Neste trabalho consideramos o problema de Navier-Stokes em RN <div style=\"width: 50%; margin: auto;\">ut = &Delta;u &mdash; &nabla;&pi; + f (t) &mdash; (u .&nabla;)u,&nbsp; &nbsp;x&isin; &Omega; <br />div(u) = 0,&nbsp; &nbsp; x &isin; &Omega; <br />u = 0,&nbsp; &nbsp;&nbsp;x &isin; &part; &Omega; <br />u(0, x) = u0 (x), onde u0 &isin; LN (&Omega;)N e &Omega; &eacute; um subconjunto aberto, limitado e suave de RN. Provamos que o problema acima &eacute; localmente bem colocado e fornecemos condi&ccedil;&otilde;es para obter que estas solu&ccedil;&otilde;es existem para todo t &ge; 0. Utilizamos t&eacute;cnicas de equa&ccedil;&otilde;es parab&oacute;licas semilineares considerando n&atilde;o linearidades com crescimento cr&iacute;tico desenvolvidas em (ARRIETA; CARVALHO, 1999). / In this work we we consider the Navier-Stokes problem on RN <div style=\"width: 50%; margin: auto;\">ut = &Delta;u &mdash; &nabla;&pi; + f (t) &mdash; (u .&nabla;)u,&nbsp; &nbsp;x&isin; &Omega; <br />div(u) = 0,&nbsp; &nbsp; x &isin; &Omega; <br />u = 0,&nbsp; &nbsp;&nbsp;x &isin; &part; &Omega; <br />u(0, x) = u0 (x), where u0 &isin; LN (&Omega;)N and &Omega; is an open, bounded and smooth subset of RN. We prove that the above problem is locally well posed and give conditions to obtain that these solutions exist for all t &ge; 0. We used techniques of semilinear parabolic equations considering nonlinearities with critical grouth developed in (ARRIETA; CARVALHO, 1999).

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In this paper we discuss the stable solvability of the inverse problem of parameter identification in a jump diffusion model. Therefore we introduce the forward operator of this inverse problem and analyze its properties. We show continuity of the forward operator and stability of the inverse problem provided that the domain is restricted in a specific manner such that techniques of compact sets can be exploited. Furthermore, we show that there is an asymptotical non-injectivity which causes instability problems whenever the jump intensity increases and the jump heights decay simultaneously.

We deal with locally ill-posed nonlinear operator equations F(x) = y in L^2(0,1), where the Fréchet derivatives A = F'(x_0) of the nonlinear forward operator F are compact linear integral operators A = M ◦ J with a multiplication operator M with integrable multiplier function m and with the simple integration operator J. In particular, we give examples of nonlinear inverse problems in natural sciences and stochastic finance that can be written in such a form with linearizations that contain multiplication operators. Moreover, we consider the corresponding ill-posed linear operator equations Ax = y and their degree of ill-posedness. In particular, we discuss the fact that the noncompact multiplication operator M has only a restricted influence on this degree of ill-posedness even if m has essential zeros of various order.

We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.

In this work we will demonstrate that the Cauchy problem associated with the Korteweg-de Vries equation, denoted by KdV, and Korteweg-de Vries modified equation, denoted by mKdV, with initial data in the space of Sobolev Hs(|R), is locally well-posed on Hs(|R), with s>3/4 for KdV and s&#8805;1/4 for mKdV, where the notion of well-posedness includes existence, uniqueness, persistence property of solution and continuous dependence of solution with respect to the initial data. This result is based on the works of Kenig, Ponce and Vega. The technique used to obtain these results is based on fixed point Banach theorem combined with the regularizantes effects of the group associated with the linear part. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Neste trabalho demonstraremos que o problema de Cauchy associado as equações de Korteweg-de Vries, denotada por KdV, e de Korteweg-de Vries modificada, denotada por mKdV, com dado inicial no espaço de Sobolev Hs(|R), é bem posto localmente em Hs(|R), com s>3/4 para a KdV e s&#8805;1/4 para a mKdV, onde a noção de boa postura inclui a existência, unicidade, a propriedade de persistência da solução e dependência contínua da solução com relação ao dado inicial. Este resultado é baseado nos trabalhos de Kenig, Ponce e Vega. A técnica utilizada para obter tais resultados se baseia no Teorema do Ponto Fixo de Banach combinada com os efeitos regularizantes do grupo associado com a parte linear.

Dans cette thèse, on étudie l'existence, l'unicité et la régularité des solutionsd'équation de type Cahn-Hilliard ainsi que son comportement asymptotiqueen termes d'existence de l'attracteur global et d'un attracteur exponentiel. Cetteéquation est considérée dans un domaine borné et régulier pour différents types denonlinéarités et de conditions au bord.D'abord, on étudie l'équation avec des conditions de type Dirichlet sur le bord etune nonlinéarité régulière. Après, on considère une perturbation du problème et ondémontre l'existence d'une famille robuste d'attracteurs exponentiels lorsque ε tendvers 0.Ensuite, on étudie l'équation avec des conditions dynamiques sur le bord. On considèretout d'abord une nonlinéarité régulière et on donne une étude théorique etnumérique. Après, on illustre ces résultats par des simulations numériques en dimensiondeux d'espace qui permettent d'étudier l'influence des différents paramètres.On termine par une étude du modèle considéré avec une nonlinéarité singulière quel'on approche par des fonctions régulières et on introduit une notion de solutionappropriée. / This thesis is devoted to the study of the existence, uniqueness andregularity of solutions for a Cahn-Hilliard type equation, as well as the asymptoticbehavior in terms of existence of the global attractor and of an exponential attractor.This equation is considered in a bounded and smooth domain under variousassumptions on the nonlinear terms and with different boundary conditions.We start by studying the equation with Dirichlet boundary conditions and a regularnonlinearity. Then, we consider a perturbation of the problem and we prove theexistence of a robust family of exponential attractors as ε tends to 0.For the equation endowed with dynamic boundary conditions, we first consider aregular nonlinearity and we treat the theoretical and numerical analysis. Then, weillustrate the results by numerical simulations in two space dimension which allow usto study the influence of different parameters. Finally, we treat the problem consideredwith a singular nonlinearity which is approximated by regular functions andwe give a suitable notion of solutions.

Orientador: Lucas Catão de Freitas Ferreira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T12:47:02Z (GMT). No. of bitstreams: 1 Lima_LidianedosSantosMonteiro_D.pdf: 2071234 bytes, checksum: 6370d2ea978f96792562cdc2e2406365 (MD5) Previous issue date: 2014 / Resumo: Nesta tese consideramos uma família de EDPs dissipativas do tipo escalar ativo cujos campos velocidades são acoplados aos escalares através de operadores multiplicadores de Fourier que podem ser de alta ordem. Na primeira parte, provamos boa-colocação global, decaimento de normas Lp, e algumas propriedades de simetria, para dados iniciais no espaço de Lebesgue crítico e sem assumir condição de pequenez. Na segunda parte, introduzimos os espaços de Fourier-Besov-Morrey, o qual parece ser novo na analise de EDPs, com o objetivo de encontrar soluções auto-similares e considerar uma classe maior de acoplamentos e dados iniciais. Condições de pequenez na norma do espaço são assumidas para estes resultados. Além disso, mostramos um resultado de estabilidade assintótica e obtemos uma classe de soluções assintoticamente auto-similares / Abstract: In this thesis we consider a family of dissipative active scalar equations whose velocity fields are coupled by means of multiplier operators that can be of high-order. In the first part, we prove global well-posedness, decay of Lp's norms and some symmetry properties of solutions for initial data in the critical Lebesgue space and without smallness condition. In the second part, we introduce the Fourie-Besov-Morrey spaces, which seems to be new in the analysis of PDEs in order to find self-similar solutions and to consider a larger class of couplings and initial data. Smallness conditions on the norm of the space are assumed for these results. Furthermore, we show an asymptotic stability result and obtain a class of asymptotically self-similar solutions / Doutorado / Matematica / Doutora em Matemática

Orientador: Andréa Cristina Prokopczyk Arita / Abstract: This work presents some results of the theory of the (a,k)-regularized resolvent families, that are the main tool used in this thesis. Related with this families, one result proved in this work is the zero-one law, providing new insights on the structural properties of the theory of (a,k)-regularized resolvent families including strongly continuous semigroups, strongly continuous cosine families, integrated semigroups, among others. Moreover, an abstract nonlinear degenerate hyperbolic equation is considered, that includes the semilinear Blackstock-Crighton-Westervelt equation. By proposing a new approach based on strongly continuous semigroups and resolvent families of operators, it is proved an explicit representation of the strong and mild solutions for the linearized model by means of a kind of variation of parameters formula. In addition, under nonlocal initial conditions, a mild solution of the nonlinear equation is established. / Resumo: Este trabalho apresenta alguns resultados da teoria de famílias resolventes (a,k)- regularizadas, que é a principal ferramenta utilizada nesta tese. Relacionado com estas famílias, um resultado provado neste trabalho é a lei zero-um, que fornece novas percepções de propriedades estruturais da teoria de famílias resolventes (a,k)- regularizadas, incluindo os semigrupos fortemente contínuos, as famílias cosseno fortemente contínuas, os semigrupos integrados, entre outras. Além disso, uma equação hiperbólica degenerada não-linear abstrata é considerada, a qual inclui a equação semilinear de Blackstock-Crighton-Westervelt. Propondo uma nova abordagem baseada em semigrupos fortemente contínuos e famílias resolvente, é demonstrada uma representação explícita das soluções forte e branda para a linearização do modelo por uma espécie de método de variação dos parâmetros. Por fim, sob condições iniciais não-locais, uma solução branda da equação não-linear é estabelecida. / Doutor

In this paper we deal with aspects of characterizing the ill-posedn ess of nonlinear inverse problems based on the discussion of specific examples. In particular, a parameter identification problem to a second order differential equation and its ill-posed linear components are under consideration. A new approach to the classification ofill-posedness degrees for multiplication operators completes the paper.