Dislocation channel-veins and persistent slip bands (PSBs) are characteristic dislocation configurations that are of interest to both industry and academia. However, existing mathematical models are not adequate to describe the mechanism of the transition between these two states. In this thesis, a series of models are proposed to give a quantitative description to such a transition. The full problem has been considered from two angles. Firstly, the general motion and instabilities of arbitrary curved dislocations have been studied both analytically and numerically. Then the law of motion and local expansions are used to track the shapes of screw segments moving through channels, which are believed to induce dislocation multiplication by cross-slip. The second approach has been to investigate the collective behavior of a large number of dislocations, both geometrically necessary and otherwise. The traditional method of multiple scales does not apply well to describe the pile-up of two arrays of dislocations of opposite signs on a pair of neighbouring glide planes in two dimensional space. Certain quantities have to be more accurately defined under the multiple-scale coordinates to capture the much more localised resultant stress caused by these dislocation pairs. Through detailed calculations, one-dimensional dipoles can be homogenised to obtain some insightful results both on a local scale where the dipole pattern is the key diagnostic and on a macroscopic scale on which density variations are of most interest. Equilibria of dislocation dipoles in a two-dimensional regular lattice have been also studied. Some natural transitions between different patterns can be found as a result of geometrical instabilities.

In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.

We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.

This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C^1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.

Angiogenesis, the process by which new vessels form from existing ones, is a key event in the development of a large and malignant vascularised tumour. A rapid expansion in in vivo and in vitro angiogenesis research in recent years has led to increased knowledge about the processes underlying angiogenesis and to promising steps forward in the development of anti-angiogenic therapies for the treatment of various cancers. However, substantial gaps in knowledge persist and the development of effective treatments remains a major challenge. In this thesis we study tumour-induced angiogenesis within the context of a highly controllable experimental environment: the cornea micropocket assay. Using a multidisciplinary approach that combines experiments, image processing and analysis, and mathematical and computational modelling, we aim to provide mechanistic insight into the action of two angiogenic factors which are known to play central roles during tumour-induced angiogenesis: vascular endothelial growth factor A (VEGF-A) and basic fibroblast growth factor (bFGF). Image analysis techniques are used to extract quantitative data, which are both spatially and temporally resolved, from experimental images. These data are then used to develop and parametrise mathematical models describing the evolution of the corneal vasculature in response to both VEGF-A and bFGF. The first models developed in this thesis are one-dimensional continuum models of angiogenesis in which VEGF-A and/or bFGF are released from a pellet implanted into a mouse cornea. We also use an object-oriented framework, designed to facilitate the re-use and extensibility of hybrid multiscale models of angiogenesis and vascular tumour growth, to develop a complementary three-dimensional hybrid model of the same system. The hybrid model incorporates a new non-local cell sensing model which facilitates the formation of well-perfused and functional vascular networks in three dimensions. The continuum models are used to assess the utility of the cornea micropocket assay as a quantitative assay for angiogenesis, to characterise proposed synergies between VEGF-A and bFGF, and to generate experimentally testable predictions regarding the effect of anti-VEGF-A therapies on bFGF-induced angiogenesis. Meanwhile, the hybrid model is used to provide context for the comparison that is drawn between the continuum models and the data, to study the relative distributions of perfused and unperfused vessels in the evolving neovasculature, and to investigate the impact of tip cell sensing dysregulation on the angiogenic response in the cornea. We have found that by exploiting a close link with quantitative data we have been able to extend the predictive and hypothesis-testing capabilities of our models. As such, this thesis demonstrates the potential for integrating mathematical modelling with image analysis techniques to increase insight into the mechanisms underlying angiogenesis.

The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence. In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in ux, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions.

A presença de onda de choque e vórtices de pequena escala exigem métodos numéricos mais sofisticados para simular escoamentos compressíveis em velocidades altas. Alguns desses métodos produzem resultados adequados para regiões com função suave, embora os mesmos não possam ser utilizados diretamente em regiões com função descontínua, resultando em oscilações espúrias. Dessa forma, métodos foram desenvolvidos para solucionar esse problema, apresentando um bom desempenho para regiões com função descontínua; entretanto, estes possuem termos de alta dissipação. Para evitar os problemas encontrados, foram desenvolvidos os métodos híbridos, onde dois métodos com características ideais para cada região são combinados através de uma função detectora que analisa numericamente a variação de uma quantidade em uma região através de fórmulas que envolvem derivadas. Um detector de descontinuidades foi desenvolvido a partir da revisão bibliográfica de diversos métodos numéricos híbridos existentes, sendo avaliadas as principais desvantagens e limitações de cada um. Diversas comparações entre o novo detector e os detectores de descontinuidades já desenvolvidos foram realizadas através da aplicação em funções unidimensionais e bidimensionais. Finalmente, o método híbrido foi aplicado para a solução das equações de Euler unidimensionais e bidimensionais. / The presence of shock and small-scale vortices require more sophisticated numerical methods to simulate compressible flows at high speeds. Some of these methods produce good results for regions with smooth function, altough they cannot be used directly in regions with discontinuous functions, resulting in spurious oscillations. Thus, methods have been developed to solve this problem, showing a good performance for regions with discontinuous functions; however, these methods contain high dissipation terms. To avoid the problems encountered, hybrid methods have been developed, where two methods with ideal characteristics for each region are combined through a function that analyze numerically the variation of a quantity in the region using formulas involving derivatives. A discontinuity detector was developed from the literature review of several existing hybrid methods, evaluating the main disadvantages and limitations of each. The new detector and other developed discontinuity detectors were compared by applying on one and two-dimensional functions. Finally, the hybrid method was applied fo the solution of one and twodimensional Euler equations.

Imagine fluid moving through a long pipe or channel, and we inject dye or solute into this pipe. What happens to the dye concentration after a long time? Initially, the dye just moves along downstream with the fluid. However, it is also slowly diffusing down the pipe and towards the edges as well. It turns out that after a long time, the combined effect of transport via the fluid and this slow diffusion results in what is effectively a much more rapid diffusion process, lengthwise down the stream. If 0 <nu << 1 is the slow diffusion coeffcient, then the effective longitudinal diffusion coeffcient is inversely proportional to 1/nu, i.e. much larger. This phenomenon is called Taylor Dispersion, first studied by GI Taylor in the 1950s, and studied subsequently by many authors since, such as Aris, Chatwin, Smith, Roberts, and others. However, none of the approaches used in the past seem to have been mathematically rigorous. I'll propose a dynamical systems explanation of this phenomenon: specifically, I'll explain how one can use a Center Manifold reduction to obtain Taylor Dispersion as the dominant term in the long-time limit, and also explain how this Center Manifold can be used to provide any finite number of correction terms to Taylor Dispersion as well.

Cette thèse à pour objet l'étude de certains aspects du problème de Cauchy pour l'équation des vagues. Dans la première partie, on utilise une formulation paradifférentielle pour prouver un critère d'explosion pour les vagues de gravités. On montre ensuite des estimations de Strichartz pour les vagues de capillarités, avant de les utiliser pour résoudre le problème de Cauchy à faible régularité. Dans la deuxième partie, on prouve des estimations a priori pour les vagues de gravité avec fond émergent. / This thesis studies some aspects of the Cauchy problem for the water waves equation. In the first part, we use a paradifferential formulation to prove a blow-up criterion for gravity waves. We then show some Strichartz estimates for capillary waves, and use them to solve the Cauchy problem at low regularity. In the second part, we prove a priori estimates for gravity waves with an emerging bottom.

Angiogenesis is the process by which new blood vessels develop from existing vessels. Angiogenesis is important in a number of conditions such as embryogenesis, wound healing and cancer. It has been modelled phenomenologically at the macroscale, using the well-known 'snail-trail' approach in which trailing endothelial cells follow the paths of other, leading endothelial cells. In this thesis, we systematically determine the collective behaviour of endothelial cells from their behaviour at the cell-level during corneal angiogenesis. We formulate an agent-based model, based on the snail-trail process, to describe the behaviour of individual cells. We incorporate cell motility through biased random walks, and include processes which produce (branching) and annihilate (anastomosis) cells to represent sprout and loop formation. We use the transition probabilities associated with the discrete model and a mean-field approximation to systematically derive a system of non-linear partial differential equations (PDEs) of population behaviour that impose physically realistic density restrictions, and are structurally different from existing snail-trail models. We use this framework to evaluate the validity of a classical snail-trail model and elucidate implicit assumptions. We then extend our framework to explicitly account for cell volume. This generates non-linear PDE models which vary in complexity depending on the extent of volume exclusion incorporated on the microscale. By comparing discrete and continuum models, we assess the extent to which continuum models, including the classical snail-trail model, account for single and multi-species exclusion processes. We also distinguish macroscale exclusion effects introduced by each cell species. Finally, we compare the predictive power of different continuum models. In summary, we develop a microscale to macroscale framework for angiogenesis based on the snail-trail process, which provides a systematic way of deriving population behaviour from individual cell behaviour and can be extended to account for more realistic and/or detailed cell interactions.