Well-posedness for the diffusive 3D Burgers equations with initial data in H1/2H^{1/2}

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski and A. Vidal-L\'opez, Cambridge University Press, 201

Causality and the Power Spectrum

We find constraints on the generation of super-causal-horizon energy perturbations from a smooth initial state, under a simple physical scheme. We quantify these constraints by placing the upper limit $\lambda_c = 3.0 d_H$ on the wavelength at which the power spectrum turns over to $k^4$ behavior. This means that sub-horizon processes can generate significant power on scales further outside the horizon than one might naively expect. The existence of this limit may have important implications for the interpretation of the small scale power spectrum of the Cosmic Microwave Background.Comment: uuencoded postscript file, 12 pages, 1 figure. Latex and postscript versions also available at http://euclid.tp.ph/Papers/index.htm

An Eulerian–Lagrangian form for the Euler equations in Sobolev spaces

In 2000 Constantin showed that the incompressible Euler equations can be written in an “Eulerian–Lagrangian” form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces C1,μ . We review the Eulerian–Lagrangian formulation of the equations and prove that given initial data in H s for n≥2 and s>n2+1 , a unique local-in-time solution exists on the n-torus that is continuous into H s and C 1 into H s-1. These solutions automatically have C 1 trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian–Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory

Nano-mapping of Surface and Subsurface Physical Properties of 2D materials

A massive interest in two-dimensional materials (2DM) triggered by graphene (GR) discovery1 is fueled by the unique electronic, mechanical and thermal properties of these few-atomic-layers-thick materials. While electronic properties of graphene and other 2DM’s such as MoS2, WS, Bi2Se3, were extensively studied, their mechanical and thermal properties, equally record-breaking, are much less explored, due to inadequate tools for nanoscale probing of physical properties of atomically thin layers. Here we overcome this by combining atomic force microscopy (AFM) with specialist nanomechanical, nanothermal and nanoelectrical probes. By applying these to the single and few layer Gr and MoS2 we were able to explore the nanomechanical interaction of 2DM’s and the substrate, including layers adhesion and stresses; observe internal defects in the few layer 2DM’s, and defect movement under applied strain; map the nanoscale distribution, and quantify electrical charges trapped at the 2DM-substrate interface; observe with microscale and nanoscale resolution local electrical and thermal transport in these materials

Micro and Nanoscale Mapping of Electrical Characterization of Graphene and Semiconductor Heterostructures.

Graphene and 2D materials have proven to be promising materials to be used to fabricate heterostructures, and also promising candidate to be used in combination with optoelectronic devices, due to their unique electronic properties1. Here, the electrical properties of heterostructures, as well as the specific contacts area onto graphene are investigated by using properly designed test patterns. We firstly use PlayStation micro-probing (PSM) for macroscopic characterisations. Secondly, High- spatial resolution current mapping needed to provide an insight into the nanoscale mechanisms of electrical transport. We use scanning probe microscopy (SPM) as well as scanning gate microscopy (SGM), with a conductive diamond tip directly access to nano-resolution structures both topographically and electrically, with illumination and dark regimes. Additionally, we plan to report measurement in 2D materials heterostructures and their photoresponse on this regimes

TRACE: A Stigmergic Crowdsourcing Platform for Intelligence Analysis

Crowdsourcing has become a frequently adopted approach to solving various tasks from conducting surveys to designing products. In the field of reasoning-support, however, crowdsourcing-related research and application have not been extensively implemented. Reasoning-support is essential in intelligence analysis to help analysts mitigate various cognitive biases, enhance deliberation, and improve report writing. In this paper, we propose a novel approach to designing a crowdsourcing platform that facilitates stigmergic coordination, awareness, and communication for intelligence analysis. We have partly materialized our proposal in the form of a crowdsourcing system which supports intelligence analysis: TRACE (Trackable Reasoning and Analysis for Collaboration and Evaluation). We introduce several stigmergic approaches integrated into TRACE and discuss the potential experimentation of these approaches. We also explain the design implications for further development of TRACE and similar crowdsourcing systems to support reasoning