A Lp-theory for fractional stationary Navier-Stokes equations

We consider the stationary (time-independent) Navier-Stokes equations in the whole threedimensional space, under the action of a source term and with the fractional Laplacian operator (--$\Delta$) $\alpha$/2 in the diffusion term. In the framework of Lebesgue and Lorentz spaces, we find some natural sufficient conditions on the external force and on the parameter $\alpha$ to prove the existence and in some cases nonexistence of solutions. Secondly, we obtain sharp pointwise decaying rates and asymptotic profiles of solutions, which strongly depend on $\alpha$. Finally, we also prove the global regularity of solutions. As a bi-product, we obtain some uniqueness theorems so-called Liouville-type results. On the other hand, our regularity result yields a new regularity criterion for the classical ( i.e. with $\alpha$ = 2) stationary Navier-Stokes equations. Content

Liouville type theorems for stationary Navier-Stokes equations with Lebesgue spaces of variable exponent

In this article we study some Liouville-type theorems for the stationary 3D Navier-Stokes equations. These results are related to the uniqueness of weak solutions for this system under some additional information over the velocity field, which is usually stated in the literature in terms of Lebesgue, Morrey or BMO^--1 spaces. Here we will consider Lebesgue spaces of variable exponent which will provide us with some interesting flexibility

Well-posedness of a nonlinear shallow water model for an oscillating water column with time-dependent air pressure

We propose in this paper a new nonlinear mathematical model of an oscillating water column. The one-dimensional shallow water equations in the presence of this device are essentially reformulated as two transmission problems: the first one is associated with a step in front of the device and the second one is related to the interaction between waves and a fixed partially-immersed structure. By taking advantage of free surface Bernoulli's equation, we close the system by deriving a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in the previous work \cite{bocchihevergara2021}. We then show that the second transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semilinear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE. Finally, the well-posedness of the transmission problem related to the wave-structure interaction in the oscillating water column is obtained as an application of the general theory.Comment: 35 pages, 1 figur

Analysis of the temporal and spatial land use change and its manifestation in the landscape of the region of Los Ríos, Southern Chile

El objetivo general de esta tesis ha sido analizar aspectos relacionados con el cambio espacial y temporal del uso del suelo y su efecto en el paisaje en la región de Los Ríos y en términos globales en la región del Bío-Bío, utilizando sistemas de información geográficos y técnicas geoestadísticas. La tesis se encuentra estructurada en tres capítulos, cada uno de ellos se ha transformado en un artículo científico, publicado en una revista indexada de corte internacional, abordando aspectos específicos, permitiendo de esta forma dar cumplimiento al objetivo general descrito previamente. El primer artículo se titula “Spatial distribution of forest plantations in southern Chile, an area with a pulp mill”. Este trabajo se centra en los siguientes aspectos: a) realizar un análisis espacial para probar la hipótesis de que los polígonos que fragmentan el uso del suelo con plantaciones forestales no se relacionan espacialmente, lo cual explicaría que la industria forestal realiza reconversiones de uso del suelo sin seguir un patrón sobre el uso existente y que los propietarios privados realizan esta acción en forma aislada; b) construir indicadores locales para determinar la relevancia de los polígonos de plantaciones con similitudes en su entorno; c) explicar el efecto de las plantaciones en la dinámica del cambio de uso del suelo, que probaría la presión a la que los diferentes usos del territorio, en especial los bosques nativos, han estado sometidos a través de la sustitución por plantaciones forestales con especies exóticas. El segundo trabajo se titula “Temporal and spatial analysis of land use change in the south central re gion of Chile”. En este artículo se evaluó a escala espacial y temporal la dinámica de los cambios de uso del suelo ocurridos en la región centro-sur de Chile, por ser la más poblada e industrializada y que presenta las tasas de forestación más alta del país. Se construyeron indicadores que permitieron analizar los cambios de la cobertura de uso del suelo, realizando un análisis de los cambios de la cobertura de uso del suelo a nivel global regional y a nivel provincial. El tercer artículo se denominó “Estimation and spatial analysis of aerial biomass and carbon capture in native forests in the south of Chile: County of Valdivia”. Según las dos investigaciones previas y dadas las condiciones en que se encuentran los bosques nativos al sur de Chile, el objetivo de esta investigación fue estimar la biomasa aérea y el contenido de carbono en bosques nativos en la comuna de Valdivia, utilizando funciones alométricas, y determinar el grado de agrupación de los polígonos con contenido de carbono, utilizando técnicas de estadística espacial. Al conocer los mapas de distribución espacial del carbono, será posible desarrollar una línea base que servirá para la implementación de futuras acciones de mitigación, reducción de emisiones por deforestación y degradación forestal, manejo sostenible de los bosques, conservación y mejoramiento en las reservas de carbono.The general objective of this thesis has been to analyze aspects related to the spatial and temporal change of land use in the Los Ríos region and its effect on the landscape, using geographic information systems and geostatistical techniques. The thesis is structured in three chapters, each of them has been transformed into a scientific article, published in an international indexed journal, addressing specific aspects, thus allowing to fulfill the general objective previously described. The first article is entitled “Spatial distribution of forest plantations in southern Chile, an area with a pulp mill”. This paper focuses on the following aspects: a) conducting a spatial analysis to test the hypothesis that the polygons fragmenting land use with forest plantations are not spatially related, which would explain that the forest industry carries out land use reconversions without following a pattern on the existing use and that private landowners carry out this action in isolation; b) to construct local indicators to determine the relevance of plantation polygons with similarities in their surroundings; c) to explain the effect of plantations on the dynamics of land use change, which would prove the pressure to which different land uses, especially native forests, have been subjected through the substitution of forest plantations with exotic species. The second paper is entitled “Temporal and spatial analysis of land use change in the south-central region of Chile”. In this article the dynamics of land use changes in the central-southern region of Chile, which is the most populated and industrialized region with the highest forestation rates in the country, was evaluated on a spatial and temporal scale. Indicators were constructed to analyze changes in land-use cover, analyzing changes in land-use cover at the global, regional and provincial levels. The third article is entitled “Estimation and spatial analysis of aerial biomass and carbon capture in native forests in the south of Chile: County of Valdivia”. According to the two previous investigations and given the conditions of native forests in southern Chile, the objective of this investigation was to estimate aerial biomass and carbon content in native forests in Valdivia County. Allometric functions and spatial statistics techniques, were used to determine the degree of clustering of polygons with carbon content. With the spatial distribution of carbon, it will be possible to develop a baseline for the implementation of future mitigation actions, reducing emissions from deforestation and forest degradation, sustainable management of forests, conservation, and improvement of carbon stocks

Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness

We consider here the stationary Micropolar fluid equations which are a particular generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one based mainly in the velocity field u and the other one based in the microrotation field $\omega$. We will study in this work some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establish, under some suitable decay at infinity conditions for the velocity field only, the uniqueness of the trivial solution

Relación inconsistente entre puntajes del Sistema de Medición de Calidad de la Educación y el promedio de notas en estudiantes de octavo básico en Chile

El Sistema de Medición de Calidad de la Educación SIMCE tiene entre sus propósitos institucionales contribuir al mejoramiento de la calidad de la educación e informar a la ciudadanía de los logros de aprendizaje del estudiantado chileno en diferentes áreas, pero existen pocas investigaciones concretas en este último ámbito. El objetivo principal de la investigación es establecer la correlación entre las notas del estudiantado al finalizar el 8° básico y los puntajes SIMCE en las pruebas de Lenguaje y Comunicación, Matemáticas y Ciencias Naturales. Se utilizaron los datos de octavo año básico, proporcionados por la Agencia de Calidad de la Educación, perteneciente al Ministerio de Educación de Chile. La información original incluyó 12.586 alumnos, nominados como zona sur, por pertenecer al censo de las cinco ciudades más importantes del sur de Chile, esto es, Temuco, Valdivia, Osorno, Puerto Montt y el caso extremo de Ancud. Para completar el análisis correlacional y despejar el enmascaramiento de los resultados al considerar solo los datos Censales, se elaboró un muestreo dirigido, seleccionando las ciudades de Temuco, por ser la ciudad más grande. Ancud por sus características de ciudad pequeña y extrema y finalmente un colegio seleccionado al azar entre las cinco ciudades. Se determinó que existe una baja relación entre las pruebas SIMCE y el promedio de notas, demostrando que el instrumento utilizado a nivel nacional para medir los aprendizajes en estudiantes de enseñanza básica, difiere notablemente de los utilizados por el profesorado en los colegios

Asymptotic behaviour of a system modelling rigid structures floating in a viscous fluid

The PDE system introduced in Maity et al. (2019) describes the interaction of surface water waves with a floating solid, and takes into account the viscosity µ of the fluid. In this work, we study the Cummins type integro-differential equation for unbounded domains, that arises when the system is linearized around equilibrium conditions. A proof of the input-output stability of the system is given, thanks to a diffusive representation of the generalized fractional operator $\sqrt{1 + \mu s}$. Moreover, relying on Matignon (1996) stability result for fractional systems, explicit solutions are established both in the frequency and the time domains, leading to an explicit knowledge of the decay rate of the solution. Finally, numerical evidence is provided of the transition between different decay rates as a function of the viscosity $\mu$

Modelling and simulation of a wave energy converter

In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions

La industria audiovisual y su incidencia en el sector de artes plásticas en la parroquia Píntag en el Distrito Metropolitano de Quito.

The main objective of this research project is to carry out an audiovisual proposal that meets an emphasis on the treatment of graphics from design, to solve the problems of Píntag artists through audiovisual graphics, helping to recognize the works and their artists. , in addition to presenting Píntag as a tourist potential to visit, distributing through mass media such as social networks to encourage tourists and other people in the country to visit and learn about everything the sector can offer.El objetivo principal del presente proyecto de investigación es realizar una propuesta audiovisual que cumpla con un énfasis en el tratamiento de la gráfica desde el diseño, para resolver la problemática de los artistas de Píntag mediante grafismo audiovisual ayudando a al reconocimiento de las obras y sus artistas, además de presentar a Píntag como un potencial turístico a visitar, distribuyendo por medios de difusión masiva como redes sociales para incentivar a los turistas y demás personas del país a visitar y conocer todo lo que puede brindar del sector