Dans cette thèse, nous étudions des écoulements caractérisés par un faible nombre de Mach. Dans une première partie, nous développons un schéma numérique permettant la résolution des équations de Navier-Stokes à faible nombre de Mach. L’équation de continuité est résolue par une méthode de volumes finis, tandis que l’équation de conservation de la quantité de mouvement et l’équation d’évolution de la température sont résolues par éléments finis. Le schéma ainsi développé assure la préservation des états constants. Dans une seconde partie, nous faisons l’analyse d’un modèle de type faible Mach spécifique, dans lequel la pression thermodynamique est considérée constante, et la viscosité est une fonction particulière de la température. Nous montrons l’existence, l’unicité et la régularité des solutions, ainsi qu’un résultat de principe du maximum pour la température. Enfin dans une troisième partie, nous développons un schéma numérique permettant de simuler les équations de ce modèle. L’accent est mis sur la discrétisation de l’équation de température, qui est de type volumes finis. Plusieurs schémas sont étudiés et comparés sur des critères de précision et de respect du principe du maximum. L’équation de conservation de la quantité de mouvement est discrétisée par éléments finis, définissant un nouveau schéma combiné. / In this thesis, we study some flows characterized by a low Mach number. In a first part, we develop a numerical scheme allowing the resolution of the Navier-Stokes equations in the low Mach number approximation. The continuityequation is solved by a finite volume method, while the momentum and temperature equations are solved by finite elements. The scheme ensures the preservation of constant states. In a second part, we analyze a specific low Mach type model, in which the thermodynamic pressure is considered constant, and the viscosity is a particular function of the temperature. We show the existence, the uniqueness and the regularity of the solutions, as well as a maximum principle result for the temperature. Finally, in a third part, we develop a numerical scheme to simulate the equations of this model. Emphasis is placed on the discretization of the temperature equation, which is of finite volume type. Several schemes are studied and compared on criteria of precision and respect of the maximum principle. The momentum equation is discretized by finite elements, defining a new combined scheme.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T16:40:40Z No. of bitstreams: 1 DÉSIO RAMIREZ DA ROCHA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 641903 bytes, checksum: 9b0b6f008468c08a7a9d581e14ef0d13 (MD5) / Made available in DSpace on 2018-07-25T16:40:41Z (GMT). No. of bitstreams: 1 DÉSIO RAMIREZ DA ROCHA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 641903 bytes, checksum: 9b0b6f008468c08a7a9d581e14ef0d13 (MD5) Previous issue date: 2010-09 / CNPq / Capes / Neste trabalho, baseado numa seqüência de artigos de David Ho , é provado um teorema sobre a existência de uma solução fraca para um problema de valor inicial envolvendo as equações de Navier-Stokes para o caso de um escoamento unidimensional de um fluido compressível. São consideradas como hipóteses básicas a ausência de forças externas e que a pressão seja uma função contínua positiva crescente da densidade, cuja derivada também seja contínua. Quanto aos dados iniciais, estes podem possuir descontinuidades do tipo salto, não necessariamente pequenos, podendo se comportar inclusive como funções constantes por partes, em particular dados de Riemann. Tal teorema é provado baseado numa seqüência de lemas e proposições que fornecem estimativas para soluções aproximadas suaves obtidas a partir de dados regularizados. A solução nal é obtida por um processo de passagem ao limite das soluções aproximadas / In this work, based on a serie of papers by David Ho , it is proved a theorem on the existence of a weak solution to the initial value problem for the Navier-Stokes equations for a one space dimension ow of a compressible uid. It is assumed the absence of external forces and that the pressure is a continuous positive increasing function of density with the derivative also continuous. Concerning the initial data, they are allowed to have large jump discontinuities, such as piecewise constant functions, in particular Riemann data. The proof of the theorem is based on a sequence of lemmas and propositions which give estimates on the approximate smooth solutions obtained under regularized data. The nal solution is obtained by a limit process on the approximate solutions.

Made available in DSpace on 2015-05-15T11:46:21Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1219498 bytes, checksum: d931fbd819e737ae31a593f5c64c55e0 (MD5) Previous issue date: 2014-09-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we study the Navier-Stokes equations in bounded domains of Rn. Initially we the case n = 2 and we show that its variational formulation is well put (in case the Hadamard). We show the existence of solution for the case n  4 . In both cases we use the Faedo-Galerkin method. / Neste trabalho estudamos as equações de Navier-Stokes em domínios limitados do Rn. Inicialmente consideramos o caso n = 2emostramos que sua formulação variacional est´a bem posta (no sentido de Hadamard). Em seguida, mostramos a existência de solução para o caso n  4. Em ambos os casos utilizamos o método de Faedo-Galerkin.

Orientador: Lucas Catão de Freitas Ferreira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T06:33:09Z (GMT). No. of bitstreams: 1 Mesquita_ClaudiaAlineAzevedodosSantos_D.pdf: 1141685 bytes, checksum: a65a24d1917c5f998314d01970bb86e3 (MD5) Previous issue date: 2013 / Resumo: Nesta tese, estudamos dois tipos de EDPs com potenciais singulares críticos, a saber, uma equação elíptica com operador poliharmônico e a equação do calor linear. Para a primeira, pesquisamos existência e propriedades qualitativas das soluções no espaço $\mathcal{H}_{k,\vec{\alpha}}$ que é uma soma de espaços $L^{\infty}$ com peso, o qual parece ser um espaço mínimo para o tipo de potencial singular considerado. Investigamos um conceito de simetria para soluções que estende o de simetria radial e satisfaz uma ideia de invariância em torno das singularidades. Para a segunda, uma estratégia baseada na transformada de Fourier é empregada para obter resultados de boa-colocação global e comportamento assintótico de soluções, sem hipóteses de pequenez e sem utilizar a desigualdade de Hardy. Em particular, obtemos boa-colocação de soluções para o caso do potencial monopolar $V(x)=\frac{\lambda}{\left\vert x\right\vert ^{2}}$ com $\left\vert \lambda\right\vert <\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. Este valor limiar é o mesmo obtido em resultados de boa-colocação global em $L^2$ que utilizam desigualdades de Hardy e estimativas de energia. Desde que não existe uma relação de inclusão entre $L^{2}$ e $PM^{k}$, nossos resultados indicam que $\lambda_{\ast}$ é intrínseco da EDP e independe de uma particular abordagem. Palavras-chave: Equações elípticas, equação do calor, potencial singular, existência, simetria, autossimilaridade, comportamento assintótico / Abstract: In this thesis, we study two types of PDEs with critical singular potentials, namely, an elliptic equation with polyharmonic operator and the linear heat equation. For the first, we obtain existence and qualitative properties of solutions in $\mathcal{H}_{k,\vec{\alpha}}$-spaces which are a sum of weighted $L^{\infty}$-spaces, and seem to be a minimal framework for the potential profile of interest. We investigate a concept of symmetry for solutions which extends radial symmetry and carries out an idea of invariance around singularities. For the second, a strategy based on the Fourier transform is employed to obtain results of global well-posedness and asymptotic behavior of solutions, without smallness hypotheses and without using Hardy inequality. In particular, well-posedness of solutions is obtained for the case of the monopolar potential $V(x)=\frac{\lambda}{\left\vert x\right\vert ^{2}}$ with $\left\vert \lambda\right\vert <\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. This threshold value is the same one obtained for the global well-posedness of $L^{2}$-solutions by means of Hardy inequalities and energy estimates. Since there is no inclusion relation between $L^{2}$ and $PM^{k}$, our results indicate that $\lambda_{\ast}$ is intrinsic of the PDE and independent of a particular approach. Keywords: Elliptic equation, heat equation, singular potential, existence, symmetry, self-similarity, asymptotic behavior / Doutorado / Matematica / Doutora em Matemática

In the beginning we outline some important properties of blood and de- scribe it from the biological point of view. In the next section we show how we derived our model based on the mixture theory. For the final model we suggest a mathematical method based on the finite element method and subject it to tests for flow in a simple domain. In the middle part we prove the existence of solution for a model with simplified constitutive relation for the stress tensor, which still includes an anisotropic model for the platelet diffusion. In the last section we show numerical results. We start with sim- ple testing computations in simple domains, followed by computations in a two-dimensional simulation of an aneurysm, and narrowed blood vessel re- spectively. In the end we also show some illustrative computations in three dimensions. 1

Dans cette thèse, nous nous intéressons à résoudre certains problèmes inverses pour des équations différentielles aux dérivées fractionnaires. Un problème inverse est généralement mal posé. Un problème mal posé est un problème qui ne répond pas à l’un des trois critères de Hadamard pour être bien posé, c’est-à-dire, soit l’existence, l’unicité ou une dépendance continue aux données n'est plus vraie, à savoir, des petits changements dans les données de mesure entraînent des changements indéfiniment importants dans la solution. La plupart des difficultés à résoudre des problèmes mal posés sont causées par l’instabilité de la solution. D’autre part, les équations différentielles fractionnaires deviennent un outil important dans la modélisation de nombreux problèmes de la vie réelle et il y a eu donc un intérêt croissant pour l’étude des problèmes inverses avec des équations différentielles fractionnaires. Le calcul fractionnaire est une branche des mathématiques qui fait référence à l’extension du concept de dérivation classique à la dérivation d’ordre non entier. Calculer une dérivée fractionnaire à un certain moment exige tous les processus précédents avec des propriétés de mémoire. C’est l’avantage principal du calcul fractionnaire d’expliquer les processus associés aux systèmes physiques complexes qui ont une mémoire à long terme et / ou des interactions spatiales à longue distance. De plus, les équations différentielles fractionnaires peuvent nous aider à réduire les erreurs découlant de paramètres négligés dans la modélisation des phénomènes physiques. / In this thesis, we are interested in solving some inverse problems for fractional differential equations. An inverse problem is usually ill-posed. The concept of an ill-posed problem is not new. While there is no universal formal definition for inverse problems, Hadamard [1923] defined a problem as being ill-posed if it violates the criteria of a well-posed problem, that is, either existence, uniqueness or continuous dependence on data is no longer true, i.e., arbitrarily small changes in the measurement data lead to indefinitely large changes in the solution. Most difficulties in solving ill-posed problems are caused by solution instability. Inverse problems come into various types, for example, inverse initial problems where initial data are unknown and inverse source problems where the source term is unknown. These unknown terms are to be determined using extra boundary data. Fractional differential equations, on the other hand, become an important tool in modeling many real-life problems and hence there has been growing interest in studying inverse problems of time fractional differential equations. The Non-Integer Order Calculus, traditionally known as Fractional Calculus is the branch of mathematics that tries to interpolate the classical derivatives and integrals and generalizes them for any orders, not necessarily integer order. The advantages of fractional derivatives are that they have a greater degree of flexibility in the model and provide an excellent instrument for the description of the reality. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time, i.e., calculating timefractional derivative at some time requires all the previous processes with memory and hereditary properties.

Orientador: José Luiz Boldrini / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-21T14:27:32Z (GMT). No. of bitstreams: 1 Entringer_ArianePiovezan_D.pdf: 26392944 bytes, checksum: d4993ec89fdc9c9a41cd6fd1e6b28dd1 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho analisaremos dois sistemas de equações diferenciais parciais não lineares de evolução associados a modelos de interação fluido-estrutura; esses sistemas foram obtidos utilizando as equações alfa-Navier-Stokes e a metodologia do campo de fases. O primeiro de tais sistemas modela um processo de mudanças de fases envolvendo solidificação e fusão de certos materiais e leva em conta tanto os fenômenos de condução do calor quanto o da convecção da fase não sólida. Esse sistema é formado pelo acoplamento das equações alfa-Navier-Stokes para fluidos viscosos incompressíveis com uma equação para a variável campo de fases, cujos valores determinam a fase do material (sólida, líquida ou mushy), e também com uma equação de balanço de energia interna, a qual determina a evolução da temperatura. O segundo sistema a ser estudado modela a dinâmica de vesículas em um fluido viscoso e incompressível. Tal sistema consiste do acoplamento das equações alfa-Navier-Stokes com uma equação para uma variável campo de fases, a qual neste caso determina a posição da membrana da vesícula que é deformada pela ação do fluido, bem como seu interior e exterior; esta última equação tem um termo descrevendo a interação do fluido com a membrana da vesícula. Para ambos os sistemas, provaremos a existência e a unicidade das soluções em espaços funcionais adequados / Abstract: In this work we analyze two systems of nonlinear evolution partial differential equations associated to models of fluid-structure interaction; such systems were obtained by using the alfa-Navier-Stokes equations and the phase field methodology. The first of such systems models a process of phase change involving solidification and fusion of certain materials and take in consideration both the phenomena of heat conduction and convection of the non-solid phase. Such a system is formed by coupling the alfa-Navier- Stokes equations for incompressible viscous fluids to an equation for the phase field variable whose values determine the phase of the material (solid, liquid or mushy), and also to an equation for the balance of internal energy, which determines the evolution of the temperature. The second system to be studied models the dynamics of vesicles in an incompressible viscous fluid. This system consists of the coupling of alfa-Navier- Stokes equation with an equation for the phase field variable, which in this case determines the position of vesicle membrane that is deformed by the action of the fluid, as well as it's interior and exterior; this last equation has a term describing the interaction of the fluid with the vesicle membrane. For both systems, we will prove the existence and uniqueness of solutions in suitable functional spaces. / Doutorado / Matematica / Doutora em Matemática

Durant les trois années de la thèse, j’ai eu le plaisir de travailler en collaboration avec à la fois des volcanologues, des physiciens de laboratoire et des mathématiciens. Ce mémoire est l’occasion de présenter la démarche et les résultats de mes recherches dans le domaine de la modélisation d’écoulements granulaires denses fluidisés. Ces derniers consistent à développer un nouveau modèle mathématique et son étude théorique et numérique. Sur la base d’observations faites lors d’expériences de laboratoire, nous proposons une façon de modéliser le changement comportemental d’un écoulement granulaire initialement fluidisé au travers de la définition de sa rhéologie viscoplastique à seuil variable. Plus précisément, le seuil de plasticité est défini par la différence entre la pression lithostatique et la pression du fluide interstitiel. La nouveauté apportée par ce modèle ouvre de nouvelles perspectives à la fois pour le champ de recherche en mathématiques et pour la compréhension des lits granulaires fluidisés et leur application à la volcanologie. Du point de vue mathématique, une étude théorique du modèle a été menée. En proposant une preuve de l’existence de solutions faibles à un problème lié à la version homogène du modèle, nous apportons une extension au champ de connaissances autour des écoulements des fluides non-newtoniens. D’autre part, dans le but de reproduire numériquement des expériences de laboratoire de chute de colonne granulaire fluidisée, nous avons développé un code de simulation numérique incluant une nouvelle méthode de résolution des équations d’écoulement de fluides à seuil. Dans ce manuscrit, je décris et justifie les différents choix stratégiques pour le développement de ce code. Par ailleurs, je présente quelques tests académiques permettant de valider le code. Enfin, je donne les résultats de simulation de chute de colonne granulaire, qu’elle soit fluidisée ou non. Une comparaison avec les données de laboratoire est effectuée afin d’évaluer les points forts et les défauts du modèle par rapport à la réalité des expériences. En conclusion, dans la continuité du travail mené dans ce projet, des perspectives d’amélioration sont proposées. / During these three years, I enjoyed to work with collaborators from volcanology, laboratory physics and mathematics. This document presents the steps and results of my research in the field of modelling of fluidised granular flows. The last consists in the development of a new mathematical model and its theoretical and numerical study. Based on observations made on experimental studies, the model focuses on the change in the behaviour of an initially fluidised granular flow through the definition of its viscoplastic rheology with variable threshold. More precisely, the threshold (aslo called yield stress) is defined via the difference between the lithostatic pressure and the pressure of the interstitial fluid. The innovation of this model opens perspectives for the mathematical research as well as for the study of fluidised granular flows and their application to volcanology. From a mathematical point of view, a theoretical study has been conducted. Proving the existence of weak solution for the homogeneous version of the model, we offer an extension in the field of knowledges of non-newtonian fluid flows. Also, we have developped a numerical code to simulate dambreak experiments with fluidised granular media. This one includes a new method to solve the flow equations of viscoplastic fluids. In this thesis, I describe and justify the numerical strategy chosen. Moreover, I present some academic tests to validate the code. At the end, I give the numerical results in the case of the dambreak simulation for dry and fluidised fluids. By comparing with experimental data, we evaluate the validity of the model and its resolution, and highlight the advantages and inconvenients. To conclude the project, I propose some perspectives of improvement for later work.