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

The Role of Thermal Alteration in Lithic Reduction Strategies at the Leftwich Site in Middle Tennessee

Research in 1978, 1980 and 1981 at the Leftwich site (40MU262), located on the Duck River in Maury County, Tennessee revealed stratigraphically separated buried Archaic cultural strata. Radiocarbon dates of 6160 and 4190 to 4130 years before present were associated with Benton and Ledbetter projectile points, respectively. A functional analysis of artifacts from a controlled surface collection and two buried Archaic components at Leftwich is undertaken. Buried Benton and Ledbetter components were stratigraphically separated based on vertical density peaks of lithic artifacts and debitage, as well as pebbles larger than 6 mm in size. The relationship of thermal alteration to lithic resource location, lithic implement manufacturing processes and settlement is investigated. Intentional thermal alteration of lithic artifacts is indicated by: (1) an overall low incidence of overheating and (2) an association between thermal alteration and small biface thinning flakes and late stage bifaces in the surface, Benton and Ledbetter assemblages. Intentional thermal alteration is also significantly correlated with cores and large biface thinning flakes. Thermal alteration analysis can yield information regarding when and where lithic reduction took place

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

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

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

Scattering and self-adjoint extensions of the Aharonov-Bohm hamiltonian

We consider the hamiltonian operator associated with planar sec- tions of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior. First, in the Sobolev space $\mathcal H^2$, we characterize all generalized boundary conditions on the solenoid bor- der compatible with quantum mechanics, i.e., the boundary conditions so that the corresponding hamiltonian operators are self-adjoint. Then we study and compare the scattering of the most usual boundary con- ditions, that is, Dirichlet, Neumann and Robin.Comment: 40 pages, 5 figure