A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid

The linear model of plane-parallel thermal convection in a viscoelastic incompressible Kelvin-Voigt material amounts to a hybrid of the Oskolkov equations and the heat equations in the Oberbeck-Boussinesq approximation on a two-dimensional region with B'enard's conditions. We study the solvability of this model with the so-called multipoint initial-final conditions. We use these conditions to reconstruct the parameters of the processes in question from the results of multiple observations at various points and times. This enables us, for instance, to predict emergency situations, including the violation of continuity of thermal convection processes as a result of breaching technology, and so forth. For thermal convection models, the solvability of Cauchy problems and initial-final value problems has been studied previously. In addition, the stability of solutions to the Cauchy problem has been discussed. We study a multipoint initial-final value problem for this model for the first time. In addition, in this article we prove a generalized decomposition theorem in the case of a relatively sectorial operator. The main result is a theorem on the unique solvability of the multipoint initial-final value problem for the linear model of plane-parallel thermal convection in a viscoelastic incompressible fluid.Линейная модель плоскопараллельной термоконвекции вязкоупругой несжимаемой среды Кельвина - Фойгта представляет собой гибрид системы уравнений Осколкова и уравнения теплопроводности в приближении Обербека - Буссинеска, заданных в двумерной области с условиями Бенара. Целью нашего исследования является разрешимость этой модели с так называемыми многоточечными начально-конечными условиями. Такие условия используются для восстановления параметров изучаемых процессов по результатам многочисленных наблюдений с различных точек и в различные моменты времени, что позволяет, например, прогнозировать аварийные ситуации, в том числе нарушение непрерывности процесса термоконвекции в результате нарушения технологии и т.п. Ранее для моделей термоконвекции изучалась разрешимость задач Коши и начально-конечной, кроме того, была рассмотрена устойчивость решений задачи Коши. Многоточечная начально-конечная задача для этой модели изучается впервые. Кроме того, в данной работе приводится доказательство обобщенной теоремы о расщеплении в случае относительно секториального оператора. Основной результат статьи - теорема об однозначной разрешимости многоточечной начально-конечной задачи для линейной модели плоскопараллельной термоконвекции вязкоупругой несжимаемой жидкости

Multipoint Initial-Final Problem for One Class of Sobolev Type Models of Higher Order with Additive "White Noise"

Sobolev type equations theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise. A new concept of "white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equations theory and provide some practical applications. The main idea is to construct 'noise' spaces using the Nelson-Gliklikh derivative. Abstract results concerning initial-final problems for higher order Sobolev type equations are applied to the Boussinesq-Love model with additive 'white noise'. We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.Теория уравнений Соболева была предметом интереса многих исследователей последние годы, при этом много внимания уделялось детерминированным уравнениям и системам. Тем не менее, существуют также математические модели, содержащие случайные возмущения, такие как белый шум. Новая концепция "белого шума", первоначально построенная для конечномерных пространств, в данной работе распространяется на случай бесконечномерных пространств. Основная цель заключается в разработке стохастической теории уравнений cоболевского типа высокого порядка и предоставлении некоторых практических приложений. Основная идея состоит в том, чтобы построить пространства 'шумов', используя производную Нельсона - Гликлиха. Абстрактные результаты, касающиеся начально-конечных задач для уравнений cоболевского типа высокого порядка, применяются к математической модели Буссинеска - Лява с аддитивным 'белым шумом'. Использован такой известный метод теории уравнений cоболевского типа, как метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства, понимаемом как фазовое пространство

Effect of Substrate Chemistry on the Bottom-Up Fabrication of Graphene Nanoribbons: Combined Core-Level Spectroscopy and STM Study

Atomically precise graphene nanoribbons (GNRs) can be fabricated via thermally induced polymerization of halogen containing molecular precursors on metal surfaces. In this paper the effect of substrate reactivity on the growth and structure of armchair GNRs (AGNRs) grown on inert Au(111) and active Cu(111) surfaces has been systematically studied by a combination of core-level X-ray spectroscopies and scanning tunneling microscopy. It is demonstrated that the activation threshold for the dehalogenation process decreases with increasing catalytic activity of the substrate. At room temperature the 10,10'-dibromo-9,9'-bianthracene (DBBA) precursor molecules on Au(111) remain intact, while on Cu(111) a complete surface-assisted dehalogenation takes place. Dehalogenation of precursor molecules on Au(111) only starts at around 80 degrees C and completes at 200 degrees C, leading to the formation of linear polymer chains. On Cu(111) tilted polymer chains appear readily at room temperature or slightly elevated temperatures. Annealing of the DBBA/Cu(111) above 100 degrees C leads to intramolecular cyclodehydrogenation and formation of flat AGNRs at 200 degrees C, while on the Au(111) surface the formation of GNRs takes place only at around 400 degrees C. In STM, nanoribbons have significantly reduced apparent height on Cu(111) as compared to Au(111), 70 +/- 11 pm versus 172 +/- 14 pm, independently of the bias voltage. Moreover, an alignment of GNRs along low-index crystallographic directions of the substrate is evident for Cu(111), while on Au(111) it is more random. Elevating the Cu(111) substrate temperature above 400 degrees C results in a dehydrogenation and subsequent decomposition of GNRs; at 750 degrees C the dehydrogenated carbon species self-organize in graphene islands. In general, our data provide evidence for a significant influence of substrate reactivity on the growth dynamics of GNRs

Comparative NEXAFS, NMR and FTIR study of various-sized nanodiamonds – as-prepared and Fluorinated

International audienceVarious 4–50 nm in size diamond nanoparticles prepared by different synthesis methods and their fluorinated derivatives were studied by NEXAFS, solid state NMR and FTIR spectroscopy. C 1s and F 1s NEXAFS spectra of as-prepared and fluorinated nanodiamonds (NDs and F-NDs) were analyzed based on a comparison with the known ones of reference compounds (graphitized carbon nanodiscs, phenol and amino acid molecules, graphite oxide and monofluoride). It has been found that all the studied diamond nanoparticles have crystalline diamond cores and their surfaces are covered with graphite-like carbon clusters. These clusters are partially amorphized and oxidized with the formation of functional groups C–OH, C═O, or O═C–OH and the properties of these surface shells depend on the synthesis method of nanodiamonds. The fluorination of diamond nanoparticles has a purely superficial character; it almost completely cleans the NDs particles from carbon clusters and saturates dangling bonds on the surface of the diamond nanoparticles with F atoms forming covalent σ(C–F) bonds. NEXAFS data are further supported by NMR and FTIR spectroscopy, leading to similar conclusions concerning the properties of various NDs and the chemical bonding between C and F atoms in F-NDs. A combination of NEXAFS, solid state NMR and FTIR spectroscopy is demonstrated to be very efficient in investigating various NDs and their functionalized derivative

From Graphene Nanoribbons on Cu(111) to Nanographene on Cu(110): Critical Role of Substrate Structure in the Bottom-Up Fabrication Strategy.

Bottom-up strategies can be effectively implemented for the fabrication of atomically precise graphene nanoribbons. Recently, using 10,10'-dibromo-9,9'-bianthracene (DBBA) as a molecular precursor to grow armchair nanoribbons on Au(111) and Cu(111), we have shown that substrate activity considerably affects the dynamics of ribbon formation, nonetheless without significant modifications in the growth mechanism. In this paper we compare the on-surface reaction pathways for DBBA molecules on Cu(111) and Cu(110). Evolution of both systems has been studied via a combination of core-level X-ray spectroscopies, scanning tunneling microscopy, and theoretical calculations. Experimental and theoretical results reveal a significant increase in reactivity for the open and anisotropic Cu(110) surface in comparison with the close-packed Cu(111). This increased reactivity results in a predominance of the molecular-substrate interaction over the intermolecular one, which has a critical impact on the transformations of DBBA on Cu(110). Unlike DBBA on Cu(111), the Ullmann coupling cannot be realized for DBBA/Cu(110) and the growth of nanoribbons via this mechanism is blocked. Instead, annealing of DBBA on Cu(110) at 250 °C results in the formation of a new structure: quasi-zero-dimensional flat nanographenes. Each nanographene unit has dehydrogenated zigzag edges bonded to the underlying Cu rows and oriented with the hydrogen-terminated armchair edge parallel to the [1-10] direction. Strong bonding of nanographene to the substrate manifests itself in a high adsorption energy of -12.7 eV and significant charge transfer of 3.46e from the copper surface. Nanographene units coordinated with bromine adatoms are able to arrange in highly regular arrays potentially suitable for nanotemplating