Dans une première partie, on s'intéresse à un système d'EDP stochastiques variant selon deux échelles de temps, et plus particulièrement à l'approximation de la composante lente à l'aide d'un schéma numérique efficace. On commence par montrer un principe de moyennisation, à savoir la convergence de la composante lente du système vers la solution d'une équation dite moyennée. Ensuite on prouve qu'un schéma numérique de type Euler fournit une bonne approximation d'un coefficient inconnu apparaissant dans cette équation moyennée. Finalement, on construit et on analyse un schéma de discrétisation du système à partir des résultats précédents, selon la méthodologie dite HMM (Heterogeneous Multiscale Method). On met en évidence l'ordre de convergence par rapport au paramètre d'échelle temporelle et aux différents paramètres du schéma numérique- on étudie les convergences au sens fort (approximation des trajectoires) et au sens faible (approximation des lois). Dans une seconde partie, on étudie une méthode d'approximation de solutions d'EDP paraboliques, en combinant une approche semi-lagrangienne et une discrétisation de type Monte-Carlo. On montre d'abord dans un cas simplifié que la variance dépend des pas de discrétisation- enfin on fournit des simulations numériques de solutions, afin de mettre en avant les applications possibles d'une telle méthode. / In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.

Stationary, spatially inhomogenous solutions of reaction-diffusion systems are studied in this thesis. These systems appears in biological models based on a Tu- ring's idea of a diffusion driven instability. In the connection, a global behaviour of bifurcation branches of these stationary solutions is analyzed. The thesis in- sists on theory of differential equations and on (particularly topological) methods of nonlinear analysis. The existence, as well as non-compatness in one-dimensional space, of a bifurcation branch of general reaction-diffusion system leading to Tu- ring's efekt is proved. Further, a priori estimates of Thomas model are derived. The results tend to theorem, that forall diffusion coefficient from the preestab- lished set there exists at least one stacionary, spacially nontrivial solution of Tho- mas model.

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.

This thesis is devoted to the study of some homogenization problems with applications in lubrication theory. It consists of an introduction, five research papers (I–V) and a complementary appendix.Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficients. Many important problems in physics with one or several microscopic scales give rise to this kind of equations, whence the need for methods that enable an efficient treatment of such problems. To this end several mathematical techniques have been devised. The main homogenization method used in this thesis is called multiscale convergence. It is a notion of weak convergence in  Lp spaces which is designed to take oscillations into account. In paper II we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with a finite number of microscopic scales. The main idea in the proof is closely related to a decomposition of vector fields due to Hermann Weyl. The Weyl decomposition is further explored in paper III.Lubrication theory is devoted to the study of fluid flows in thin domains. More generally, tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. The mathematical foundations of lubrication theory are given by the Navier–Stokes equation which describes the motion of a viscous fluid. In thin domains several simplifications are possible, as shown in the introduction of this thesis. The resulting equation is named after Osborne Reynolds and is much simpler to analyze than the Navier--Stokes equation.The Reynolds equation is widely used by engineers today. For extremely thin films, it is well-known that the surface micro-topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelengths upon the solution of the Reynolds equation. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm for computing the solution of the homogenized equation. Another problem consists of showing, on introducing the appropriate mathematical definitions, that the homogenized equation is the correct method of averaging. Papers I, II, IV and V investigate the effects of surface roughness by homogenization techniques in various situations of hydrodynamic lubrication. To compare the homogenized solution with the solution of the deterministic Reynolds equation, some numerical examples are also included. / Godkänd; 2011; 20110408 (johfab); DISPUTATION Ämnesområde: Matematik/Mathematics Opponent: Professor Guy Bayada, Institut National des Sciences Appliquées de Lyon (INSA-LYON), Lyon, France, Ordförande: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Tid: Tisdag den 7 juni 2011, kl 10.00 Plats: D2214/15, Luleå tekniska universitet

Finite amplitude perturbations in compressible media are ubiquitous in scientific and engineering applications such as gas-turbine engines, rocket propulsion systems, combustion instabilities, inhomogeneous solids, and traffic flow prediction models, to name a few. Small amplitude waves in compressible fluids propagate as sound and are very well described by linear theory. On the other hand, the theory of nonlinear acoustics, concerning high-amplitude wave propagation (Mach<2) is relatively underdeveloped. Most of the theoretical development in nonlinear acoustics has focused on wave steepening and has been centered around the Burgers' equation, which can be extended to nonlinear acoustics only for purely one-way traveling waves. In this dissertation, theoretical and computational developments are discussed with the objective of advancing the multi-fidelity modeling of nonlinear acoustics, ranging from quasi one-dimensional high-amplitude waves to combustion-induced detonation waves. <br> <br> We begin with the theoretical study of spectral energy cascade due to the propagation of high amplitude sound in the absence of thermal sources. To this end, a first-principles-based system of governing equations, correct up to second order in perturbation variables is derived. The exact energy corollary of such second-order system of equations is then formulated and used to elucidate the spectral energy dynamics of nonlinear acoustic waves. We then extend this analysis to thermoacoustically unstable waves -- i.e. amplified as a result of thermoacoustic instability. We drive such instability up until the generation of shock waves. We further study the nonlinear wave propagation in geometrically complex case of waves induced by the spark plasma between the electrodes. This case adds the geometrical complexity of a curved, three-dimensional shock, yielding vorticity production due to baroclinic torque. Finally, detonation waves are simulated by using a low-order approach, in a periodic setup subjected to high pressure inlet and exhaust of combustible gaseous mixture. An order adaptive fully compressible and unstructured Navier Stokes solver is currently under development to enable higher fidelity studies of both the spark plasma and detonation wave problem in the future. <br>

[Non fourni] / Acute Myeloid Leukemia (AML) is a cancer of white cells characterized by a quick proliferation of immature cells, that invade the circulating blood and become more present than mature blood cells. This thesis is devoted to the study of two mathematical models of AML. In the first model studied, the cell dynamics are represented by PDE’s for the phases G₀, G₁, S, G₂ and M. We also consider a new phase called Ğ₀, between the exit of the M phase and the beginning of the G₁ phase, which models the fast self-renewal effect of cancerous cells. Then, by analyzing the solutions of these PDE’s, the model has been transformed into a form of two coupled nonlinear systems involving distributed delays. An equilibrium analysis is done, the characteristic equation for the linearized system is obtained and a stability analysis is performed. The second model that we propose deals with a coupled model for healthy and cancerous cells dynamics in AML consisting of two stages of maturation for cancerous cells and three stages of maturation for healthy cells. The cell dynamics are modelled by nonlinear partial differential equations. Applying the method of characteristics enable us to reduce the PDE model to a nonlinear distributed delay system. For an equilibrium point of interest, necessary and sufficient conditions of local asymptotic stability are given. Finally, we derive stability conditions for both mathematical models by using a Lyapunov approach for the systems of PDEs that describe the cell dynamics.

Alguns problemas elípticos assintoticamente lineares são considerados e é provada a existência de solução. Os principais resultados são estabelecidos de dois modos distintos e as provas são baseadas em resultados clássicos da teoria de pontos críticos, a saber: minimização, princípio variacional de Ekeland, grau topológico, teorema do ponto de sela e o teorema do passo da montanha / Some asymptotically linear elliptic problems are considered and solutions are proved to exist. The main results are proved in two different ways. The proofs rely on some classical results in Critical Point Theory such as minimization, Ekeland variational principle, topological degree, saddle point theorem and mountain pass theorem

Neste trabalho estabelecemos o Teorema Central do Limite para o modelo O(N) de Heisenberg hierárquico na criticalidade via equação a derivadas parciais no limite N -> infinito. Por simplicidade consideramos apenas o caso d = 4, sendo o teorema também válido para d > 4. Pelo estudo de uma dada equação a derivadas parciais (EDP) determinamos a temperatura inversa crítica do modelo esférico hierárquico contínuo para um d > 2 qualquer, havendo conexão entre criticalidade e o ponto fixo da EDP. Por meio de uma análise geométrica da trajetória crítica obtemos informações sobre a dinâmica e distribuição dos zeros de Lee-Yang. / In this work we stablish the Central Limit Theorem for the hierarchical O(N) Heisenberg model at criticality via partial differential equation in the limit N -> infinity. For simplicity we only treat the d = 4 case but the theorem is still valid for d > 4. By studying a given partial differential equation (PDE) we determine for any d > 2 the critical inverse temperature of the continuum hierarchical spherical model, and we show a connection between criticality and the fixed point of PDE. By means of a geometric analysis of the critical trajectory we obtain some informations about Lee-Yang zeros´s dynamics and distribution.

Este trabalho trata de problemas de contorno definidos no plano. O problema central desta tese é chamado Problema de Riemann-Hilbert, o qual pode ser descrito como segue. Seja L um campo vetorial complexo não singular definido em uma vizinhança do fecho de um aberto simplesmente conexo do plano com fronteira suave. O Problema de Riemann-Hilbert para o campo L consiste em obter uma solução para a equação Lu = F(x, y, u) no aberto em estudo, sendo dada uma função F mensurável. Pede-se também que a solução tenha extensão contínua até a fronteira e que satisfaça lá uma condição adicional; trabalha-se aqui no contexto das funções Hölder contínuas. Foram obtidos resultados para o problema acima no caso em que L pertence a uma classe de campos hipocomplexos. O caso clássico conhecido é quando o campo vetorial é o operador de Cauchy-Riemann, ou, mais geralmente, quando é um campo elítico. / This work deals with boundary problems in the plane. The central problem in this thesis is the so-called Riemann-Hilbert problem, which may be described as follows. Let L be a non-singular complex vector field defined on a neighborhood of the closure of a simply connected open subset of the plane having smooth boundary. The Riemann-Hilbert problem for the vector field L consists in finding a solution to the equation Lu = F(x, y, u) on the open set under study, where the given function F is measurable. It is also required that the solution have a continuous extension up to the boundary and satisfy an additional condition there. Results were obtained for the above problem when L belongs to a class of hypocomplex vector fields. The well-known classical case is the one in which the vector field under study is the Cauchy-Riemann operator, or more generally when it is an elliptic vector field.

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma equação de placas extensíveis acoplada a uma equação de calor do tipo hiperbólico. O problema corresponde a um modelo de termo-elasticidade baseado em teorias de calor do tipo não-Fourier. Considerando que efeitos de inércia de rotação estão presentes no modelo, mostramos que o efeito dissipativo do calor e suficiente para estabilizar exponencialmente o sistema, sem dissipações adicionais. Além disso, provamos que o sistema possui um atrator global de dimensão fractal finita e também atratores exponenciais. Nossos resultados generalizam e complementam diversos trabalhos existentes / This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. It corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system exponentially and guarantee the existence of a finite-dimensional global attractor. In addition the existence of an exponential attractor and some further properties are also considered. Our results complements several existing results