ranges. This is likely because many traits that determine a species range size, such as dispersal ability or body size, are autocorrelated. As such, it is rare to find groups that exhibit both large and small ranges. Once identified, however, these groups provide a powerful system for isolating specific traits that influence species distributions. One group of terrestrial vertebrates, gecko lizards, tends to exhibit small geographic ranges. Despite one exception, this applies to the Neotropical dwarf geckos of the genus Gonatodes. This exception, Gonatodes humeralis, has a geographic distribution almost 1,000,000 km2 larger than the combined ranges of its 30 congeners. As the smallest member of its genus and a gecko lizard more generally, G. humeralis is an unlikely candidate to be a wide-ranged Amazonian taxon. To test whether or not G. humeralis is one or more species, we generated molecular genetic data using restriction-site associated sequencing (RADseq) and traditional Sanger methods for samples from across its range and conducted a phylogeographic study. We conclude that G. humeralis is, in fact, a single species across its contiguous range in South America. Thus, Gonatodes is a unique clade among Neotropical taxa, containing both wide-ranged and range-restricted taxa, which provides empiricists with a powerful model system to correlate complex species traits and distributions. Additionally, we provide evidence to support species-level divergence of the allopatric population from Trinidad and we resurrect the name Gonatodes ferrugineus from synonymy for this population

A phase-field approximation of the Willmore flow with volume and area constraints

The well-posedness of a phase-field approximation to the Willmore flow with area and volume constraints is established when the functional approximating the area has no critical point satisfying the two constraints. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by a variational minimization principle. The main difficulty stems from the nonlinearity of the area constraint