Instabilities in fluid mechanics and convex integration

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-07-2021In this dissertation we analyze several problems related to unstable interfaces in uid mechanics: the vortex sheet problem and both the fully unstable and partially unstable Muskat problem. Moreover, we present a quantitative homotopy principle with applications to evolution equations within the framework of the technique we will employed, the convex integration method. In chapter 1 we introduce the partial di erential equations that we will study, the problems that have motivated this dissertation, the results we have obtained in [26, 103, 102, 27] and, nally, the convex integration method and the homotopy principle. The chapter 2 is devoted to gather some common aspects of the problems we analyze. This allows us to introduce the \turbulence zone" where the uid behaves wildly. This zone starts to grow around the unstable region of the interface. Moreover, we present some lemmas related to the Birkho -Rott and Muskat operators, which will be crucial within the proofs of the results in chapters 4-6. These lemmas can be found in [103], joint work with L aszl o Sz ekelyhidi, and in [27], joint work with Angel Castro and Daniel Faraco. In chapter 3 we present a quantitative version of the homotopy principle for a class of evolution equations. In the previous versions ([52]) weak solutions are recovered from a \subsolution" through a convex integration scheme. This quantitative version recovers also microscopic and macroscopic information. On the one hand, it shows that the uid can behave wildly inside the turbulence zone. On the other hand, it measures the proximity, in terms of weak -continuous quantities, between the weak solutions obtained and the subsolution. This allows to select those which retain more information from the subsolution, thus emphasizing the role of the subsolution as the candidate for the macroscopic solution. These results are based on [26], joint work with Angel Castro and Daniel Faraco. In chapter 4 we study the vortex sheet problem for the incompressible Euler equation. We construct in nitely many admissible solutions starting from vorticities without xed sign and concentrated on non-analytic curves. The existence of weak solutions for vortex sheet data with mixed sign was an open problem from the work of Delort ([56]) and was only known for particular initial data ([94]). These weak solutions are smooth outside a turbulence zone which grows linearly in time around the vortex sheet. Furthermore, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality, and measures the maximal initial dissipation rate in terms of the vortex sheet strength. These results appear in [103], joint work with L aszl o Sz ekelyhidi. In chapter 5 we prove a homotopy principle for the incompressible porous media (IPM) equation with density-viscosity jump. As a rst example, non-trivial weak solutions with compact support in time are obtained (see [39] for the case of constant viscosity). Secondly, we construct mixing solutions to the unstable Muskat problem with initial at interface. As a byproduct, we check that the connection, established by Sz ekelyhidi for the case of constant viscosity ([126]), between the subsolution and the Lagrangian relaxed solution of Otto ([116]), holds for the case of viscosity jump as well. In this case we show how a pinch singularity in the relaxation prevents the two uids from mixing wherever there is neither Rayleigh-Taylor instability nor vorticity at the interface. In chapter 6 we construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [24, 23]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method. These results appear in [27], joint work with Angel Castro and Daniel Farac

Arqueo-acarología: potencialidades y limitaciones de una prácticamente inédita subdisciplina arqueozoólogica

El análisis de faunas de yacimientos arqueológicos viene tradicionalmente concibiéndose como un estudio sobre osamentas y conchas, esto es, sobre vertebrados y moluscos. Sin negar el peso específico de estos grupos, conviene no olvidar que muchos otros, entre los que destacan equinodermos (erizos de mar y afines) y artrópodos (insectos, crustáceos, etc ... ), ofrecen óptimas potencialidades analíticas así como una amplia gama de aplicaciones en arqueología. En el presente trabajo presentamos lo que pretendemos se consolide como una nueva línea de investigación en el seno del Laboratorio de Arqueozoología de la UAM, la arqueo-acarología, al tiempo que esbozamos algunas de las ventajas y de los inconvenientes que conllevan los estudios sobre este peculiar grupo zoológico.Faunal analysis in archaeological sites has been traditionally conceived as the study of bones and shells, in other words, of vertebrates and molluscs. Without denying the importance of both phyla, one must remind scholars that many others, including echinoderms (starfish, sea urchins and the like) or arthropods (insects, crabs and the like) offer a wide variety of analytical treatments in archaeology worth exploring. Our paper deals with what we hope will soon become an established line of research in oUT Laboratory at U.A.M., the study of mite remains, whose advantages and drawbacks are briefly discussed in the context of archaeofaunal studies

La Dolores : drama lírico en tres actos

Copia digital. Valladolid : Junta de Castilla y León. Consejería de Cultura y Turismo, 2009-201

Potential energy surface and spectroscopy of clusters of rare-gas atoms with cyclopropane

Analytical empirical potential energy surfaces describing the van der Waals interaction between rare-gas atoms and cyclopropane are presented. The functional form is based on pairwise Lennard-Jones-type potentials which have been widely used to describe rare-gas-benzene complexes, also studied in this work in order to check our theoretical method and for comparison. The parameters have been chosen in order to accurately fit the high resolution microwave spectra recently reported by Xu and Jäger [J. Chem. Phys. 106, 7968 (1997)]. The observed splitting in the microwave spectra of Ne-cyclopropane, associated with rotational tunneling, is well reproduced. Moreover, such tunneling is also important for complexes of Ar and Kr in excited van der Waals states. These phenomena involve a high delocalization of the wave functions and, therefore, intermolecular spectroscopy techniques would provide a good check of the potential energy surface over a broad region of the configuration space. © 1998 American Institute of Physics.This work has been supported by Consejería de Educación, Cultura y Deportes del Gobierno de Canarias under Grant No. PI 2/95, DGYCIT (Spain) under Grant No. PB95- 0071, and by the European TMR network Contract No. ERBFMRX-CT96-0088.Peer Reviewe

BogotáViva, guía de ocio alternativo

El proyecto surge a partir de una percepción nuestra al identificar que gran cantidad de bogotanos gastan su tiempo libre en los mismo lugares y actividades comúnmente conocidas. Por esta razón, propusimos una plataforma donde los habitantes de Bogotá sean los principales beneficiados y puedan encontrar en esta actividades de ocio alternativo. BogotáViva, es una guía de ocio alternativo, entendido alternativo como algo no convencional y poco conocido. La página web cuenta con apoyos audiovisuales, factor diferencial del resto de guías de ocio. Además, cada actividad incluye una amplia galería fotográfica y toda la información necesaria para dar a conocer la actividad.The project arises from our perception when we identify that the great majority of Bogota citizens spend their free time in the same places and doing commonly known activities. For this reason, we proposed a platform where the habitants of Bogotá are the main beneficiaries and can find alternative activities to spend their leisure time. BogotáViva, is an alternative leisure guide, understanding alternative as something unconventional and unknown. The differential factor of our website is the audiovisual material, which is not found in the other ones. Besides, each activity includes a large photo gallery and all the necessary information to visit the place.Comunicador (a) SocialPregrad

Álgebras de Banach y transformadas integrales

El objetivo de este trabajo es de carácter introductorio en el campo de las álgebras de Banach, y tiene como finalidad la aproximación a una serie de transformadas integrales muy conocidas desde el contexto de estos espacios. En el primer capítulo se desarrolla la teoría de representación de Gelfand (cuyo propósito es construir la transformada de Gelfand), la cual hemos ampliado al incorporar al caso original de la teoría (las álgebras de Banach complejas conmutativas) el caso de las álgebras de Banach reales y el caso no conmutativo. El segundo capítulo se centra en la obtención de la transformada de Gelfand sobre un espacio de Banach que equipamos con un producto de convolución (con el que es álgebra de Banach); con éste producto hemos conseguido unificar varios productos de convolución muy conocidos (hablamos de la convolución de Volterra, Fourier, Laplace, Zeta y Dirichlet). Gracias a este punto de vista global que nos otorga el producto generalizado, hemos podido entender mejor algunas relaciones que se apreciaban entre las transformadas de Gelfand (cuya expresión es integral) que dan los productos individuales en grupos y semigrupos, obteniéndose también algunos resultados conocidos por una vía novedosa. También estudiaremos otros productos de convolución con los que veremos otra forma de obtener las transformadas correspondientes; entre éstas se encuentra la transformada coseno. Para las últimas páginas hemos escogido algunas aplicaciones de estas transformadas integrales en la resolución de ecuaciones diferenciales y en diferencias. Con ello se pretende mostrar como las propiedades algebraicas de estas transformadas sobre espacios funcionales, pueden utilizarse para transformar un problema difícil en un espacio de partida, en otro sencillo en un espacio de llegada; lo que permite, bajo ciertas condiciones, obtener la solución del problema inicial

Meta-heurística ACO (Ant Colony Optimization) para la resolución de problemas en líneas de producción

La meta -heurística ACO (Ant Colony Optimization) es un procedimiento heurístico para la resolución de problemas de optimización di screta basado en el comportamiento de las hormigas. Sus principales características son: (1) la utilización de "feed -back" positivo, (2) computación distribuida (la estructura de estos algoritmos permite su paralelización de forma muy simple y natural), y (3) el uso de heurísticas Greedy constructivas (ayuda a encontrar soluciones aceptables en las primeras etapas del proceso de exploración). En este artículo se presenta la aplicación de dichas meta -heurísticas a la resolución de problemas de producción co mo equilibrado de líneas de montaje o la secuenciación de unidades en un sistema que no permite esperasPostprint (published version

Examination of the Feynman-Hibbs Approach in the Study of NeN_N-Coronene Clusters at Low Temperatures

Feynman-Hibbs (FH) effective potentials constitute an appealing approach for investigations of many-body systems at thermal equilibrium since they allow us to easily include quantum corrections within standard classical simulations. In this work we apply the FH formulation to the study of Ne$_N$-coronene clusters ($N=$ 1-4, 14) in the 2-14 K temperature range. Quadratic (FH2) and quartic (FH4) contributions to the effective potentials are built upon Ne-Ne and Ne-coronene analytical potentials. In particular, a new corrected expression for the FH4 effective potential is reported. FH2 and FH4 cluster energies and structures -obtained from energy optimization through a basin-hoping algorithm as well as classical Monte Carlo simulations- are reported and compared with reference path integral Monte Carlo calculations. For temperatures $T> 4$ K, both FH2 and FH4 potentials are able to correct the purely classical calculations in a consistent way. However, the FH approach fails at lower temperatures, especially the quartic correction. It is thus crucial to assess the range of applicability of this formulation and, in particular, to apply the FH4 potentials with great caution. A simple model of $N$ isotropic harmonic oscillators allows us to propose a means of estimating the cut-off temperature for the validity of the method, which is found to increase with the number of atoms adsorbed on the coronene molecule