Application of real and functional analysis to solve boundary value problems

This thesis is about using appropriate tools in functional analysis arid classical analysis to tackle the problem of existence and uniqueness of nonlinear partial differential equations. There being no unified strategy to deal with these equations, one approaches each equation with an appropriate method, depending on the characteristics of the equation. The correct setting of the problem in appropriate function spaces is the first important part on the road to the solution. Here, we choose the setting of Sobolev spaces. The second essential part is to choose the correct tool for each equation. In the first part of this thesis (Chapters 3 and 4) we consider a variety of nonlinear hyperbolic partial differential equations with mixed boundary and initial conditions. The methods of compactness and monotonicity are used to prove existence and uniqueness of the solution (Chapter 3). Finding a priori estimates is the main task in this analysis. For some types of nonlinearity, these estimates cannot be easily obtained, arid so these two methods cannot be applied directly. In this case, we first linearise the equation, using linear recurrence (Chapter 4). In the second part of the thesis (Chapter 5), by using an appropriate tool in functional analysis (the Sobolev Imbedding Theorem), we are able to improve previous results on a posteriori error estimates for the finite element method of lines applied to nonlinear parabolic equations. These estimates are crucial in the design of adaptive algorithms for the method, and previous analysis relies on, what we show to be, unnecessary assumptions which limit the application of the algorithms. Our analysis does not require these assumptions. In the last part of the thesis (Chapter 6), staying with the theme of choosing the most suitable tools, we show that using classical analysis in a proper way is in some cases sufficient to obtain considerable results. We study in this chapter nonexistence of positive solutions to Laplace's equation with nonlinear Neumann boundary condition. This problem arises when one wants to study the blow-up at finite time of the solution of the corresponding parabolic problem, which models the heating of a substance by radiation. We generalise known results which were obtained by using more abstract methods.

Scope for further analytical solutions for constant flux infiltration into a semi-infinite soil profile or redistribution in a finite soil profile.

We attempt to generate new solutions for the moisture content form of the one-dimensional Richards' [1931] equation using the Lisle [1992] equivalence mapping. This mapping is used as no more general set of transformations exists for mapping the one-dimensional Richards' equation into itself. Starting from a given solution, the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions. We first seek new analytical solutions satisfying Richards' equation subject to a constant flux surface boundary condition for a semi-infinite dry soil, starting with the Burgers model. The first iteration produces an existing solution, while subsequent iterations are shown to endlessly reproduce this same solution. Next, we briefly consider the problem of redistribution in a finite-length soil. In this case, Lisle's equivalence mapping is generalized to account for arbitrary initial conditions. As was the case for infiltration, however, it is found that new analytical solutions are not generated using the equivalence mapping, although existing solutions are recovered.

Integral approach and numerical improvement to calculate the carbon concentration profiles in carburising process

In the present work, the carbon diffusion in steel, where the carbon diffusivity varies with the carbon content, was solved with the integral methods under the third boundary condition. The variation of carbon diffusivity in steel with the carbon content was described with two different functions ie. linear dependence and exponential dependence. The integral approximation for both cases was improved with the numerical computation to more accurately predict the carbon profiles. The integral solution is more accurate than the formulation based on the assumption of a constant diffusivity or those based on the assumption of a constant diffusivity and/or constant carbon content at part surface. It is also more easily used in practice than the numerical method to describe the carburising process and predict the carbon content at steel surface and carbon profiles in treated layer.

Transient responses of a wetting film to mechanical and electrical perturbations

This article reports real-time observations and detailed modeling of the transient response of thin aqueous films bounded by a deformable surface to external mechanical and electrical perturbations. Such films, tens to hundreds of nanometers thick, are confined between a molecularly smooth mica plate and a deformable mercury/electrolyte interface on a protuberant drop at a sealed capillary tube. When the mercury is negatively charged, the water forms a wetting film on mica, stabilized by electrical double layer forces. Mechanical perturbations are produced by driving the mica plate toward or by retracting the mica plate from the mercury surface. Electrical perturbations are applied to change the electrical double layer interaction between the mica and the mercury by imposing a step change of the bias voltage between the mercury and the bulk electrolyte. A theoretical model has been developed that can account for these observations quantitatively. Comparison between experiments and theory indicates that a no-slip hydrodynamic boundary condition holds at the molecularly smooth mica/electrolyte surface and at the deformable mercury/electrolyte interface. An analysis of the transient response based on the model elucidates the complex interplay between disjoining pressure, hydrodynamic forces, and surface deformations. This study also provides insight into the mechanism and process of droplet coalescence and reveals a novel, counterintuitive mechanism that can lead to film instability and collapse when an attempt is made to thicken the film by pulling the bounding mercury and mica phases apart.

Integral approach and numerical improvement to calculate carbon concentration profiles in carburising

The carbon diffusion in steel, where the carbon diffusivity varies with the carbon content, was solved with the integral methods under the third boundary condition. The variation of carbon diffusivity in steel with the carbon content was described with two different functions, linear dependence and exponential dependence. The integral approximation for both cases was improved with the numerical computation to more accurately predict the carbon profiles. The integral solution is more accurate than the formulation based on the assumption of a constant diffusivity or those based on the assumption of a constant diffusivity and/or constant carbon content at part surface. It is also more easily used in practice than the numerical method to describe the carburising process and predict the carbon content at steel surface and carbon profiles in treated layer.

Consumer perceptions of bundles and time-limited promotion deals: Do contracts matter?

Marketers use various types of deals to positively influence consumers' product evaluations. Across two experiments, we manipulated print advertisements to examine whether the commonly used deal content of both bundling and time-limited promotions affect consumers' perceived confusion, risk and value. In study 1, the influence of this content was tested in the context of a 2-year telecommunications (telco) contract. Here, consumers associated a three-item bundle with greater perceived value than a single item, but perceived value was reduced and risk heightened when it was only available for a limited time. We speculate that this is because of the long-term nature of the contract. Study 2 removed the contract restriction, examining the bundling of a video game console and game(s), again with a time-limited promotion. However, in this context, we failed to locate any interaction effects. It appears that consumers further appraise the drawbacks of a long-term telco contract when accompanied by a time-limited promotion and may perceive the switching costs for study 1 three-item telco bundle to be particularly risky. Our studies represent the first empirical investigation of the effect on consumers' perceptions of offering a bundle in conjunction with a time-limited promotion. Testing these effects in contract and no contract conditions adds to the contribution of our studies by delineating a boundary condition. From a managerial perspective, our findings are thought-provoking in respect to information integration, or how consumers process different deal content together.

A practical condition for migration dynamic recrystallization

A readily evaluated condition for migration dynamic recrystallization is developed. It is based on the postulate that “the distance traversed by the boundary of a hypothetical growing grain in the time taken for the attainment of a recovery steady state must exceed the size of a critical nucleus”. A method for estimating the boundary mobility based on the kinetics of static recrystallization is also developed to facilitate evaluation of the condition. The derivation focuses first on developing an upper limit for the dynamically recrystallized grain size. This upper limit is only slightly higher than experimental values. The critical condition also agrees well with a limited set of experimental data. These data include the occurrence (and, in two cases, suppression) of dynamic recrystallization in Cu, Ni, Mg, α-Fe, γ-Fe and Al.

The boundary lubrication of chemically grafted and cross-linked hyaluronic acid in phosphate buffered saline and lipid solutions measured by the surface forces apparatus

High molecular weight hyaluronic acid (HA) is present in articular joints and synovial fluid at high concentrations; yet despite numerous studies, the role of HA in joint lubrication is still not clear. Free HA in solution does not appear to be a good lubricant, being negatively charged and therefore repelled from most biological, including cartilage, surfaces. Recent enzymatic experiments suggested that mechanically or physically (rather than chemically) trapped HA could function as an “adaptive” or “emergency” boundary lubricant to eliminate wear damage in shearing cartilage surfaces. In this work, HA was chemically grafted to a layer of self-assembled amino-propyl-triethoxy-silane (APTES) on mica and then cross-linked. The boundary lubrication behavior of APTES and of chemically grafted and cross-linked HA in both electrolyte and lipid 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) solutions was tested with a surface forces apparatus (SFA). Despite the high coefficient of friction (COF) of μ ≈ 0.50, the chemically grafted HA gel significantly improved the lubrication behavior of HA, particularly the wear resistance, in comparison to free HA. Adding more DOPC lipid to the solution did not improve the lubrication of the chemically grafted and cross-linked HA layer. Damage of the underlying mica surface became visible at higher loads (pressure >2 MPa) after prolonged sliding times. It has generally been assumed that damage caused by or during sliding, also known as “abrasive friction”, which is the main biomedical/clinical/morphological manifestation of arthritis, is due to a high friction force and, therefore, a large COF, and that to prevent surface damage or wear (abrasion) one should therefore aim to reduce the COF, which has been the traditional focus of basic research in biolubrication, particularly in cartilage and joint lubrication. Here we combine our results with previous ones on grafted and cross-linked HA on lipid bilayers, and lubricin-mediated lubrication, and conclude that for cartilage surfaces, a high COF can be associated with good wear protection, while a low COF can have poor wear resistance. Both of these properties depend on how the lubricating molecules are attached to and organized at the surfaces, as well as the structure and mechanical, viscoelastic, elastic, and physical properties of the surfaces, but the two phenomena are not directly or simply related. We also conclude that to provide both the low COF and good wear protection of joints under physiological conditions, some or all of the four major components of joints—HA, lubricin, lipids, and the cartilage fibrils—must act synergistically in ways (physisorbed, chemisorbed, grafted and/or cross-linked) that are still to be determined.

Molecular aspects of boundary lubrication by human lubricin : effect of disulfide bonds and enzymatic digestion

Lubricin (LUB) is a glycoprotein of the synovial cavity of human articular joints, where it serves as an antiadhesive, boundary lubricant, and regulating factor for the cartilage surface. It has been proposed that these properties are related to the presence of a long, extended, heavily glycosylated and highly hydrated mucinous domain in the central part of the LUB molecule. In this work, we show that LUB has a contour length of 220 ± 30 nm and a persistence length of ≤10 nm. LUB molecules aggregate in oligomers where the protein extremities are linked by disulfide bonds. We have studied the effect of proteolytic digestion by chymotrypsin and removal of the disulfide bonds, both of which mainly affect the N− and C− terminals of the protein, on the adsorption, normal forces, friction (lubrication) forces, and wear of LUB layers adsorbed on smooth, negatively charged mica surfaces, where the protein naturally forms lubricating polymer brush-like layers. After in situ digestion, the surface coverage was drastically reduced, the normal forces were altered, and both the coefficient of friction and the wear were dramatically increased (the COF increased to μ = 1.1−1.9), indicating that the mucinous domain was removed from the surface. Removal of disulfide bonds did not change the surface coverage or the overall features of the normal forces; however, we find an increase in the friction coefficient from μ = 0.02−0.04 to μ = 0.13−1.17 in the pressure regime below 6 atm, which we attribute to a higher affinity of the protein terminals for the surface. The necessary condition for LUB to be a good lubricant is that the protein be adsorbed to the surface via its terminals, leaving the central mucin domain free to form a low-friction, surface-protecting layer. Our results suggest that this “end-anchoring” has to be strong enough to impart the layer a sufficient resistance to shear, but without excessively restricting the conformational freedom of the adsorbed proteins.

Fluctuations of redox conditions across the Permian–Triassic boundary : new evidence from the GSSP section in Meishan of South China

Oceanic anoxia has long been considered as one of the main causes for the end-Permian mass extinction. However, the results obtained by different researchers are rather divergent from different sections, or even on the same section using the same redox proxy. This study aims to examine the causes for some of these divergent results using high-resolution pyrite framboid sampling at the Meishan GSSP section in South China. Detailed microfacies analysis shows that the uppermost Late Permian strata comprises two significantly different sedimentary facies: one characterized by silicious muddy limestone and recognized as representing autochthonous background sediments; the other distinguished by bioclastic grainstone, interpreted to be allochthonous in origin and have been transported from the nearby platform margin. These two different sedimentary facies represent two distinctly different redox conditions. Together with the facies analysis, a statistical analysis of pyrite framboids was carried out to evaluate the redox evolution across the Permian-Triassic boundary. Abundant framboids with average diameters of about 6μm are found in background sediments beneath the extinction boundary, indicating generally anoxic bottom water conditions. But this condition was punctuated by transient intervals of rapid oxygenation interpreted to have been caused by intrusion of intermittent turbidity flows. Our study also showed that anoxic conditions persisted into the immediate aftermath of the mass extinction, thereafter it was quickly followed by a relatively long period of oxic conditions (with rare framboids). However, the redox conditions returned to anoxia (with abundant pyrite framboids averaging about 5μm in diameter), accompanied by a rapid global transgression. The oxygenation manifested near the Permian-Triassic boundary coincides with the negative excursion of carbon isotope. This would imply that, contrary to previous interpretations, this great δ13C negative excursion was probably not caused by the upwelling of anoxic deep ocean waters.