The problem of establishing local existence and uniqueness of solutions to systems of differential equations is well understood and has a long history. However, the problem of proving global existence and uniqueness is more difficult and fails even for some very simple ordinary differential equations. It is still not known if the 3D Navier-Stokes equation have global unique solutions and this open problem is one of the Millennium Prize Problems. However, many of these mathematical models are extremely useful in the understanding of complex physical systems. For years people have considered methods for modifying these equations in order to obtain models that still capture the observed fundamental physics, but for which one can rigorously establish global results. In this thesis we focus on a taming method to achieve this goal and apply taming to modeling and numerical problems. The method is also applied to a class of nonlinear differential equations with conservative nonlinearities and to Burgers’ Equation with Neumann boundary conditions. Numerical results are presented to illustrate the ideas. / Master of Science

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On présente quelques résultats de contrôlabilité et de contrôlabilité lagrangienne sur des équations aux dérivées partielles.

Orientador: Igor Leite Freire / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T07:51:21Z (GMT). No. of bitstreams: 1 Soares_JuniorCesarAlves_M.pdf: 448504 bytes, checksum: 3bdbb23b41bf8a05b373b9117cd9aa9b (MD5) Previous issue date: 2011 / Resumo: Neste trabalho, é estudada uma generalização da equação de Burgers do ponto de vista da teoria de simetrias de Lie / Abstract: In this work, a generalization of Burgers equation is studied from the point of view of Lie point symmetry theory / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

Ce travail porte sur l’étude de mésophases lamellaires lyotropes qui contiennent des polymères hydrosolubles confinés dans les lamelles d’eau de l’empilement. L’effet du polymère confiné sur la stabilité de la structure lamellaire est en particulier étudié grâce à une mesure directe des interactions entre les membranes de tensioactif grâce la technique du Surface Force Apparatus (SFA). Les systèmes étudiés sont des fluides complexes et très visqueux ce qui a donné lieu au développement d’une nouvelle méthode pour analyser les données collectées. En l’absence de polymère dissous dans l’empilement lamellaire, la valeur du module de compressibilité élastique mesurée est remarquablement interprétée avec la théorie électrostatique corrigée des corrélations des contre-ions. En présence de polymère, au fur et à mesure que les macromolécules remplacent le contenu en eau, le module élastique de compressibilité de l’empilement lamellaire chute,signature d’une interaction attractive due à la présence des macromolécules. Néanmoins, les développements théoriques proposés ne parviennent pas à interpréter quantitativement cette décroissance. De plus, des comportements très intéressants et inattendus ont été mis en évidence lors de l’approche d’un point critique : la présence de dislocations de très grands vecteurs de Burgers à grande séparation ainsi que des phénomènes d’avalanches. En particulier, des énergies de nucléation de dislocation ont pu être extraites. / This work deals with lyotropic lamellar mesophases which contain confined hydrosoluble polymers in the water layers of the stack. In particular the effect of confined polymers on the stability of the lamellar structure is investigated through a direct measurement of the interactions between the surfactant membranes thanks to the Surface Force Apparatus (SFA) technique. The investigated systems are complex fluids and samples are extremely viscous, this is why a new procedure for analyzing the collected data has been developed. In the absence of dissolved polymer in the lamellar stack, the measured value of the elastic compressibility modulus is remarkably interpreted with the electrostatic interaction corrected from the counterions correlations. In the presence of polymer, as long as the macromolecules replace the water content the elastic compressibilitymodulus of the lamellar stack decreases, evidencing an attractive interaction due to the macromolecules. Nevertheless, the proposed theoretical developments fail to quantitatively interpret this drop. Moreover, very interesting and unexpected phenomena have been evidenced on the approach of one critical point: dislocations of large Burgers vectors at large separations and avalanche phenomena. In particular dislocation nucleation energies have been extracted.

L’objet de ce travail est l’évaluation du processus de vieillissement et de la durabilité des tissus techniques utilisés en ingénierie de construction. Deux types de tissu en polyester enduit de PVC ont été testés: VALMEX vieilli naturellement et AF9032 soumis au vieillissement accéléré. L'impact du processus de vieillissement a été observé en analysant les changements des valeurs des paramètres des modèles constitutifs: linéaire par morceaux, viscoélastique de Burgers et viscoplastique de Bodner-Partom. La comparaison de tissu VALMEX utilisé pendant 20 ans à l’Opéra de la Forêt à Sopot et du même tissu stocké à l’abri a montré que la matière, après 20 années de fonctionnement, a conservé ses propriétés mécaniques et pourrait être encore utilisée pendant plusieurs années. Le vieillissement dû aux conditions environnementales d’exploitation a provoqué une augmentation de la rigidité du revêtement (PVC), alors que la raideur des fibres en polyester est restée pratiquement inchangée. Le tissu AF9032 a été soumis au processus de vieillissement accéléré. Les échantillons de tissu ont été placés dans une chambre thermique à 80 et 90 °C durant 12 semaines. Les variations des paramètres des modèles constitutifs en fonction de temps de vieillissement ont pu être exprimées par des relations linéaires. Les résultats de vieillissement thermique accéléré ont été extrapolés d’après la méthode d’Arrhenius pour obtenir des valeurs correspondantes au vieillissement naturel. Pour les deux cas de vieillissement, les changements plus importants se sont produits dans la direction de la trame. Pour les fils de chaîne, les différences sont beaucoup plus petites et parfois négligeables. / This thesis describes the investigation on the durability of technical fabrics that are commonly used as the constructional-covering material for engineering structures. Two types of polyester reinforced PVC coated fabrics have been subjected to the exploitation ageing (VALMEX fabric) and the artificial, laboratory ageing (AF9032 fabric). The influence of the ageing process has been presented as the parameters evolution of the following constitutive formulations: the piecewise, the viscoelastic Burgers and the viscoplastic Bodner-Partom models. The comparative analysis between two types of the VALMEX fabric (the material used for 20 years on the structure of the Forest Opera roof and the not used one – kept as a spare material) have been conducted. The obtained results have shown the very good performance of the material used outdoor and indicated that it could still work satisfactorily for the next several years. The accelerated ageing of the AF9032 fabric has been performed at the elevated temperature of 80°C and 90°C in a thermal chamber for up to 12 weeks. The achieved parameters values versus ageing time have been approximated by linear functions. The results of the thermal ageing have been extrapolated using the Arrhenius methodology. The both types of ageing have significant influence on the mechanical properties of the tested fabrics for the fill direction while the warp direction is free of this effect. As a result of the performed identifications the parameters of three constitutive models have been obtained. They can be implemented for the numerical simulation of the polyester reinforced PVC coated fabrics performance and help to predict their life span under service conditions.

Cette thèse est consacrée à l'étude du contrôle de quelques équations aux dérivées partielles non linéaires issues de la mécanique des fluides. On s'intéresse notamment à l'équation de Burgers et à l'équation de Navier-Stokes. L'objectif principal est de démontrer des résultats de contrôle globaux en temps petit y compris en présence de couches limites. On montre que cela est possible en introduisant une nouvelle méthode dite "de la dissipation bien préparée". Cette méthode consiste à procéder en deux phases : une phase très courte non visqueuse suivi d'une phase plus longue d'auto-dissipation de la couche limite. Aussi bien pour Burgers que pour Navier-Stokes avec des conditions au bord de glissement avec frottement, on démontre que cette dissipation est suffisante si elle a été bien préparée. De plus, on étudie une question de contrôlabilité locale pour l'équation de Burgers lorsqu'un seul contrôle scalaire est utilisé. On démontre en améliorant une technique de noyau quadratique que le système n'est pas localement contrôlable en temps petit. / This thesis is devoted to the study of the controllability of non linear partial differential equations in fluid mechanics. We are mostly interested in Burgers equation and Navier-Stokes equation. Our main goal is to prove small-time global results, even in the presence of boundary layers. We prove that it is possible to obtain such results by introducing a new method named: ``well prepared dissipation''. This method proceeds in two phases: first, a quick phase using the inviscid behavior of the system, then a longer phase during which the boundary layer dissipates all by itself. Both for Burgers and for Navier-Stokes with Navier slip-with-friction boundary conditions, we prove that this dissipation is sufficient if it has been well prepared. Moreover, we study a question of local null controllability for the Burgers equation with a single scalar control. We prove by enhancing a second order kernel approach that the system is not small time locally null controllable.

The main objective of this research work is to apply the discontinuous Galerkin method to a classical partial differential equation to investigate the properties of the numerical solution and compare the numerical solution to the analytical solution by using discontinuous Galerkin method. This scheme is applied to 1-D non-linear conservation equation (Burgers equation) in which the governing differential equation is simplified model of the inviscid Navier-stokes equations. In this work three cases are studied. They are sinusoidal wave profile, initial shock discontinuity and initial linear distribution. A grid and time step refinement is performed. Riemann fluxes at each element interfaces are calculated. This scheme is applied to forward differentiation method (Euler's method) and to second order Runge-kutta method of this work.

A irradiação de carnes e seus derivados têm sido empregada em alguns países como medida de tratamento para garantir a segurança microbiológica destes produtos evitando infecções alimentares. A irradiação pode provocar algumas mudanças indesejáveis na composição química dos alimentos tal como a oxidação lipídica. Na produção de derivados de carnes, o perfil lipídico está diretamente relacionado com a qualidade sensorial e nutricional do produto. Para retardar o processo da oxidação lipídica são utilizados antioxidantes, que podem ser sintéticos ou naturais. Atualmente, maior atenção tem sido dada aos antioxidantes naturais derivados de ervas e especiarias, tais como o alecrim e o orégano. O objetivo deste trabalho foi avaliar o potencial antioxidante dos extratos destas duas especiarias em hambúrgueres bovinos submetidos à irradiação por fonte de 60CO com a dose de 6, 7 e 8 kGy e aceleradores de elétrons com dose de 3,5 e 7 kGy e estocados sob congelamento por períodos pré-estabelecidos de 0, 45 e 90 dias. Os resultados demonstraram que o alecrim apresentou maior proteção antioxidante em uma matriz alimentícia heterogênea como o hambúrguer, porém o orégano quando utilizado em sinergismo com o alecrim ou com o BHT/BHA conseguiu retardar a oxidação lipídica durante todo o período de análise. Apesar de terem ocorrido alterações no perfil de ácidos graxos não foi possível demonstrar uma linearidade dependente da dose de irradiação e/ou do tempo de estocagem. A análise sensorial mostrou que dentre as amostras formuladas com antioxidantes naturais, o orégano recebeu maiores notas entre os provadores. A amostra irradiada formulada com alecrim recebeu maior nota em relação a não-irradiada. O uso de especiarias com atividade antioxidante com o objetivo de inibir a degragação oxidativa em alimentos gordurosos mostra-se promissor para a aplicação industrial. / Radiation processing has been employed in some countries as a mean of treatment to assure microbiological safety of meat and meat products, avoiding the occurrence of food-borne disease. The ionizing radiation may cause some undesirable changes on chemistry composition of food and the lipid oxidation is one of the main reactions. In meat products processing industry, the lipid composition is directly related to nutritional and sensory quality of the product. For preventing oxidation, use of antioxidants which can be synthetic or natural, has been practically applied in some products. Currently, most attention has been given to natural antioxidants from herbs and spices like rosemary and oregano. The aim this study was to assess the antioxidant effects of either rosemary and oregano extract in beef burgers submitted to irradiation in 60Co source with dose 6, 7 e 8 kGy, electron beams with dose 3,5 e 7 kGy and storage under freeze along 0, 45 e 90 days. The results showed that rosemary extract has the major antioxidant effects when it is used on heterogeneous food matrix like beef burger, but oregano extract was better efficient to delay lipid oxidation along storage time when it is used in synergism with rosemary and/or BHT/BHA. Although to have occurred changes in the fatty acids composition it was not possible to demonstrate a straight dependence of irradiation dose and/or storage time. Sensory analysis showed that between the samples prepared with natural antioxidants, the beef burger prepared with oregano has received better scores by panelists. Irradiated beef burger prepared with rosemary has received better scores when compared to non-irradiated one. The use of spices with antioxidant activity to avoid the oxidative damage in foods that contain fats in their formulation is thought to be promising to application in food facilities.

This thesis considers a model of a scalar partial differential equation in the presence of a singular source term, modeling the interaction between an inviscid fluid represented by the Burgers equation and an arbitrary, finite amount of particles moving inside the fluid, each one acting as a point-wise drag force with a particle related friction constant. \begin{align*} \partial_t u + \partial_x (u^2/2) &= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t)) \end{align*} The model was introduced for the case of a single particle by Lagoutière, Seguin and Takahashi, is a first step towards a better understanding of interaction between fluids and solids on the level of partial differential equations and has the unique property of considering entropy admissible solutions and the interaction with shockwaves. The model is extended to an arbitrary, finite number of particles and interactions like merging, splitting and crossing of particle paths are considered. The theory of entropy admissibility is revisited for the cases of interfaces and discontinuous flux conservation laws, existing results are summarized and compared, and adapted for regions of particle interactions. To this goal, the theory of germs introduced by Andreianov, Karlsen and Risebro is extended to this case of non-conservative interface coupling. Exact solutions for the Riemann Problem of particles drifting apart are computed and analysis on the behavior of entropy solutions across the particle related interfaces is used to determine physically relevant and consistent behavior for merging and splitting of particles. Well-posedness of entropy solutions to the Cauchy problem is proven, using an explicit construction method, L-infinity bounds, an approximation of the particle paths and compactness arguments to obtain existence of entropy solutions. Uniqueness is shown in the class of weak entropy solutions using almost classical Kruzkov-type analysis and the notion of L1-dissipative germs. Necessary fundamentals of hyperbolic conservation laws, including weak solutions, shocks and rarefaction waves and the Rankine-Hugoniot condition are briefly recapitulated. / Diese Arbeit befasst sich mit dem Modell einer skalaren partiellen Differentialgleichung mit singulärem Quellterm, das die Interaktion zwischen einem reibungsfreiem Fluid, dargestellt durch die Burgers Gleichung, und einer gegebenen, endlichen Menge von sich in dem Fluid bewegenden Partikeln beschreibt, die eine punktweise Zugkraft auf das Fluid auswirken und durch eine entsprechende Reibungskonstante charakterisiert sind. \begin{align*} \partial_t u + \partial_x (u^2/2) &= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t)) \end{align*} Das Modell wurde für den Fall der Interaktion mit einem einzelnen Partikel durch Lagoutière, Seguin and Takahashi eingeführt, stellt einen ersten Schritt zu einem besseren Verständnis der Interaktion zwischen einem Fluid und Festkörpern auf dem Level der partiellen Differentialgleichungen dar und hat die einzigartige Eigenschaft, dass Entropielösungen und die Interaktion mit Schockwellen berücksichtigt werden. Das Modell wird zu einer beliebigen, endlichen Anzahl von Partikeln erweitert und Interaktionen wie das Verschmelzen und Spaltung von Partikeln werden behandelt. Existierende Theorie der Entropie-Zulässigkeit im Hinblick auf Interfaces und Erhaltungsgleichungen mit unstetiger Flussfunktion wird zusammengefasst, die Resultate werden verglichen und für die Regionen mit Partikelinteraktionen angepasst. Zu diesem Zweck wird die Theorie der Germs, eingeführt von Andreianov, Karlsen und Risebro, auf den vorliegenden Fall eines nicht-erhaltenden Interfaces erweitert. Für das Riemann Problem von auseinanderdriftenden Partikeln werden die exakten Lösungen berechnet und eine Analyse des Verhaltens von Entropielösungen über die von den Partikeln erzeugten Interface wird genutzt, um ein physikalisch sinnvolles und mit der Theorie eines einzelnen Partikels konsistentes Verhalten beim Verschmelzen und Spalten von Partikeln herzuleiten. Mit Hilfe einer expliziten Konstruktionsmethode, hergeleiteten L-infinity Beschränkungen, einer Approximation der Partikelpfade und Kompaktheitsargumenten wird gezeigt, dass das entsprechende Cauchy Problem wohlgestellt ist. Eindeutigkeit im Raum der schwachen Entropielösungen wird mit beinahe klassischen Argumenten der Theorie von Kruzkov sowie der Theorie von L1-dissipativen Germs gezeigt. Notwendige Grundlagen zu hyperbolischen Erhaltungsgleichungen, unter anderem die Theorie schwacher Lösungen, Schock- und Verdünnungswellen sowie die Rankine-Hugoniot Bedingung, werden in Grundzügen am Anfang der Arbeit wiederholt.