Tesis (Análisis Matemático)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10812

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Operators on Banach spaces of Dirichlet series and semigroups of analytic functions
(2024-07-24) Gómez Cabello, Carlos; Contreras Márquez, Manuel Domingo; Rodríguez Piazza, Luis; Universidad de Sevilla. Departamento de Análisis Matemático; Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI)
The two main objects studied in this thesis are some operators acting on Banach spaces of Dirichlet series and continuous semigroups of analytic functions. The main part of thesis is mostly oriented towards the study of the first ones, although the first chapter is entirely devoted to continuous semigroups. Nevertheless, at several points along the thesis, both notions are often mixed together. Regarding the continuous semigroups {Φt}t≥0 of analytic functions, these objects are considered in the classical setting of the unit disc D and in the right half-plane C+. We characterise the continuous semigroups in C+ with Denjoy-Wolff point ∞ converging uniformly to the identity in the whole right half-plane as t → 0+. Concerning the continuous semigroups in D, we provide a quantitative version of the well-known fact that these semigroups converge to the identity uniformly in the whole unit disc. We prove that the rate of convergence is always O(√ t), as t → 0+, and this order of convergence is sharp. In the part of the thesis devoted to Banach spaces of Dirichlet series several problems are considered. We begin by characterising the strongly continuous semigroups of composition operators in the Hardy spaces of Dirichlet series Hp. This is done in terms of the continuous semigroups in the so-called Gordon- Hedenmalm class G. This class consists on the analytic functions Φ : C+ → C+ giving rise to bounded composition operators on H2. The existence of a rich variety of such semigroups is ensured thanks to the description of the infinitesimal generators of such continuous semigroups. Namely, these infinitesimal generators are those Dirichlet series sending the right half-plane into its closure. Then, we move on to the algebra of Dirichlet series, this is, the bounded Dirichlet series in C+ which are uniformly continuous there. We characterise the bounded composition operators CΦ acting on this algebra. We also show, for CΦ, the equivalence between compactness and weak compactness and provide several characterisations of this property. A description of the strongly continuous semigroups of composition operators in the algebra is also given. We conclude with the consideration of a third class of Banach spaces of Dirichlet series: a family of Bergman type spaces. Two main problems are considered in this context. First, the estimate of the norm of the evaluation functionals for a certain collection of these Bergman spaces. Second, we carry out a detailed study of the Volterra operator Tg acting on these Bergman type spaces of Dirichlet series.
Evolution of Navier-Stokes and Euler vortex filaments
(2024-07-02) Hidalgo Torné, Antonio; Gancedo García, Francisco; Universidad de Sevilla. Departamento de Análisis Matemático
Understanding the Euler and Navier-Stokes equations is essential for the mathematical development of fluid mechanics, as they serve as pillars for various models within this field. On the other hand, the study of vortex filaments is motivated by their resemblance to some phenomena in nature, but our current knowledge about them is limited. Part of the difficulty in the mathematical study of these structures lies in their singularity. In particular, the energy around the filament and the effective velocity on it are infinite, making the formulation of their evolution a non-trivial problem. The results found in the literature regarding this matter require smallness in the data or time of existence, some form of symmetry, modifications to the equations, or regularization of the initial data. As a result, the study of vortex filaments in the Euler and Navier-Stokes equations is a hot topic in the literature. This thesis is devoted to the development of two new results on vortex filament dynamics. In chapter 1, the Euler and Navier-Stokes equations are introduced. This is followed by considerations on vortex filaments, while reviewing the existing literature on the subject. The chapter 2 is based on the submitted paper [52]. In it, the Cauchy problem of a helical vortex filament in the Navier-Stokes equations is studied. Specifically, we prove the global-in-time existence, uniqueness and regularization of solutions for this problem. Since the Navier-Stokes equations do not preserve the null helical swirl, this is the first global-in-time existence result for vortex filaments without size restriction in the presence of vortex stretching. To obtain such a result, it is first shown that the evolution of periodic vortex filaments in one direction is well-posed for short times. Next, it is proved that if the vortex filament is initially helical, then the solution obtained is also helical. In general, as occurs in two dimensions, a vorticity with helical symmetry generates a slowly decaying velocity with infinite energy. By adapting results from local energy weak solutions, together with a new estimate using helical symmetry in non-helical domains, we show that the solution obtained for short times can be uniquely extended globally in time while keeping the symmetry. The chapter 3 is based on the submitted paper [53]. In it, the Cauchy problem of a circular vortex filament in the Euler equations is studied. Specifically, the first existence result of weak solutions to the Euler equations with velocity in C([0, T], L2−) is obtained. With our approach, there is no need to regularize the initial data or rescale the time variable, as it is usual in the literature. By applying convex integration, a solution with finite and decreasing energy is obtained for positive times, with vorticity supported inside a ring that moves and thickens with time. After simplifying the system using axisymmetric coordinates without swirl, a subsolution with the desired properties is constructed. The proof concludes with an adaptation of the h-principle to a function space that includes the vortex filament as initial data.
Diferenciación en espacios vectoriales topológicos
(1973) Arias de Reyna Martínez, Juan; Castro Brzezicki, Antonio de; Universidad de Sevilla. Departamento de Análisis Matemático
Free boundary and turbulence for incompressible viscous fluids
(2023-06-29) Salguero Quirós, Elena; Gancedo García, Francisco; Universidad de Sevilla. Departamento de Análisis Matemático
The mathematical bases of the dynamics of viscous fluids are given by the classical Navier- Stokes equations, which model the motion of a viscous incompressible fluid. We can consider a wide variety of scenarios involving these type of fluids. In particular, one can classify the motion of fluids in two general regimes: laminar and turbulent. The Reynolds number is a constant intimately related to this behavior, which associates the viscosity forces acting on the fluid (causing friction between particles), with the inertial forces (causing acceleration of the fluid). In this thesis, we study the dynamics of viscous fluids from two very different perspectives. On the one hand, we study the scenario where the Reynolds number is vanishingly small, giving rise to the Stokes system. We describe the behavior of two different two-dimensional fluids which evolve in time, and we analyze the properties of the interface between them. This problem lies in the class of free boundary problems. On the other hand, we consider a drastically different scenario, where the Reynolds number is large and turbulence is developed. We study the motion of a two or three-dimensional fully developed homogeneous isotropic turbulent fluid, through the Kolmogorov two-equation model of turbulence. This thesis is divided into two parts, each of them devoted to one of the problems. The first part of the thesis contains an introduction and two chapters. In the first chapter, we present the model which describes the dynamics of two incompressible immiscible viscous fluids in the Stokes regime, filling a 2D horizontally periodic strip. We assume that the fluids are subject to the gravity force and they have different densities. This framework is chosen motivated by the lack of results in the density jump setting with an infinitely deep geometry and a non-integrable density. In this scenario, the density jump induces the dynamics of the free interface arising between the two fluids. One of the classical methods to deal with free boundary problems is to use potential theory to furnish explicit solutions for the system. Using this approach, we derive a contour dynamics formulation for this problem, through a x1-periodic version of the Stokeslet. This technique yields explicit solutions of the system, even for more general forcing terms than the one used in our analysis (the gravity force). Furthermore, this formulation of the velocity consist of a non-local and strongly non-linear equation. As a first approach, we analyze the linear operator inside the explicit solution, which shows what we call a weak damping effect in the stable stratification of the densities, when the lighter fluid lies above the denser one. This type of operator shows a contrast between this and other related free boundary problems, whose linear operators are of parabolic type. In our case, the weak damping effect suggests that the solutions do not gain regularization in time, hence the nature of the problem is hyperbolic. Having this in mind, we study the full non-linear equation and we show local-in-time well-posedness when the initial interface is described by a curve with no self-intersections and C1+γ H¨older regularity, with 0 < γ < 1. According to the expected hyperbolic behavior, the solution does not gain any regularity, it is C1+γ in space. This well-posedness result holds regardless of the Rayleigh-Taylor stability of the physical system, i.e., the system is well-posed even when the denser fluid lies above the lighter one. This behavior is due to the viscosity of the fluids. In the second chapter, we study the long time behavior of solutions when the initial data is small and described by the graph of a function. The techniques used exploit the properties of the linear semi-group and the so-called weak damping effect. With these techniques, we prove the global-in-time existence for the Rayleigh-Taylor stable case of the densities (the lighter fluid lies above the denser fluid). The proof relies on a priori energy estimates on suitable Sobolev spaces and the careful study of the singular kernels appearing. We also prove stability of the flat interface, i.e., the decay of the free interface to the flat steady state. In particular, we prove existence and uniqueness of global interfaces with H3 regularity and polynomial decay of the interface. Moreover, we can extend this global-in-time existence result to analytic solutions in suitable Wiener spaces. We use Fourier techniques of the contour dynamics equation and the properties of the linear semi-group in Wiener algebras to obtain global-in-time existence and exponential decay to the flat interface. Finally, in the Rayleigh-Taylor unstable regime, we construct a wide family of smooth solutions with exponential in time growth for an arbitrarily large interval of existence, showing that the free boundaries can grow exponentially. The second part of the thesis contains an introduction and one chapter. In this chapter, we establish a local well-posedness result for the Kolmogorov two-equation model of turbulence. This model belongs to the k-ε models, and describes the dynamics of an homogeneous and isotropic fully-developed turbulent flow. We generalize the previous results letting the turbulent kinetic energy vanish, in order to cover a wider range of phenomena. Consequently, we lose the parabolicity of the system, and an accurate analysis is needed to find the existence and uniqueness of solutions. We prove local well-posedness in critical Sobolev spaces Hs for s > 1 + d/2, for the cases of two and three dimensional fluids, in a periodic box Td. We consider fractional regularity, and consequently, our study involves paradifferential calculus, passing through Littlewood-Paley decomposition, in order to have a priori high order energy bounds.
Geometry in metric spaces and fixed point theorems for classes of nonexpansive type mappings
(2023-03-27) Parasio Sobreira de Souza, Daniel; Espínola García, Rafael; Japón Pineda, María de los Ángeles; Universidad de Sevilla. Departamento de Análisis Matemático
The main purpose of this work is to present recent results obtained on the existence of Fxed points for nonexpansive mappings and orbit-nonexpansive mappings in the general context of metric spaces. Additionally, our tech- niques will allow us to deduce the existence of common xed points for groups of such mappings based on features of the closed balls of the metric space. In order to do that, the concepts of normal structure and uniform normal structure will be analyzed and extended from the Banach space framework to the more general environment of metric spaces. Applications to important families of metric spaces without linear structure will be dis- played. The work is divided into three chapters which are subdivided into sec- tions. In Chapter 1 we present the basic concepts and results that we believe are necessary for reading and understanding the other chapters. In Chapter 2 and Chapter 3 we present the aforemetioned results most of which can be found in the articles [1] and [2] by Rafael Espínola García, María Japón and myself.
Spaces of Analytic Functions With Average Radial Integrability and Integration Operators
(2021-10-04) Aguilar-Hernández, Tanausú; Contreras Márquez, Manuel Domingo; Rodríguez Piazza, Luis; Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI); Universidad de Sevilla. Departamento de Análisis Matemático
In this thesis, we introduce the family of spaces of holomorphic functions in the unit disc with average radial integrability RM(p; q), 0 < p; q 1. This family contains the classical Hardy spaces Hq (when p = 1) and Bergman spaces Ap (when p = q). We characterize the inclusion between RM(p1; q1) and RM(p2; q2) depending on the parameters. For 1 < p; q < 1, we provide a description of the dual spaces of RM(p; q) by means of the boundedness of the Bergman projection. We show that RM(p; q) is separable if and only if q < 1. In fact, we provide a method to build isomorphic copies of `1 in RM(p;1). In the second half, we study integration operators Tg(f)(z) = z 0 f(w)g0(w) dw acting on RM(p; q) spaces. When we consider the operator Tg between the same RM(p; q) space, we provide a characterization of the boundedness, compactness, and weak compactness. When considering the action of Tg between different spaces, which is already an involved situation, we only characterize its boundedness. For the first case, we develop different tools such as a description of the bidual of RM(p; 0) and estimates of the norm of these spaces using the derivative of the functions, a family of results that we call Littlewood-Paley type inequalities. For the second case, we solve a problem of Carleson type measures for tent spaces of analytic functions ATq p in the unit disc. These spaces consist of those analytic functions of the tent spaces spaces Tq p introduced by Coifman, Meyer, and Stein, and it turns out that in many cases RM(p; q) = ATq p . This Carleson type problem was originally posed by Luecking.
Estudio de espacios localmente convexos sobre H. mediante subyacentes reales
(1976-05) Carmona Álvarez, José; Castro Brzezicki, Antonio de; Arias de Reyna Martínez, Juan; Universidad de Sevilla. Departamento de Análisis Matemático
La teoría del punto fijo en espacios funcionales modulares
(2001-02-09) Samadi, Sedki; Domínguez Benavides, Tomás; Amine Khamsi, Mohamed; Universidad de Sevilla. Departamento de Análisis Matemático
La teoría del punto fijo ha sido extensamente desarrollada en los espacios de Banach y los espacios métricos, Los espacios funcionales modulares no están incluidos en los espacios anteriores aunque el modular comparte algunas propiedades con la métrica. En 1990,Khamsi, Kozlowski y Reich iniciaron la teoría del punto fijo en los espacios funcionales modulares, estudiando las aplicaciones contractivas y las aplicaciones no-expansivas. Nosotros hemos seguido esta misma vía de investigación extendiendo el estudio de la teoría del punto fijo en los espacios funcionales modulares a las siguientes aplicaciones: -Las aplicaciones p-asintómaticamente regulares. -Las aplicaciones p-uniformemente Lipschitzianas. -Las aplicaciones p-asintóticamente no-expansivas. Se considera un subconjunto C convexo, cerrado, acotado y p-a.e. Secuencialmente compacto de un espacio funcional modular Lp, y una aplicación T: C C de alguno de los tipos anteriores. Bajo hipótesis muy generales probamos la existencia de un punto fijo para estas aplicaciones. De esta forma hemos conseguido contestar a algunos problemas abiertos y al mismo tiempo hemos abierto nuevas vias de investigación como por ejemplo, extender nuestros resultados a familias conmutativas de aplicaciones y a aplicaciones multivaluadas.
Relaciones entre operadores asociados a distintas medidas de no compacidad
(1993-09-13) Rodríguez Álvarez, Ramón Jaime; Domínguez Benavides, Tomás; Universidad de Sevilla. Departamento de Análisis Matemático
El objeto fundamental del trabajo es investigar las relaciones existentes entre operadores contractivos para distintas medidas de no compacidad en base a las propiedades geométricas del espacio subyacente, para ello se estudian varias medidas de no compacidad y algunos coeficientes geométricos que pueden definirse en relación con ellas en espacios métricos o de Banach.
Conjuntos de unicidad de sistemas de funciones independientes. Quantum derivadas
(2001-09-28) Ríos Collantes de Terán, Ricardo; Freniche Ibáñez, Francisco José; Universidad de Sevilla. Departamento de Análisis Matemático
En la primera parte de la tesis se estudian los conjuntos de unicidad de sucesiones (Pn) de funciones independientes definidas en [0,1] de media cero, varianza uno y acotadas, por M>-1, Se calcula la mejor constante Cm que verifica que los conjuntos de medida menor que Cm son conjuntos de una unidad para todas las sucesiones. Ademas generalizando los resultados de Stechkin y Ul¿yanov sobre el sistema de Rademacher, se demuestra qe los conjuntos de medida no total son conjuntos de unicidad debil. En la segunda parte se demuestran los mismos resultados que demostraron Zygmund y Marcinkiewicz para la Derivada de Riemann, para el caso de la Quantum Derivada. Y se obtienen, tambien los mismos resultados para la primeras y segundas Derivadas Generalizadas, basandonos en las ideas del trabajo de Ash sobre las Derivadas de Riemann Generalizadas.
Retículos de espacios invariantes de operadores de composición
(2008-01-24) Ponce Escudero, Manuel; Montes Rodríguez, Alfonso; Universidad de Sevilla. Departamento de Análisis Matemático
Un estudio, con aplicaciones, de las fórmulas de cuadratura matriciales.
(2002-07-08) Polo García, Beatriz; Durán Guardeño, Antonio José; Universidad de Sevilla. Departamento de Análisis Matemático
Polinomios matriciales ortonormales
(1995-11-13) López Rodríguez, Pedro; Durán Guardeño, Antonio José; Universidad de Sevilla. Departamento de Análisis Matemático
Sobre un método iterado para la resolución de sistemas no lineales y rectangulares de ecuaciones
(1984-02-02) Castillejos Toledano, Felipe; Cortés Gallego, José; Universidad de Sevilla. Departamento de Análisis Matemático
• Después de hacer un capítulo introductorio relativo al estudio del error para un conjunto de métodos iterados para sistemas rectangulares se plantea el que hemos denominado método de afinamiento, a continuación se expone el algoritmo los teoremas de convergencia y el error del mencionado método para soluciones simples de sistemas no lineales y rectangulares de ecuaciones. En un tercer capítulo se tiene el mismo esquema mencionado anteriormente pero aplicado a soluciones múltiples de sistemas. Por último se estudia el método para sistemas lineales donde resulta un proceso finito pues siempre se obtiene una solución en un numero finito de iteraciones. El trabajo se acompaña de dos apéndices de resultados numéricos de ejemplos (lineales y no) resueltos en el vax 11 de esta universidad.
Algunos módulos para la propiedad (B) de Rolewicz y otras propiedades geométricas de los espacios de Banach
(1997-05-06) Francisco Cutillas, Salvador; Domínguez Benavides, Tomás; Ayerbe Toledano, José María; Universidad de Sevilla. Departamento de Análisis Matemático
Estudio asintótico de polinomios matriciales ortogonales
(2003-05-07) Daneri Vías, Enrique; Durán Guardeño, Antonio José; Universidad de Sevilla. Departamento de Análisis Matemático
La memoria de esta tesis doctoral presenta una serie de investigaciones realizadas sobre la teoría de la ortogonalidad matricial relativas al comportamiento asintótico de los polinomios matriciales ortogonales. El contenido original se divide en tres partes, según el problema que se resuelve: 1,- Cociente asintótico. Estudio de la asintótica del cociente para polinomios matriciales ortogonales con coeficientes de recurrencia no acotados. También se prueba el cociente asintótico para polinomios matriciales ortogonales con coeficientes de recurrencia asintóticamente periódicos. Ejemplos. 2,- Convergencia débil. Estudio de convergencia débil para familias uniparamétricas de polinomios matriciales ortogonales. Ilustración de resultados en numerosos ejemplos. 3,- Extensión del Teorema de Markov para medidas solución del problema de momentos matricial completamente indeterminado, diferenciado el caso de Hamburger y el de Stieltjes.
Linear and algebraic structures in function sequence spaces
(2020-06-18) Gerlach Mena, Pablo José; Calderón Moreno, María del Carmen; Prado Bassas, José Antonio; Universidad de Sevilla. Departamento de Análisis Matemático
Historically, many mathematicians of all ages have been attracted and fascinated by the existence of large algebraic structures that satisfy certain properties that, a priori, contradict the mathematical intuition. The aim of the present Dissertation is the study of the lineability of certain families of sequences of functions with very specific properties. The Dissertation is divided in 6 chapters, where Chapters 1, 2 and 3 focus on introducing the basic notation and main terminology of the theory of Lineability and modes of convergence that will be used along this Dissertation. In Chapter 4 we begin with the study of the algebraic size of two families of sequences of functions with different modes of convergence in the closed unit interval [0, 1]: convergence in measure but pointwise almost everywhere and pointwise but not uniform convergence. In Chapter 5 we focus our attention on the setting of (Lebesgue) integrable functions. We start with sequences of integrable functions with different modes of convergence in comparison to the L1-convergence, and finish the chapter with the algebraic size of the family of unbounded, continuous and integrable functions on [0, +∞) and sequences of them. Finally, in Chapter 6 we turn into the setting of series of functions, obtaining positive results about the linear and algebraic size of the family of sequences of functions whose series converges absolutely and uniformly but does not verify the hypothesis of the Weierstrass M-test.
Propiedades de oscilación en ecuaciones diferenciales autoadjuntas
(1973-02-24) Couce Calvo, Julio; Castro Brzezicki, Antonio de; Universidad de Sevilla. Departamento de Análisis Matemático
Resolución de ecuaciones funcionales planteadas mediante operadores lineales
(1970-11-09) Rodríguez Cano, José Juan; Castro Brzezicki, Antonio de; Universidad de Sevilla. Departamento de Análisis Matemático
Local distribution of Rademacher series and function spaces
(2017-05) Carrillo Alanís, Francisco Javier; Curbera Costello, Guillermo; Universidad de Sevilla. Departamento de Análisis Matemático

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