The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\`ere type arises here. / Mathematics

Cette thèse est consacrée à l'étude de la dynamique des gaz à faible nombre de Mach. Le modèle étudié provient des équations de Navier-Stokes complètes lorsque le nombre de Mach tend vers zéro. On cherche à montrer que le problème de Cauchy correspondant est bien posé. Les cas visqueux et non visqueux sont tous deux considérés. Les coefficients physiques peuvent dépendre de la densité (ou de la température) inconnue. En particulier, nous prenons en compte les effets de conductivité thermique et on autorise de grandes variations d'entropie. Rappelons qu'en absence de diffusion thermique, la limite à faible nombre de Mach implique la condition d'incompressibilité. Dans le cadre étudié ici, en introduisant un nouveau champ de vitesses à divergence nulle, le système devient un couplage non linéaire entre une équation quasi-parabolique pour la densité et un système de type Navier-Stokes (ou Euler) pour la vitesse et la pression. Pour le cas avec viscosité, on établit le résultat classique, à savoir qu'il existe une solution forte existant localement (resp. globalement) en temps pour des données initiales grandes (resp. petites). On considère ici le problème de Cauchy avec données initiales dans des espaces de Besov critiques. Lorsque les coefficients physiques du système vérifient une relation spéciale, le système se simplifie considérablement, et on peut alors établir qu'il existe des solutions faibles globales en temps à énergie finie. Par un argument d'unicité fort-faible, on en déduit que les solutions fortes à énergie finie existent pour tous les temps positifs en dimension deux. Pour le cas sans viscosité, on montre d'abord le caractère bien posé dans des espaces de Besov limites, qui s'injectent dans l'espace des fonctions lipschitziennes. Des critères de prolongement et des estimations du temps de vie sont établis. Si l'on suppose la donnée initiale à énergie finie dans l'espace de Besov limite à exposant de Lebesgue infini, on a également un résultat d'existence locale. En dimension deux, le temps de vie tend vers l'infini quand la densité tend vers une constante positive. Des estimations de produits et de commutateurs, ainsi que des estimations a priori pour les équations paraboliques et pour le système de Stokes (ou d'Euler) à coefficients variables, se trouvent dans l'annexe. Ces estimations reposent sur la théorie de Littlewood-Paley et le calcul paradifférentiel / This thesis is devoted to the study of the dynamics of the gases with small Mach number. The model comes from the complete Navier-Stokes equations when the Mach number goes to zero, and we aim at showing that it is well-posed. The viscous and inviscid cases are both considered. The physical coefficients may depend on the unknown density (or on the unknown temperature).In particular, we consider the effects of the thermal conductivity and hence large variations of entropy are allowed. Recall that if there is no thermal diffusion, then the low Mach number limit just implies the incompressibility condition. In the framework considered here, by introducing a new solenoidal velocity field, the system becomes a nonlinear coupling between a quasi-parabolic equation for the density and an evolutionary Stokes (or Euler) system for the velocity and the pressure. For the case with viscosity, we establish classical results, namely the strong solutions exist locally (resp. globally) in time for big (resp. small) initial data. We consider the Cauchy problem in the critical Besov spaces with the lowest regularity. Under a special relationship between the two physical coefficients, the system recasts in a simpler form and one may prove that there exist weak solutions with finite energy. In dimension two, this implies that strong solutions with finite energy exist for all positive times. In the inviscid case, we first prove the well-posedness result in endpoint Besov spaces, which can be embedded into the set of Lipschitzian functions. Continuation criterions and estimates for the lifespan are both established.If we suppose the initial data to be in the borderline Besov spaces with infinite Lebesgue exponent and to be of finite energy, we also have a local existence result. In dimension two, the lifespan goes to infinity when the density tends to a positive constant. Estimates for products and commutators, together with a priori estimates for the parabolic equations and the Stokes (or Euler) system with variable coefficients, are postponed in the appendix. These estimates are based on the Littlewood-Paley theory and the paradifferential calculus

Made available in DSpace on 2015-05-15T11:46:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 810707 bytes, checksum: 33aad19ba73032d41420617a7e402095 (MD5) Previous issue date: 2010-08-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we'll prove existence of weak solutions to a coupled mixed problem of nonlinear partial diferential equation in the class of systems of nonlinear Klein- Gordon equations involving pseudo-Laplacian operator. For proving existence of weak solutions we use Faedo-Galerkin's method with compacity and monotonicity properties. / Neste trabalho provaremos a existência de soluções fracas para um problema misto de equações diferenciais parciais n~ao-lineares do tipo Klein-Gordon envolvendo o operador pseudo-Laplaciano. Com esse fim, usaremos o método de Faedo-Galerkin juntamente com argumentos de compacidade e monotonicidade.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we are interested in establishing some variational methods, together with applications, that determine the existence and non uniqueness of weak solutions for the nonlinear elliptic partial differential equation −div (K(x)-u) = K(x)f(u) + h, x E R2, where K is an exponential weight, h is a linear functional and f is the nonlinearity that presents critical exponential growth. First of all, for the sake of convenience of the reader, this study shows detailed proofs of some classic results of the theory that involves these methods as, for example, the deformation and mountain pass theorems; and Ekeland’s variational principle. Second of all, we work with a Trudinger-Moser inequality that is related to a Sobolev space with weight K in order to achieve our aim. / Neste trabalho, estamos interessados em apresentar alguns Métodos Variacionais, juntamente com aplicações, que determinam existência e a não unicidade de soluções fracas para uma específica Equação Diferencial Parcial Elíptica não linear −div (K(x)-u) = K(x)f(u) + h, x E R2, onde K é um peso exponencial, h é um funcional linear e f é a não linearidade que apresenta crescimento exponencial crítico. Em um primeiro momento, para uma maior comodidade do leitor, estabelecemos provas detalhadas de alguns resultados clássicos da teoria que contém esses métodos como, por exemplo, os Teoremas da Deformação e do Passo da Montanha; e o Princípio Variacional de Ekeland. Em seguida, trabalhamos com uma Desigualdade do tipo Trudinger-Moser em um Espaço de Sobolev com peso K com o objetivo de alcançarmos nossa meta.

The aim of this work is to provide new results of global existence for dif- ferent evolution equations of fluid mechanics. We are in general interested in finding weak solutions without restrictions on the size of initial data. The proofs of existence are based on several different approaches including en- ergy methods, convergence analysis of finite numerical methods and convex integration. All these techniques significantly exploit results of mathematical analysis and other branches of mathematics. 1

Stochastic partial differential equations have proven useful in many applied areas of mathematics, such as physics or mathematical finance. A major part of such equations consists of linear equations with additive noise. In certain cases, however, the drift part of the differential equation additionally contains a possibly problematic non-linear term, which makes it unsolvable by the standard methods and even a solution in the mild sense may be out of reach. In such situations, we may still find a solution in the weak sense by employing a suitable transformation of the probability space. This thesis deals with semilinear stochastic evolution equations in a separable Hilbert space, where the driving process is an element of a large class of processes - so called Volterra processes, which can be understood as a generalisation of the Wiener process and may be of use to model a wide range of phenomena. The weak solutions, however, have been studied so far only for equations with the cylindrical fractional Brownian motion as the driving process. In this thesis, we introduce a generalisation of the Girsanov theorem for cylindrical Gaussian Volterra processes and give, in full generality, sufficient conditions for the existence of a weak solution and the uniqueness of the equation in law. Further, we introduce...

Dans cette thèse, nous nous intéressons à l'étude de trois équations aux dérivées partielles et d'évolution non-locales en espace et en temps. Les solutions de ces trois solutions peuvent exploser en temps fini. Dans une première partie de cette thèse, nous considérons l'équation de la chaleur nonlinéaire avec une puissance fractionnaire du laplacien, et obtenons notamment que, dans le cas d'exposant sur-critique, le comportement asymptotique de la solution lorsque $t\rightarrow+\infty$ est déterminé par le terme de diffusion anormale. D'autre part, dans le cas d'exposant sous-critique, l'effet du terme non-linéaire domine. Dans une deuxième partie, nous étudions une équation parabolique avec le laplacien fractionnaire et un terme non-linéaire et non-local en temps. On montre que la solution est globale dans le cas sur-critique pour toute donnée initiale ayant une mesure assez petite, tandis que dans le cas sous-critique, on montre que la solution explose en temps fini $T_{\max}>0$ pour toute condition initiale positive et non-triviale. Dans ce dernier cas, on cherche le comportement de la norme $L^1$ de la solution en précisant le taux d'explosion lorsque $t$ s'approche du temps d'explosion $T_{\max}.$ Nous cherchons encore les conditions nécessaires à l'existence locale et globale de la solution. Une toisième partie est consacré à une généralisation de la deuxième partie au cas de systèmes $2\times 2$ avec le laplacien ordinaire. On étudie l'existence locale de la solution ainsi qu'un résultat sur l'explosion de la solution avec les mêmes propriétés étudiées dans le troisième chapitre. Dans la dernière partie, nous étudions une équation hyperbolique dans $\mathbb{R}^N,$ pour tout $N\geq2,$ avec un terme non-linéaire non-local en temps. Nous obtenons un résultat d'existence locale de la solution sous des conditions restrictives sur les données initiales, la dimension de l'espace et les exposants du terme non-linéaire. De plus on obtient, sous certaines conditions sur les exposants, que la solution explose en temps fini, pour toute condition initiale ayant de moyenne strictement positive.

Title: Flows of incompressible fluids with pressure-dependent viscosity (and their application to modelling the flow in journal bearing) Author: Martin Lanzendörfer Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Josef Málek, DSc. Abstract: The viscosity of the fluids involved in hydrodynamic lubrication typically depends on pressure and shear rate. The thesis is concerned with steady isothermal flows of such fluds. Generalizing the recent results achieved in the case of homogeneous Dirichlet boundary conditions, the existence and uniqueness of weak solutions subject to the boundary conditions employed in practical applications will be established. The second part is concerned with numerical simulations of the lubrication flow. The experiments indicate that the presented finite element method is successful as long as certain restrictions on the constitutive model are met. Both the restrictions involved in the theoretical results and those indicated by the numerical experiments allow to accurately model real-world lubricants in certain ranges of pressures and shear rates. The last part quantifies those ranges for three representative lubricants. Keywords: existence and uniqueness of weak solutions, finite element method, pressure- thickening, shear-thinning, incompressible fluids,...

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T17:30:30Z No. of bitstreams: 1 ceilimarcolinomoreira.pdf: 628590 bytes, checksum: 89404f2fdb6f6a266713327a91a21c05 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T19:02:09Z (GMT) No. of bitstreams: 1 ceilimarcolinomoreira.pdf: 628590 bytes, checksum: 89404f2fdb6f6a266713327a91a21c05 (MD5) / Made available in DSpace on 2017-05-29T19:02:09Z (GMT). No. of bitstreams: 1 ceilimarcolinomoreira.pdf: 628590 bytes, checksum: 89404f2fdb6f6a266713327a91a21c05 (MD5) Previous issue date: 2014-03-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, apresentamos métodos envolvendo sub e supersolução para estudar a existência de solução, no sentido fraco, para três classes de problemas elípticos de segunda ordem com condição de fronteira de Dirichlet homogênea. Nos dois primeiros casos encontramos solução em W1,2 0 (Ω) e no terceiro caso encontramos solução em L1(Ω) com algumas restrições. / This paper presents methods involving sub and supersolution in order to learn the existence of weak solutions of three classes of second order elliptic problems with homogeneous Dirichlet boundary conditions. In the first two cases we find solution in W1,2 0 (Ω) and in the third case we find solution in L1(Ω) with some restrictions.

Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales. Les solutions de ces quatre équations peuvent exploser en temps fini. Dans la théorie des équations d'évolution non-linéaires, une solution est qualifiée de globale si elle est définie pour tout temps positif. Au contraire, si une solution existe seulement sur un intervalle de temps [0; T) borné, elle est dite locale. Dans ce dernier cas et quand le temps maximal d'existence est relié à une alternative d'explosion, on dit aussi que la solution explose en temps fini. Dans un premier travail, nous considérons l'équation de Schrödinger non-linéaire avec une puissance fractionnaire du laplacien, et nous obtenons l'explosion de la solution en temps fini Tmax > 0 pour toute condition initiale positive et non-triviale dans le cas d'exposant sous-critique. Ensuite, nous étudions une équation des ondes amorties avec un potentiel d'espace-temps et un terme non-linéaire et non-local en temps. Nous obtenons un résultat d'existence locale d'une solution dans l'espace d'énergie sous des conditions restrictives sur les données initiales, la dimension de l'espace et la croissance du terme non-linéaire. De plus, nous obtenons l'explosion de la solution en temps fini pour toute condition initiale de moyenne strictement positive. De plus, nous étudions un problème de Cauchy pour l'équation d'évolution avec un p- Laplacien avec une non linéarité non-locale en temps. Dans ce cadre, nous nous intéressons à l'étude de l'existence locale d'une solution de cette équation ainsi qu'un résultat de non-existence de solution globale. Finalement, nous étudions l'intervalle maximal d'existence des solutions de l'équation des milieux poreux avec un terme non-linéaire non-local en temps. / In this thesis, we study four non-local evolution equations. The solutions of these four equations can blow up in finite time. In the theory of nonlinear evolution equations, a solution is qualified as global if it isdefined for any time. Otherwise, if a solution exists only on a bounded interval [0; T), it is called local solution. In this case and when the maximum time of existence is related to a blow up alternative, we say that the solution blows up in finite time. First, we consider the nonlinear Schröodinger equation with a fractional power of the Laplacien operator, and we get a blow up result in finite time Tmax > 0 for any non-trivial non-negative initial condition in the case of sub-critical exponent. Next, we study a damped wave equation with a space-time potential and a non-local in time non-linear term. We obtain a result of local existence of a solution in the energy space under some restrictions on the initial data, the dimension of the space and the growth of nonlinear term. Additionally, we get a blow up result of the solution in finite time for any initial condition positive on average. In addition, we study a Cauchy problem for the evolution p-Laplacien equation with nonlinear memory. We study the local existence of a solution of this equation as well as a result of non-existence of global solution. Finally, we study the maximum interval of existence of solutions of the porous medium equation with a nonlinear non-local in time term.