This work analyses the phenomenon of diffusion in crystalline Si. Basic diffusion mechanisms and equations are described in the basic diffusion characteristics (diffusion coefficient, electrical conductivity and the concentration of vacancies) are analyzed by means of vacancies, what are generated by the beams of x – rays. The obtained facts suggest the makings of diffusion in crystalline Si at room temperature.

In this thesis, we study the nonlinear diffusion equation especially Porous Medium Equation (PME). u_t= \Delta(u^m) + f(u), Parameter m>1 in the case of slow diffusion, m=1 means linear model and $0

Research in the biomedical sciences is incredibly diverse and often involves the interaction of specialists in a variety of fields. In particular, quantitative, mathematical, and computational methods are increasingly playing significant roles in studying problems arising in biomedical science. This is particularly exciting for mathematical modeling as the complexity of biological systems poses new challenges to modelers and leads to interesting mathematical problems. On the other hand mathematical modeling can provide considerable insight to laboratory and clinical researchers. In this thesis we develop mathematical representations for three biological processes that are of current interest in biomedical science. A deeper understanding of these processes is desirable not only from the standpoint of basic science, but also because of the connections these processes have with certain diseases. The processes we consider are collective cell motility, bone remodeling, and injury response in articular cartilage. Our goals are to develop mathematical representations of these processes that can provide a conceptual framework for understanding the processes at a fundamental level, that make rigorous the intuition biological researchers have developed about these processes, and that help to translate theoretical and experimental work into information that can be used in clinical settings where the primary concern is in treating diseases associated with the process.

This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

Retracer l’histoire d’un bassin est un préalable essentiel à toute recherche d’hydrocarbures. Pour cela, on a recours à un modèle stratigraphique, qui simule l'évolution des bassins sédimentaires sur de grandes échelles de temps (millions d'années) et d'espace (centaines de kilomètres). Le logiciel Dionisos, développé à IFPEN depuis 1992 et très apprécié par les compagnies pétrolières, permet d’effectuer ce type de calculs en prenant en compte deux grands processus physiques : (1) le transport gravitaire des sédiments dû à l’inclinaison du sol ; (2) l’écoulement de l’eau provenant des fleuves et des précipitations. Le transport gravitaire est décrit par une équation de diffusion dans laquelle le flux de sédiments dépend de la pente du sol. Initialement, cette dépendance est linéaire. Pour mieux s’approcher des observations réelles, on souhaite la rendre non-linéaire par l’intermédiaire d’un p-Laplacien. Ce changement nécessite la conception d’une nouvelle méthode de résolution numérique, qui doit offrir non seulement une grande rapidité d’exécution, mais aussi des garanties de robustesse et de précision des résultats. De plus, elle doit être compatible avec une contrainte sur le taux d’érosion présente dans le modèle. L’ajout de l’écoulement de l’eau est aussi une sophistication récente du modèle physique de Dionisos. Il se traduit par l’introduction d’une nouvelle équation aux dérivées partielles, couplée à celle du transport. Là encore, il est important d’élaborer une stratégie de résolution numérique innovante, en ce sens qu’elle doit être à la fois performante et bien adaptée au fort couplage de ces deux phénomènes. L'objectif de cette thèse est de moderniser le cœur numérique de Dionisos afin de traiter plus adéquatement les processus physiques ci-dessus. On cherche notamment à élaborer un schéma implicite par rapport à toutes les inconnues qui étend et améliore le schéma actuel. Les méthodologies retenues serviront de base à la prochaine génération du calculateur. / An essential prerequisite to finding hydrocarbons is to trace back the history of a basin. To this end, geologists resort to a stratigraphic model, which simulates the evolution of sedimentary basins over large time scales (million years) and space (hundreds of kilometers). The Dionisos software, developed by IFPEN since 1992 and highly praised by oil companies, makes this type of calculation possible by accounting for two main physical processes: (1) the sediment transport due to gravity; (2) the flow of water from rivers and rains. The gravity transport is described by a diffusion equation in which the sediment flow depends on the slope of the ground. Initially, this dependence is linear. To better match experimental observations, we wish to make it nonlinear by means of a p-Laplacian. This upgrade requires to design a dedicated numerical method which should not only run fast but also provide guarantees of robustness and accuracy. In addition, it must be compatible with a constraint on the erosion rate in the present model. The water flow due to rivers and rains is also a recent enhancement brought to the physical model of Dionisos. This is achieved by introducing a new partial differential equation, coupled with that of sediment transport. Again, it is capital to work out an innovative numerical strategy, in the sense that it must be both efficient and well suited to the strong coupling of these two phenomena. The objective of this thesis is to rejuvenate the numerical schemes that lie at the heart of Dionisos in order to deal more adequately with the physical processes above. In particular, we look for an implicit scheme with respect to all the unknowns that extends and improves the current scheme. The methodologies investigated in this work will serve as a basis for the next generation of stratigraphic modelling softwares.

A variation of the excite-and-probe technique is used to measure the picosecond evolution of laser-induced transient gratings that are produced in germanium by the direct absorption of 40 psec optical pulses at 1.06-μm. Grating lifetimes are determined for free carrier densities between 10¹⁸ cm⁻³ and 10²¹ cm⁻³ . For carrier densities less than 10¹⁹ cm⁻³ , a linear diffusion-recombination model for the grating provides a good fit to the experimental data and allows the extraction of the diffusion coefficient and an estimation of the linear recombination lifetime. Above carrier densities of approximately 10²⁰ cm⁻³ , the density dependence of the diffusion coefficient and nonlinear recombination processes must be considered. Numerical solutions to the resulting nonlinear partial differential equation are obtained that allow extraction of information concerning the high density diffusion coefficient and the nonlinear recombination rates.

This thesis is dedicated to the so-called Dirichlet-to-Neumann map associated with the weighted 𝑝-Laplace operator. In Chapter 1, we begin by deriving the Dirichlet-to-Neumann map by using classical modelling and outline why it is interesting to study this boundary operator. In the remaining part of Chapter 1, we dedicate each section an overview about the content of one chapter and summarize the main results. Chapter 2 is dedicated to the Poisson problem and the inverse of the Dirichlet-to-Neumann map. Chapter 3 provides the first main application of the Dirichlet-to-Neumann map, namely, it generates a strongly continuous semigroup of contractions on the Lebesgue space 𝐿2 and this contraction can be extrapolated to a contraction on 𝐿q for all 1 ≤ 𝑞 ≤ ∞. In Chapter 4, we develop an abstract theory to establish global 𝐿𝑞-𝐿∞ regularization estimates satisfied by the semigroup generated by the negative Dirichlet-to-Neumannmap. Chapter 5 is concerned with 𝐿1 and point-wise estimates on the time-derivative of the semigroup generated by the neagtive Dirichlet-to- Neumann map, which are known in the literatur as Aronson-Bénilan type estimates. In Chapter 6, we outline the theory of 𝑗-functional and its application to evolution problems. This theory allows us to study the Dirichlet problem on general open sets Ω, and to realize the Dirichlet-to-Neumann map as an operator in 𝐿2 (𝜕Ω). In Chapter 7, we consider the limit case 𝑝 = 1, which corresponds to the Dirichlet-to-Neumann map associated with the (unweighted) 1-Laplace operator. Each chapter covers parts of the authors papers mentioned in the references.:Chapter 1 Introduction................................................... 1 1.1 Motivation-physical background ............................. 2 1.2 The Dirichlet-to-Neumann map - an analyst’s perspective . . . . . . . . . 5 1.2.1 Step1. The Dirichlet problem.......................... 5 1.2.2 Step2. The Neumann boundary operator ................ 8 2 1.3 The Dirichlet-to-Neumann map on 𝐿2 ......................... 9 1.4 The Dirichlet-to-Neumann map and Leray-Lions operators . . . . . . . . 11 1.5 The Dirichlet-to-Neumann map is a nonlocal operator . . . . . . . . . . . . 12 1.6 The Dirichlet-to-Neumann map on open sets.................... 13 1.6.1 𝑗-elliptic functionals and their 𝑗-subgradient . . . . . . . . . . . . . 13 1.6.2 The construction of a weak trace on open sets ............ 15 1.6.3 Construction of the Dirichlet-to-Neumann map . . . . . . . . . . . 17 1.7 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . 19 1.8 Evolution problems governed by the Dirichlet-to-Neumann map . . . 21 1.9 𝐿𝑞-𝐿∞ regularization and decay estimates...................... 27 1.10 Aronson-Bénilantypeestimates .............................. 30 1.11 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . 33 Chapter 2 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . . . 45 2.1 The Poisson problem........................................ 45 2.2 Preliminaries .............................................. 46 2.3 The Dirichlet problem....................................... 48 2.4 The Dirichlet-to-Neumann map............................... 51 2.5 Proof of Theorem 2.1 ....................................... 56 2.5.1 Proof of claim (1) of Theorem 2.1 ...................... 56 2.5.2 Preliminaries for the proof of claim (2) of Theorem 2.1 . . . . 58 2.5.3 Proof of claim( 2) of Theorem 2.1 ...................... 60 Chapter 3 Nonlinear elliptic-parabolic evolution problems.................... 61 3.1 Main result................................................ 61 3.2 Preliminaries .............................................. 64 3.2.1 Some function spaces................................. 64 3.2.2 Nonlinear semigroupt heory - Part I..................... 65 3.2.3 Homogeneous operators - Part I ........................ 75 2 3.3 The Dirichlet-to-Neumann map on 𝐿2 ...................... 77 3.4 The Dirichlet-to-Neumann map on 𝐿1, 𝐿𝜓 and C................ 82 3.5 Proof of Theorem 3.1 ....................................... 84 Chapter 4 𝑳𝒒-𝑳∞ regularization and decay estimates ........................ 89 4.1 Main results............................................... 89 4.2 Preliminaries .............................................. 91 4.3 Sobolev implies 𝐿𝑞 -𝐿𝑟 regularization estimates ................. 92 4.4 Extrapolation towards 𝐿1 .................................... 98 4.5 A nonlinear interpolation theorem.............................100 4.6 Extrapolation towards 𝐿∞ via interpolation of the semigroup . . . . . . 107 4.7 Proof of Theorem 4.1 .......................................115 Chapter 5 Aronson-Bénilan type estimates..................................117 5.1 Main results ...............................................117 5.2 Preliminaries ..............................................119 5.2.1 Nonlinearsemigrouptheory-PartII ....................119 5.2.2 Homogeneousaccretiveoperators ......................130 5.2.3 Homogeneous completely accretive operators . . . . . . . . . . . . 138 5.3 Proof of Theorem 5.1 .......................................141 Chapter 6 The Dirichlet-to-Neumann map on open sets ......................143 6.1 Main results ...............................................143 6.2 The 𝑗-subgradient and basic properties ........................146 6.2.1 Definition and characterisation as a classical gradient . . . . . . 146 6.2.2 Ellipticextensions ...................................151 H 6.2.3 Identification of 𝜑 ..................................152 6.2.4 The case when 𝑗 is a weakly closed operator .............155 6.3 Semigroups and invariance of convex sets ......................156 6.3.1 Positive semigroups ..................................160 6.3.2 Comparison and domination of semigroups ..............161 6.3.3 𝐿∞-contractivity and extrapolation of semigroups . . . . . . . . . 163 6.4 Application:The Dirichlet-to-Neumann map....................168 Chapter 7 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . . . 171 7.1 Preliminaries ..............................................171 7.1.1 Functions of bounded variation.........................171 7.1.2 Nonlinear semigroup theory - Part III ...................178 7.2 The Dirichlet problem for the 1-Laplace operator................180 7.3 A Robin-type problem for the 1-Laplace operator................187 7.4 Proofs of the main results....................................189 7.4.1 The Dirichlet-to-Neumann operator in 𝐿1 ................189 7.4.2 The Dirichlet-to-Neumann operator in 𝐿2 ................200 7.4.3 The Dirichlet-to-Neumann operator in 𝐿1 (continued)...........204 7.4.4 Long-timestability...................................206 Appendix A Weighted Sobolev Spaces........................................213 A.1 p-admissible weights........................................213 B Mean spaces by Lions and Peetre ................................215 B.1 The connection between mean spaces and 𝐿p spaces.............215 References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index .............................................................227

Cette thèse est consacrée à l’étude des systèmes d’équations différentielles ordinaires, et ceux aux dérivées partielles paraboliques issus de modèles de dynamique des populations et de la biologie. L’objectif principal est de faire l’analyse mathématique, la simulation numérique ainsi que l’identification des diffusions croisées dans les modèles construits. Nous présentons d’abord un système de réaction-diffusion modélisant la croissance de plantes en compétition spatiale dans un milieu saturé. Nous effectuons par la suite l’étude théorique et numérique de tels systèmes, ainsi que l’étude des problèmes d’identification des termes de diffusions croisées. Ensuite, nous proposons un modèle proie-prédateur de type Leslie-Gower modifié avec une fonction de réponse de type Crowley-Martin. Nous étudions dans un premier temps la dynamique temporelle globale du modèle considéré, et nous présentons des simulations numériques pour illustrer les résultats théoriques. En outre, nous introduisons la dimension spatiale dans le modèle dynamique considéré, et nous effectuons une analyse théorique complète de la dynamique spatio-temporelle du modèle. / This thesis is devoted to the study of ordinary differential systems, and systems of non linear parabolic PDEs resulting from models of population dynamics and biology. The main objective is to perform mathematical analysis, numerical simulations, and identification of cross-diffusion in built models. We first present a reaction-diffusion system that models the spatial competition of plants in a saturated environment. We then perform a theoretical and a numerical study of such systems, and handle the identification of cross-diffusion problem. Secondly, we propose a modified Leslie-Gower-type predator-prey model with a Crowley-Martin type functional response. Within this context, we study the global temporal dynamics of the considered model, and present numerical simulations as illustration of the theoretical results. Finally, we introduce the spatial dimension in the previous dynamical model, and perform a comprehensive theoretical analysis of the spatio-temporal model.

Cette thèse est consacrée à l’étude d’une classe d’équations de réaction-diffusion avec advection non-locale. La motivation vient du mouvement cellulaire avec le phénomène de ségrégation observé dans des expérimentations de co-culture cellulaire. La première partie de la thèse développe principalement le cadre théorique de notre modèle, à savoir le caractère bien posé du problème et le comportement asymptotique des solutions dans les cas d'une ou plusieurs espèces.Dans le Chapitre 1, nous montrons qu'une équation scalaire avec un noyau non-local ayant la forme d'une fonction étagée, peut induire des bifurcations de Turing et de Turing-Hopf avec le nombre d’ondes dominant aussi grand que souhaité. Nous montrons que les propriétés de bifurcation de l'état stable homogène sont intimement liées aux coefficients de Fourier du noyau non-local.Dans le Chapitre 2, nous étudions un modèle d'advection non-local à deux espèces avec inhibition de contact lorsque la viscosité est égale à zéro. En employant la notion de solution intégrée le long des caractéristiques, nous pouvons rigoureusement démontrer le caractère bien posé du problème ainsi que la propriété de ségrégation d'un tel système. Par ailleurs, dans le cadre de la théorie des mesures de Young, nous étudions le comportement asymptotique des solutions. D'un point de vue numérique, nous constatons que sous l'effet de la ségrégation, le modèle d'advection non-locale admet un principe d'exclusion.Dans le dernier Chapitre de la thèse, nous nous intéressons à l'application de nos modèles aux expérimentations de co-culture cellulaire. Pour cela, nous choisissons un modèle hyperbolique de Keller-Segel sur un domaine borné. En utilisant les données expérimentales, nous simulons un processus de croissance cellulaire durant 6 jours dans une boîte de pétri circulaire et nous discutons de l’impact de la propriété de ségrégation et des distributions initiales sur les proportions de la population finale. / This thesis is devoted to the study for a class of reaction-diffusion equations with nonlocal advection. The motivation comes from the cell movement with segregation phenomenon observed in cell co-culture experiments. The first part of the thesis mainly develops the theoretical framework of our model, namely the well-posedness and asymptotic behavior of solutions in both single-species and multi-species cases.In Chapter 1, we show a single scalar equation with a step function kernel may display Turing and Turing-Hopf bifurcations with the dominant wavenumber as large as we want. We find the bifurcation properties of the homogeneous steady state is closed related to the Fourier coefficients of the nonlocal kernel.In Chapter 2, we study a two-species nonlocal advection model with contact inhibition when the viscosity equals zero. By employing the notion of the solution integrated along the characteristics, we rigorously prove the well-posedness and segregation property of such a hyperbolic nonlocal advection system. Besides, under the framework of Young measure theory, we investigate the asymptotic behavior of solutions. From a numerical perspective, we find that under the effect of segregation, the nonlocal advection model admits a competitive exclusion principle.In the last Chapter, we are interested in applying our models to a cell co-culturing experiment. To that aim, we choose a hyperbolic Keller-Segel model on a bounded domain. By utilizing the experimental data, we simulate a 6-day process of cell growth in a circular petri dish and discuss the impact of both the segregation property and initial distributions on the finial population proportions.

[pt] É comum observar na natureza a emergência de comportamentos coletivos em populações biológicas, como formação de padrão. Neste trabalho, estamos interessados em caracterizar a distribuição de uma população de espécie única (como alguns tipos de bactérias ou de vegetação), a partir de modelos matemáticos que descrevem a evolução espaço-temporal, governados por processos elementares como: dispersão, crescimento e competição não-local por recursos. Primeiramente, utilizando uma generalização da equação de FKPP, analisamos numérica e analiticamente, o impacto de mecanismos de regulação dependentes da densidade, tanto na difusão quanto no crescimento. Tais mecanismos representam processos internos de retroalimentação, que modelam a resposta do sistema à superlotação ou rarefação da população. Mostramos que, dependendo do tipo de resposta em ação, os indivíduos podem se auto-organizar em subpopulações desconectadas (fragmentação), mesmo na ausência de restrições externas, ou seja, em uma paisagem homogênea. Discutimos o papel crucial que a dependência com a densidade tem na forma dos padrões, particularmente na fragmentação, o que pode trazer consequências importantes para processos de contato como disseminação de epidemias. Tendo compreendido esse fenômeno em um meio homogêneo, estudamos o papel que um ambiente heterogêneo tem na organização espacial de uma população, que representamos através de uma taxa de crescimento que varia com a posição. Investigamos as estruturas que emergem próximo a fronteira de um meio para o outro. Descobrimos que, dependendo da forma de interação nãolocal e de outros parâmetros do modelo, três perfis diferentes podem emergir a partir da interface: (i) oscilações não-atenuadas (ou padrões espaciais, sem decaimento da amplitude); (ii) oscilações atenuadas (com amplitude decaindo a partir da interface); (iii) decaimento exponencial (sem oscilações) a um perfil homogêneo. Relacionamos o comprimento de onda e a taxa de decaimento das oscilações com os parâmetros das interações (comprimento característico e forma de decaimento com a distância). Discutimos como as heterogeneidades do ambiente permitem acessar informações (ocultas no caso homogêneo) sobre os fenômenos biológicos do sistema, tais como os que mediam interações competitivas. / [en] It is common to observe in nature the emergence of collective behavior in biological populations, such as pattern formation. In this work, we are interested in characterizing the distribution of a single-species population (such as some bacteria or vegetation), based on mathematical models that describe the spatio-temporal evolution, and governed by elementary processes, such as: dispersion, growth, and nonlocal competition by resources. First, using a generalization of the FKPP equation, we analyze numerically and analytically the impact of density-dependent regulatory mechanisms, both on diffusion and growth. Such mechanisms represent processes of internal feedback, which shape the system s response to population overcrowding or rarefaction. We show that, depending on the type of the response in action, some individuals can organize themselves in disconnected sub-populations (fragmentation), even in the absence of external restrictions, that is in a homogeneous landscape. We discuss the crucial role that density-dependence has in the form of patterns, particularly in fragmentation, which can have important consequences for contact processes, such as the spread of epidemics. After understanding this phenomenon in a homogeneous environment, we study the role that a heterogeneous environment has in the spatial organization of a population, which was presented as a growth rate that varies with position. We investigate the structures that emerge near the border from one environment to the other. We found that, depending on the shape of nonlocal interaction and other model parameters, three different profiles can emerge from the interface: (i) sustained oscillations (or spatial patterns, without amplitude decay); (ii) attenuated oscillations (with amplitude decreasing from the interface); (iii) exponential decay (without oscillations) to a homogeneous profile. We related the wavelength and the rate of decay of oscillations with the parameters of the interaction (characteristic length and form of decay with distance). We discussed how the heterogeneities of the environment allow access to information (hidden in the homogeneous case) about the biological phenomena of the system, such as those that mediate competitive interactions.