Coercivity and stability results for an extended Navier-Stokes system

In this article we study a system of equations that is known to {\em extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of controlling the nonlinear terms. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Even in two space dimensions, global existence is open in general, and remains so, primarily due to the lack of a self-contained $L^2$ energy estimate. However, through use of new $H^1$ coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure

Diversity and Security in UK Electricity Generation: The Influence of Low Carbon Objectives

We explore the relationship between low carbon objectives and the strategic security of electricity in the context of the UK Electricity System. We consider diversity of fuel source mix to represent one dimension of security - robustness against interruptions of any one source - and apply two different diversity indices to the range of electricity system scenarios produced by the UK government and independent researchers. Using data on wind generation we also consider whether a second dimension of security - the reliability of generation availability - is compromised by intermittency of renewable generation. Our results show that low carbon objectives are uniformly associated with greater long-term diversity in UK electricity. We discuss reasons for this result, explore sensitivities, and briefly discuss possible policy instruments associated with diversity and their limitations.Diversity, Security, Low Carbon, Wind Generation, Electricity

Smeared heat-kernel coefficients on the ball and generalized cone

We consider smeared zeta functions and heat-kernel coefficients on the bounded, generalized cone in arbitrary dimensions. The specific case of a ball is analysed in detail and used to restrict the form of the heat-kernel coefficients $A_n$ on smooth manifolds with boundary. Supplemented by conformal transformation techniques, it is used to provide an effective scheme for the calculation of the $A_n$. As an application, the complete $A_{5/2}$ coefficient is given.Comment: 23 pages, JyTe

Realizations of Differential Operators on Conic Manifolds with Boundary

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index.Comment: 41 pages, 1 figur

Driver Response to Simulated Intersections: An Analysis of Workload-Related Variables

A roadway intersection driving simulation was created to investigate driver information processing at intersections. Research participants were provided a visual simulation of approaching intersections using a video display with a 120 degree visual field. Six groups, each containing 12 subjects, were formed according to age and gender, with age ranging from 18 to 74 years. All participants viewed 14 separate intersections, which varied according to types of traffic control signs and signals. Individual workload was assessed in three categories of response: performance, subjective, and physiological. A MANOVA was performed on six dependent variables in the 3 (age) by 2 (gender) design. Results indicate significant main effects for both age and gender. The three significant dependent variables were pedal response errors, speed of response, and heart rate reactivity to each intersection. The responses suggest greater workloads for older drivers and female drivers. In addition to age and gender, a number of driver information processing characteristics were measured. Stepwise regressions indicated that performance decrements to the simulated driving situations could best be predicted by subjects\u27 scores for field dependency, visual acuity, and depth perception. However, age alone, accounted for more variance in performance than any single information processing variable

Global Theory of Quantum Boundary Conditions and Topology Change

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold $M$ with regular boundary $\Gamma=\partial M$. The space \CM of self-adjoint extensions of the covariant Laplacian on $M$ is shown to have interesting geometrical and topological properties which are related to the different topological closures of $M$. In this sense, the change of topology of $M$ is connected with the non-trivial structure of \CM. The space \CM itself can be identified with the unitary group \CU(L^2(\Gamma,\C^N)) of the Hilbert space of boundary data L^2(\Gamma,\C^N). A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, \CC_-\cap \CC_+ (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary condition reaches the Cayley submanifold \CC_-. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space \CM is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self--adjoint boundary conditions, the space \CC_- can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold \CC_- is dual of the Maslov class of \CM.Comment: 29 pages, 2 figures, harvma

A class of well-posed parabolic final value problems

This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax--Milgram operators in vector distribution spaces constitute the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.Comment: 16 pages. To appear in "Applied and numerical harmonic analysis"; a reference update. Conference contribution, based on arXiv:1707.02136, with some further development