Cette thèse est consacrée à l'étude d'équations de réaction-diffusion dans un environnement périodique en temps. Ces équations modélisent l'évolution d'une tumeur cancéreuse en présence d'un traitement qui correspond à une immunothérapie dans la première partie du manuscrit, et à une chimiothérapie cytotoxique dans la suite.On considère dans un premier temps des nonlinéarités périodiques en temps pour lesquelles 0 et 1 sont des états d'équilibre linéairement stables. On étudie l'unicité, la monotonie et la stabilité de fronts pulsatoires. On exhibe également des cas d'existence et de non-existence de telles solutions. Dans la deuxième partie de la thèse, on commence par travailler sur des nonlinéarités périodiques en temps qui sont la somme d'une fonction positive traduisant la croissance de la tumeur et d'un terme de mort de cellules cancéreuses du au traitement. On s'intéresse aux états d'équilibres de telles nonlinéarités, et on va déduire de cette étude des propriétés de propagation de perturbations et l'existence de fronts pulsatoires. On raffine ensuite le modèle en considérant des nonlinéarités qui sont la somme d'une fonction asymptotiquement périodique en temps et d'un terme perturbatif. On prouve notamment que les propriétés relatives à la propagation de perturbations restent valables dans ce cadre là. Pour finir, on s'intéresse à l'influence du protocole de traitement. / This phD thesis investigates reaction-diffusion equations in a time periodic environment. These equations model the evolution of a cancerous tumor in the presence of a treatment that corresponds to an immunotherapy in the firs part of the manuscript, and to a cytotoxic chemotherapy after. We begin by considering time-periodic nonlinearities for which 0 and 1 are linearly stable equilibrium states. We study uniqueness, monotonicity and stability of pulsating fronts. We also provide some conditions for the existence and non-existence of such solutions.In the second part of the manuscript, we begin by working on time-periodic nonlinearities which are the sum of a positive function which stands for the growth of the tumor in the absence of treatment and of a death term of cancerous cells due to treatment. We are interested in equilibrium states of such nonlinearities, and we will infer from this study spreading properties and existence of pulsating fronts. We then refine the model by considering nonlinearities which are the sum of an asymptotic periodic nonlinearity and of a small perturbation. In particular we prove that the spreading properties remain valid in this case. To finish, we are interested in the influence of the protocol of the treatment.

Cette thèse concerne les phénomènes de propagation de certaines équations d'évolution dans des habitats périodiques. Dans la première partie, nous étudions les phénomènes d'expansion de certaines équations d'intégro-différence spatialement périodiques. Tout d'abord, nous établissons une théorie générale sur l'existence des vitesses de propagation pour des systèmes d'évolution noncompacts, sous l'hypothèse que les systèmes linéarisés ont des valeurs propres principales. Ensuite, nous introduisons la notion d'irréductibilité uniforme des mesures de Radon finies sur le cercle. On démontre que tout opérateur de convolution généré par une telle mesure admet une valeur propre principale. Enfin, nous prouvons l'existence de vitesses de propagation pour certains équations d'intégro-différence avec des noyaux de dispersion uniformément irréductibles. Dans la deuxième partie, nous étudions les phénomènes de propagation de front pour des équations de réaction-diffusion spatialement périodiques avec des non-linéarités bistables. Nous nous concentrons d'abord sur les solutions de type fronts pulsatoires. Sous diverses hypothèses, il est prouvé que les fronts pulsatoires existent lorsque la période spatiale est petite ou grande. Nous caractérisons aussi le signe des vitesses et nous montrons la stabilité exponentielle globale des fronts pulsatoires de vitesse non nulle. Nous étudions ensuite les solutions de type fronts de transition. Sous des hypothèses convenables, on prouve que les fronts de transition se ramènent aux fronts pulsatoires avec une vitesse non nulle. Mais nous montrons aussi l'existence de nouveaux types de fronts de transition qui ne sont pas des fronts pulsatoires. / This dissertation is concerned with propagation phenomena of some evolution equations in periodic habitats. The main results consist of the following two parts. In the first part, we investigate the spatial spreading phenomena of some spatially periodic integro-difference equations. Firstly, we establish a general theory on the existence of spreading speeds for noncompact evolution systems, under the hypothesis that the linearized systems have principal eigenvalues. Secondly, we introduce the notion of uniform irreducibility for finite Radon measures on the circle. It is shown that, any generalized convolution operator generated by such a measure admits a principal eigenvalue. Finally, applying the above general theories, we prove the existence of spreading speeds for some integro-difference equations with uniformly irreducible dispersal kernels. In the second part, we study the front propagation phenomena of spatially periodic reaction-diffusion equations with bistable nonlinearities. Firstly, we focus on the propagation solutions in the class of pulsating fronts. It is proved that, under various assumptions on the reaction terms, pulsating fronts exist when the spatial period is small or large. We also characterize the sign of the front speeds and we show the global exponential stability of the pulsating fronts with nonzero speed. Secondly, we investigate the propagation solutions in the larger class of transition fronts. It is shown that, under suitable assumptions, transition fronts are reduced to pulsating fronts with nonzero speed. But we also prove the existence of new types of transition fronts which are not pulsating fronts.

Dans cette thèse nous étudions l'influence de l'environnement sur le comportement d'une cellule dans deux situations différentes. Dans chacune de ces deux situations, apparaît un couplage non-linéaire sur le champ d'advection lié à un terme non-local provenant du bord du domaine. Dans une première partie, nous modélisons la polarisation cellulaire durant la conjugaison de la cellule de levure. Nous utilisons un modèle de type convection-diffusion avec un terme de convection non-linéaire et non-local. Ce modèle présente des similarités avec le modèle de Keller-Segel, la source du potentiel attractif étant sur le bord du domaine. Nous étudions le cas de la dimension un en utilisant des inégalités de Sobolev logarithmiques et HWI. En nous appuyant sur un raisonnement heuristique, nous ramenons l'étude de notre modèle en dimension deux au bord du domaine. Nous validons le modèle à l'aide des résultats expérimentaux obtenus par M. Piel en utilisant un bruit dynamique dans nos simulations numériques. Nous étudions ensuite le problème du dialogue cellulaire entre cellules de levure de sexe opposé. Dans une seconde partie, nous étudions la réaction immunitaire durant l'athérosclérose. Nous construisons puis développons un modèle structuré en âge pour décrire l'inflammation. Pour des paramètres particuliers, nous déterminons le comportement en temps long de notre système en utilisant une fonctionnelle de Lyapunov. / We investigate the influence of the environment on the behaviour of a cell in two different situations. In each of these situations, there is a non-linear coupling of the drift due to a non-local term coming from the boundary of the domain.The first part focuses on the modeling of cell polarisation during the mating of yeast. We use a convection-diffusion model with a non-linear and non-local drift. This model is similar to the Keller-Segel model, the source of the attractive potential comes from the boundary of the domain. We study the long time behaviour of the one-dimensional case by using logarithmic Sobolev and HWI inequalities.By relying on a heuristic, we reduce the study of our model in the two-dimensional case to the boundary of the domain. We validate the model with data provided by M. Piel. This validation requires adding a dynamical noise in our numerical simulations. We study then the cell discussion between yeast of opposite gender. In the second part we study the immune response in atherosclerosis. We build and then develop an age structured model in order to describe the inflammation. For specific parameters, we investigate the long time behaviour of our system by using a Lyapunov functional.

Le but de cette thèse est de développer des méthodes de démonstration d’existence et d’unicité de solutions d’équations d’évolution stochastiques locales ou non locales dans les problèmes de transition de phase. Au chapitre 1, nous étudions un problème à valeur initiale pour une ´équation de réaction-diffusion stochastique non locale avec des conditions aux limites de Neumann homogènes dans un ouvert borné de ℝn de frontière suffisamment régulière. On considère le cas d’un opérateur elliptique non linéaire assez général et on suppose que le bruit est additif et induit par un processus Q-Wiener. Le problème déterministe modélise la séparation de phases dans des alliages binaires. La démonstration d’existence de la solution du problème stochastique est basée sur un changement de fonction qui fait intervenir la solution de l’équation de la chaleur stochastique avec un terme de diffusion non linéaire. On est ainsi conduit à l'étude d’un problème sans terme de bruit, ce qui facilite l’application de la méthode de monotonie pour identifier la limite des termes non linéaires. Au chapitre 2, nous démontrons l’existence et l’unicité de la solution d’un système de champ de phase stochastique avec des bruits multiplicatifs induits par des processus Q-Wiener. Les problèmes de champ de phase sont utilisés pour d´écrire des modèles où deux phases distinctes interviennent comme par exemple l’eau et la glace. Dans ce but, nous appliquons la méthode de Galerkin et nous établissons des estimations a priori pour la solution approchée. Nous nous appuyons ensuite sur la méthode de monotonie stochastique pour identifier la limite du terme non linéaire. Finalement, au chapitre 3, nous démontrons l’existence et l’unicité d’une solution trajectorielle en dimension d’espace d ≤ 6 pour l’équation d’Allen-Cahn non locale stochastique avec un bruit multiplicatif induit par un processus Q-Wiener. La présence d’une variable supplémentaire empêche l’application des théorèmes de compacité usuels utilisés dans les problèmes déterministes. C’est ce qui nous amène à appliquer la méthode de compacité stochastique. / The aim of this thesis is to develop methods for proving the existence and uniqueness of solutionsof local and nonlocal stochastic evolution equations in phase transition problems. In chapter 1, we studyan initial value problem for a nonlocal stochastic reaction-diffusion equation with homogeneous Neumannboundary conditions in an open bounded set of ℝn, with a smooth boundary. We consider the case of ageneral nonlinear elliptic operator and we suppose that the noise is additive and induced by a Q-Wiener process.The deterministic problem with a linear diffusion term is used to model phase separation in a binarymixture. The proof of existence for the stochastic problem is based on a change of function which involvesthe solution of the stochastic heat equation with a nonlinear diffusion term. We obtain a problem withoutthe noise term. This simplifies the application of the monotonicity method, which we use to identify thelimit of the nonlinear terms. In chapter 2, we prove the existence and uniqueness of the solution for a phasefield problem with multiplicative noises induced by Q-Wiener processes. This problem models for instancethe process of melting and solidification. To that purpose we apply the Galerkin method and derive a prioriestimates for the approximate solutions. The last step is to identify the limit of the nonlinear terms whichwe do by the so-called stochastic monotonicity method. Finally, in chapter 3, we prove the existence anduniqueness of a pathwise solution in space dimension up to 6 for the stochastic nonlocal Allen-Cahn equationwith a multiplicative noise induced by a Q-Wiener process. The usual compactness method for deterministicproblems cannot be applied in a stochastic context because of the additional probability variable. Therefore,we apply the stochastic compactness method.

The Master Thesis performs search concerning advanced small and medium power light-water reactors’ designs, including different possibilities to gain a license for their development and operation. The work covers the principal theory about the area of neutronics calculations, principal equations and simplifications. There are several different methods for solution of neutronics calculations. The thesis gives an overview of two principal groups of codes – deterministic methods and Monte Carlo method. The survey shows computational codes examples based on mentioned methods. The computational code PARCS is chosen for further study, which contained description of the input and output file, process of the model creation and conditions for neutronics calculation the of selected reactor design. Based on these facts, the transient calculation has been prepared within the thesis. Thee analyses are described – reactor emergency shutdown, reactor shutdown with stuck group of control and emergency shutdown rods and reactor shutdown with faulty reaction of emergency shutdown rods.

Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25

When the finite element method is used to solve boundary value problems, the corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters. There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation. Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment. In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown. Numerical experiments for the equilibrated residual method, for the hierarchical error estimator and for the adaptive algorithm confirm the theory. The adaptive algorithm shows its potential by creating the anisotropic mesh for the problem with the boundary layer starting with a very coarse isotropic mesh.

This thesis is concerned with analysis and implementation of Time discontinuous Galerkin method. Important part of it is constructing of algorithm for solving nonlinear convection-diffusion equations, which combines Discontinuous Galerkin method in space (DGFEM) with Time discontinuous Galerkin method (TDG). Nonlinearity of the problem is overcome by damped Newton-like method. This approach provides easy adaptivity manipulation as well as high order approximation with respect to both space and time variables. The second part of the thesis is focused on Time discontinuous Galerkin method, applied to ordinary differential equations. It is shown that the solution of Time discontinuous Galerkin equals the solution obtained by Radau IIA implicit Runge-Kutta method in the roots of right Radau Quadrature. By virtue of this relation, error estimates of the order higher by one than the standard order can be obtained in these points. Furthermore, almost two times higher order can be achieved in the endpoints of the intervals of time discretization. Finally, the thesis deals with the phenomenon of stiffness, which may dramatically decrease the order of the applied method. The theoretical results are verified by numerical experiments. Powered by TCPDF (www.tcpdf.org)

During the development of multicellular organisms, organs grow to well-defined shapes and sizes. The proper size and patterning of tissues are ensured by signaling molecules as e.g. morphogens. Secreted from a restricted source, morphogens spread into the adjacent target tissue where they form a graded concentration profile. Upon binding of the morphogens to receptors on the cell surfaces, the morphogenetic signal is transduced inside the cell via the phosphorylation of transcription factors, which subsequently regulate the expression of different target genes. Thus, cell fates are determined by the local concentration of such morphogens. In this work, we investigate three key aspects of morphogenetic signaling processes in growing tissues. First, we study the mechanics of tissue growth via cell division and cell death. We examine the rearrangements of cells on large scales and times by developing a continuum theory which describes the growing tissue as an active complex fluid. In our description we include anisotropic stresses generated by oriented cell division, and we show that average cellular trajectories exhibit anisotropic scaling behaviors. Our description is used to study experimentally measured shape changes of the developing wing disk of the fruit fly Drosophila melanogaster. Second, we focus on the spreading of morphogens in growing tissues. We show that the flow field of cell movements due to oriented cell division and cell death causes a drift term in the morphogen transport equation, which captures the stretching and dilution of the concentration profile. Comparing our theoretical results to recent experiments in the Drosophila wing disk, we find that the transport of the morphogen Dpp is mainly intracellular. We moreover show that the decay length of the Dpp gradient increases during development as a result of changing degradation rate and diffusion coefficient, whereas the drift of molecules due to growth is negligible. Thus growth does not affect the decay length of the gradient, but the decay length of the gradient might affect the tissue growth rate as discussed in this work. Finally, we develop a microscopic theoretical description of the intracellular transduction machinery of morphogenetic signals within an individual cell. Our description captures the kinetics of the trafficking of proteins between different cellular compartments in response to receptor-bound signaling molecules. Analyzing experimental data at the Drosophila neuromuscular junction, we show that the morphogenetic signaling is modulated by synaptic signaling via neuronal action potentials.

Disertační práce se zabývá modelováním znečištění ovzduší, jeho transportních a disperzních procesů ve spodní části atmosféry a zejména numerickými metodami, které slouží k řešení těchto modelů. Modelování znečištění ovzduší je velmi důležité pro předpověď kontaminace a pomáhá porozumět samotnému procesu a eliminaci následků. Hlavním tématem práce jsou metody pro řešení modelů popsaných parciálními diferenciálními rovnicemi, přesněji advekčně-difúzní rovnicí. Polovina práce je zaměřena na známou metodu přímek a je zde ukázáno, že tato metoda je vhodná k řešení určitých konkrétních problémů. Dále bylo navrženo a otestováno řešení paralelizace metody přímek, jež ukazuje, že metoda má velký potenciál pro akceleraci na současných grafických kartách a tím pádem i zvětšení přesnosti výpočtu. Druhá polovina práce se zabývá poměrně mladou metodou ELLAM a její aplikací pro řešení atmosférických advekčně-difúzních rovnic. Byla otestována konkrétní forma metody ELLAM společně s navrženými adaptacemi. Z výsledků je zřejmé, že v mnoha případech ELLAM překonává současné používané metody.