Dans cette thèse, on s'intéresse au problème de Cauchy pour des équations quasi-linéaires dispersives. Pour une telle équation, l'enjeu est de montrer l'existence et l'unicité d'une solution de l'équation avec une donnée initiale prescrite dans un espace fonctionnel le plus large possible. Nous étudierons deux modèles décrivant l'évolution de la surface d'un fluide satisfaisant certaines conditions physiques.La première partie est consacrée à l'étude de l'équation de Kadomtsev-Petviashvili avec forte tension de surface (KP-I). Cette équation possède une structure Hamiltonienne et admet donc une fonctionnelle d'énergie préservée par le flot. Afin d'obtenir des solutions définies globalement en temps, on cherche donc à construire un flot dans l'espace de Banach naturellement associé à cette énergie. De plus, on se restreint à des espaces contenant des solutions particulières (les solitons linéaires de KdV), on impose donc une condition de périodicité dans la direction transverse à la propagation du fluide.On commence par illustrer le caractère quasi-linéaire de l'équation en montrant a priori que le flot dans cet espace ne peut pas être très régulier. Ceci restreint l'éventail des méthodes connues pour résoudre ce type de problème. On a donc recours à la méthode dite de restriction de la transformée de Fourier en temps petits développée récemment par Ionescu, Kenig et Tataru pour traiter ce même modèle sans condition de périodicité. On obtient ainsi l'existence globale et l'unicité de la solution du problème de Cauchy dans l'espace d'énergie. Enfin, on montre que le flot ainsi construit est continu mais pas uniformément continu sur les ensembles bornés de l'espace d'énergie.Une application intéressante de la construction d'un flot global sur l'espace d'énergie contenant les solitons linéaires est de lever une restriction sur les perturbations admissibles dans un résultat de Rousset-Tzvetkov sur la stabilité orbitale des solitons linéaires de faible vitesse.Dans la deuxième partie de la thèse, on s'intéresse à l'équation KP-I d'ordre cinq, qui est une alternative au modèle précédent dans le cas d'une tension de surface avoisinant une valeur critique pour laquelle l'effet dispersif devient plus faible. Pour cette équation, le comportement quasi-linéaire ne se manifeste que pour des données périodiques dans la direction transverse, et les autres cas avaient été étudiés précédemment dans les travaux de Saut et Tzvetkov. On considère ici des données également périodiques dans la direction de propagation. On montre que pour certains choix de périodes, le flot ne peut pas être régulier. Afin de traiter le problème indifféremment des périodes spatiales, on utilise donc une nouvelle fois la méthode précédente pour construire un flot global dans l'espace associé au Hamiltonien de ce modèle. / This thesis investigates the Cauchy problem for some quasilinear dispersive equations. Being given such an equation, the goal is then to construct a unique solution to this equation with a prescribed initial data belonging in a function space as large as possible. We will study two models describing the time evolution of the surface of a fluid in a particular regime.The first part of this thesis is devoted to the study of the Kadomtsev-Petviashvili equation in the case of strong surface tension (KP-I). This equation has a Hamiltonian structure, so it admits an energy functional which is preserved under the flow. In order to recover solutions which are globally defined in time, we thus seek to construct a flow map in the Banach sace naturally associated with the energy. In addition, we restrict ourself to spaces including some special solutions (the KdV line soliton), so we require the functions to be periodic in the transverse direction.We start by illustrating the quasilinear behaviour of the equation : we show that a flow map defined on this space cannot be too regular. This limits the range of applicable methods known to solve this kind of problem. We thus use the so-called small times Fourier restriction norm method recently developped by Ionescu, Kenig and Tataru to deal with the same model without the periodicity assumption. We thereby obtain the global existence and uniqueness of a solution to the Cauchy problem in the energy space. At last, we prove that the flow map constructed this way is continuous yet not uniformly continuous on the bounded sets of the energy space.An interesting application of the construction of a global flow on the energy space containing the line solitons is to get rid of an extra condition on admissible perturbations in a result of Rousset-Tzvetkov on the orbital stability of the small speed line solitons.In the second part of the thesis, we turn to the fifth-order KP-I equation, which is an alternative to the previous model should the tension surface come close to a critical value in which the dispersive effect becomes weaker. Regarding this equation, the quasilinear behaviour only manifests when solutions are periodic in the transverse direction, and the other cases were treated in the work of Saut and Tzvetkov. We study the case of functions which are also periodic in the direction of propagation, and we show that at least for some choice of periods the flow map fails to be smooth. In order to treat the problem regardless of the periods, we make another use of the method above to construct a global flow in the space associated to the Hamiltonian of the equation.

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Tsunamis impose significant threat to human life and coastal infrastructure. The goal of my dissertation is to develop a robust, accurate, and computationally efficient numerical model for quantitative hazard assessment of tsunamis. The length scale of the physical domain of interest ranges from hundreds of kilometers, in the case of landslide-generated tsunamis, to thousands of kilometers, in the case of far-field tsunamis, while the water depth varies from couple of kilometers, in deep ocean, to few centimeters, in the vicinity of shoreline. The large multi-scale computational domain leads to challenging and expensive numerical simulations. I present and compare the numerical results for different important problems --- such as tsunami hazard mitigation due to presence of coastal vegetation, boulder dislodgement and displacement by long waves, and tsunamis generated by an asteroid impact --- in risk assessment of tsunamis. I employ depth-integrated shallow water equations and Serre-Green-Naghdi equations for solving the problems and compare them to available three-dimensional results obtained by mesh-free smoothed particle hydrodynamics and volume of fluid methods. My results suggest that depth-integrated equations, given the current hardware computational capacities and the large scales of the problems in hand, can produce results as accurate as three-dimensional schemes while being computationally more efficient by at least an order of a magnitude. / Ph. D. / A tsunami is a series of long waves that can travel for hundreds of kilometers. They can be initiated by an earthquake, a landslide, a volcanic eruption, a meteorological source, or even an asteroid impact. They impose significant threat to human life and coastal infrastructure. This dissertation presents numerical simulations of tsunamis. The length scale of the physical domain of interest ranges from hundreds of kilometers, in the case of landslide-generated tsunamis, to thousands of kilometers, in the case of far-field tsunamis, while the water depth varies from couple of kilometers, in deep ocean, to few centimeters, in the vicinity of shoreline. The large multi-scale computational domain leads to challenging and expensive numerical simulations. I present and compare the numerical results for different important problems — such as tsunami hazard mitigation due to presence of coastal vegetation, boulder dislodgement and displacement by long waves, and tsunamis generated by an asteroid impact — in risk assessment of tsunamis. I employ two-dimensional governing equations for solving the problems and compare them to available three-dimensional results obtained by mesh-free smoothed particle hydrodynamics and volume of fluid methods. My results suggest that twodimensional equations, given the current hardware computational capacities and the large scales of the problems in hand, can produce results as accurate as three-dimensional schemes while being computationally more efficient by at least an order of a magnitude.

We investigate the local and global well-posedness of a variety of nonlinear Dirac type equations with null structure on R1+1. In particular, we prove global existence in L2 for a nonlinear Dirac equation known as the Thirring model. Local existence in Hs for s > 0, and global existence for s > 1/2 , has recently been proven by Selberg-Tesfahun where they used Xs,b spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara-Nakanishi-Tsugawa, we prove local existence in the scale invariant class L2 by using null coordinates. Moreover, again using null coordinates, we prove almost optimal local wellposedness for the Chern-Simons-Dirac equation which extends recent work of Huh. To prove global well-posedness for the Thirring model, we introduce a decomposition which shows the solution is linear (up to gauge transforms in U(1)), with an error term that can be controlled in L∞. This decomposition is also applied to prove global existence for the Chern-Simons-Dirac equation. This thesis also contains a study of bilinear estimates in Xs,b± (R2) spaces. These estimates are often used in the theory of nonlinear Dirac equations on R1+1. We prove estimates that are optimal up to endpoints by using dyadic decomposition together with some simplifications due to Tao. As an application, by using the I-method of Colliander-Keel-Staffilani-Takaoka-Tao, we extend the work of Tesfahun on global existence below the charge class for the Dirac-Klein- Gordon equation on R1+1. The final result contained in this thesis concerns the space-time Monopole equation. Recent work of Czubak showed that the space-time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in Hs(R2) for s > 1/4 . Here we show that the Monopole equation has null structure in Lorenz gauge, and use this to prove local well-posedness for large initial data in Hs(R2) with s > 1/4.

This thesis evaluates the utility of a geochemical technique for the investigation of palaeoenvironmental change in south west England. The method, EDMA (Energy Dispersive X-ray Micro Analysis), is a rapid, non-destructive analysis tool, capable of detecting a large range of geochemical elements. This research examines the most appropriate method of sample preparation for organic soils and peats, and investigates the reliability of results gained from EDMA with respect to conventional bulk geochemical techniques. A detailed study focused on a range of different sedimentary sites in south west England where a variety of palaeoenvironmental changes were thought to occur. Pollen analysis was undertaken on the same sedimentary material, and provided complementary information on the nature and scale of vegetation change through time. Sediments from a coastal valley mire near North Sands, Salcombe, revealed information relating to the processes of sea-level change in this part of south Devon and the subsequent autogenic processes as the sediment accumulated through time. A range of sites were located on the granitic upland of Dartmoor. A raised bog, Tor Royal, provided data relating to the changing nature of the central upland landscape from late Mesolithic times to the present day. Two soligenous sites, Upper Merrivale and Piles Copse, sought to investigate the activities of postulated anthropogenic activity at a much smaller spatial scale, with particular interest placed upon the evidence for deforestation activity and the utilisation of the local mineral resources. The last site, Crift Down, a lowland spring fed valley mire utilised geochemical and palynological fluxes within the peat to investigate processes and activities associated with archaeological evidence for Medieval tinworking in this area of Cornwall. The results from the EDMA investigations, and comparable studies using other geochemical methods including EMMA, AAS and flame photometry, suggest the technique to have greatest applicability as a first stage tool in the analysis of general activities of past environmental change. The technique was found to yield reliable results for the major elements (Si, Al, 5, Fe, Ca, K, Na and Mg), but is generally incapable of providing useful data on heavy metal elements. The data from south west England suggest the method to reflect activity at a range of different scales, and as part of a structured programme of analysis may contribute information to allow a more holistic environmental reconstruction to be made.

Dispersal influences several ecological and evolutionary processes, such as intraspecific competition, genetic drift and inbreeding. It can lead to phenotypic mismatch with the habitat when a locally adapted individual winds up in an environment with a divergent selection regime compared to the source habitat. The aim of this project was to compare dispersive traits in the freshwater isopod Asellus aquaticus from a cave habitat, with surface dwelling isopods collected upstream and downstream from the cave system. The subterranean stream (cave) represents a rare, geographically limited habitat which has a divergent selective pressure compared to the surrounding habitats. Experiments on dispersal were performed in the laboratory, in darkness with IR-equipment for visualization. Displacement was measured using one-dimensional test arenas. Compared to the surface phenotype, the cave phenotype was expected to have reduced fitness outside of the cave and unlikely to successfully disperse to new areas of similar suitable conditions. The results did not follow my main hypothesis that isopods from the cave would be less dispersive than individuals from the surface. The inconclusive results might derive from large variation in the data and divergent adaptations which yield similar expression of dispersal.

Este trabalho apresenta um novo formalismo bi-dimensional em diferenças finitas no domínio do tempo (2D-FDTD) para a simulação de estruturas baseadas em metamateriais. A natureza dispersiva destes meios é levada em consideração de forma precisa pela inclusão dos modelos materiais de Drude para os tensores permissividade elétrica e permeabilidade magnética. Todos os elementos dos tensores são considerados neste formalismo, o que o torna muito atraente para a modelagem de uma classe geral de estruturas eletromagnéticas. Dois efeitos de enorme impacto são analisados em detalhes, sendo eles a cobertura de invisibilidade e o rotacionamento de campo. Ambos os efeitos requerem a utilização de técnicas de transformação de coordenadas a qual deve ser aplicada apenas na região onde os campos eletromagnéticos precisam ser manipulados, tirando vantagem da invariância das equações de Maxwell quanto a estas operações. Esta técnica redefine localmente os parâmetros de permissividade e permeabilidade do meio transformado. O formalismo implementado apresentou grande estabilidade e precisão, uma conseqüência direta da natureza dispersiva dos modelos materiais de Drude, o que o caracteriza como uma boa contribuição para uma completa compreensão da fenomenologia por trás destes efeitos fascinantes. Os resultados numéricos apresentaram boa concordância com os disponíveis na literatura. Foi também observado que ambas estruturas são muito sensíveis a variações de freqüência do campo de excitação. / This work introduces an extended two-dimensional finite difference time domain method (2D-FDTD) for the simulation of metamaterial based structures. The dispersive nature of these media is accurately taken into account through the inclusion of the Drude material models for the permittivity and permeability tensors. All tensor elements are properly accounted for, making the formalism quite attractive for the modeling of a general class of electromagnetic structures. Two striking effects are investigated with the proposed model, namely, the invisibility cloaking and the field rotation effects. Both effects require the utilization of a coordinate transformation technique which must be applied only in the region where the electromagnetic field needs to be manipulated, taking advantage of the invariance of Maxwell\'s equations with respect to these operations. This technique locally redefines the permittivity and permeability parameters of the transformed media. The implemented formalism has proved to be quite stable and accurate, a direct consequence of the dispersive nature of the Drude material model, which characterizes it as a good contribution to fully understand the phenomenology behind these fascinating effects. The numerical results are in good agreement with those available in the literature. It was also verified that both structures are very sensitive to frequency variations of the excitation field.

In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution Ψlin to the dispersive equation with no non-linearity and initial data Ψ+ there exists a unique solution Ψ to the non-linear equation with initial data ΨO such that Ψ behaves as Ψlin as t→ ∞. Then the wave operator is the map W + that takes Ψ+/sub; to Ψ0. By its definition, W+ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts. The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the #8220; semi-relativistic ” Schrëdinger equation as well as the Klein-Gordon-Schrëdinger system on R2. In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.

Rubber and plastics companies are using mixing equipment (‘internal mixers') which was invented by Banbury in 1916 and which has hardly evolved since then. There is an urgent need for the modernization of such equipment and the market is demanding higher and higher performances for rubber goods. The physics of the dispersion of porous or fibrous agglomerates in a flow field has not been widely addressed in the past, despite of its importance. This is mainly due to the technical difficulties associated with the observations of the kinetics of this disagglomeration and the wide range of size that must be probed. Two mechanisms are recognized : erosion and rupture. Actually, different software solutions to simulate the 3D transient behavior of a flow in internal batch mixer are available. In all existing codes, it is assumed that mixing and flow calculations are decoupled : the analysis of the mixing (distributive or dispersive mixing) is performed after the calculation of the flow. To sum-up, hierarchical modeling including micro-macro models is considered in this work. In this thesis, we developed new distributive tools and new dispersive mathematical model. We compared the numerical prediction with several experiments. Finally, we use this model to design a new rotor shape in the framework of a European project.