Geometric investigations of a vorticity model equation

This article consists of a detailed geometric study of the one-dimensional vorticity model equation $$\omega_{t} + u\omega_{x} + 2\omega u_{x} = 0, \qquad \omega = H u_{x}, \qquad t\in\mathbb{R},\; x\in S^{1}\,,$$ which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on $\operatorname{Diff}(S^{1})$ when the latter is endowed with the right-invariant homogeneous $\dot{H}^{1/2}$-metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-C\'ordoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to C\'ordoba-C\'ordoba.Comment: 30 pages; added references; corrected typo

Allocation of Land at the Rural-Urban Fringe Using a Spatially-Realistic Ecosystem Constraint

Development in rural-urban fringe communities is increasing with the potential to damage healthy ecosystems and endanger the long-term persistence of resident flora and fauna. The environmental impacts of development include loss, degradation, and fragmentation of wildlife habitat, increased air and water pollution, increased soil erosion, and decreased aesthetic appeal of the landscape. Current land use policies rarely incorporate features of landscape-scale ecosystem health. This paper develops a model that combines ecological and economic constructs to determine the optimal allocation of development across a spatially-realistic landscape. The land allocation model establishes links between long-term metapopulation persistence and development through an ecosystem constraint. A social planner seeks to maximize the benefits of development while guaranteeing a certain likelihood of long-term metapopulation persistence across the landscape that accounts for the changes to habitat patches and species dispersal success brought about by development. It is shown that in an economically homogeneous environment, the allocation of land to developed uses is determined solely by ecological elements (landscape structure and species parameters). The amount of land remaining in each habitat patch is the same regardless of their initial sizes or initial levels of development. The cost to society of meeting the ecological objective for metapopulation persistence depends on the land rent, the level of the safe-minimum-standard, the area of the landscape management unit, the distance between habitat patches, the dispersal ability of the focal species, and the species-specific area scaling parameter. Cost is not affected by the initial conditions of the habitat patches or the amount of development that has already taken place in the landscape management unit. When heterogeneity is introduced, the allocation of land is also determined by the differential land rents. More development occurs in habitat patches and landscape management units with higher land rents compared with the homogeneous case. In the heterogeneous land use case, where different land uses have different intensities of damages, the development intensity parameters are factors in the solution with more development occurring in areas zoned for less intensive land uses and the cost to society of achieving the ecological objective is a function of initial habitat patch sizes.Land Economics/Use,

Isometric Immersions and the Waving of Flags

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values into $\mathbb R^3$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. The submanifold result is then used to derive the equations of motion, after equipping the space of isometric immersions with its natural kinetic energy. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.Comment: 25 pages, 1 figur