An aggregation equation with a nonlocal flux

In this paper we study an aggregation equation with a general nonlocal flux. We study the local well-posedness and some conditions ensuring global existence. We are also interested in the differences arising when the nonlinearity in the flux changes. Thus, we perform some numerics corresponding to different convexities for the nonlinearity in the equation

The confined Muskat problem: differences with the deep water regime

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip \RR\times[-l,l]. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The main goal of this paper is to compare the qualitative properties between the model when the fluids move without boundaries and the model when the fluids are confined. We find that, in a precise sense, the boundaries decrease the diffusion rate and the system becomes more singular.Comment: Revised version. 32 pages, 4 figure

Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations

We address a spectral problem for the Dirichlet-Laplace operator in a waveguideis obtained from repsilon an unbounded two-dimensional strip ?? which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, O(1)O(1) and O(?)O(?) respectively, where 0<??10<??1. We look at the band-gap structure of the spectrum . We derive asymptotic formulas for the endpoints of the spectral bands and show that has a large number of short bands of length ) which alternate with wide gaps of width O(1)O(1)

Spectral gaps in a double-periodic perforated Neumann waveguide

We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter ε>0. The periodicity cell itself contains a string of holes at a distance O(ε) between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as ε→0. We demonstrate that, for ε small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a 'fast Floquet variable' and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed.The work has been partially supported by MICINN through PGC2018-098178-B-I00, PID2020-114703GB-I00 and Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S)

Análisis asintótico y homogeneización de ecuaciones en derivadas parciales

Tesis Doctoral inédita, leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-06-200

Semana Multidisciplinar en la U.A.M.

La Semana Multidisciplinar es una nueva actividad que se va a desarrollar en el Campus de Cantoblanco del 12 al 22 de Julio. Organizada por miembros del Departamento de Matemáticas de la UAM y del Instituto de Ciencias Matemáticas, instituto mixto CSICUAMUC3M- UCM. La Semana Multidisciplinar pretende reunir a estudiantes de últimos años de licenciatura o de posgrado con grupos de investigación de distintas áreas (Astrofísica, Biología, Matemáticas, Medicina, Telecomunicaciones,…) con el objetivo de plantear y resolver con diferentes herramientas matemáticas las problemáticas de distintas áreas de la Ciencia que se plateen. A continuación vamos a presentar una descripción de las actividades a realizar durante este mes de julio de 2010 así como los objetivos que nos planteamo

A note on interface dynamics for convection in porous media

We study the fluid interface problem through porous media given by two incompressible 2-D fluids of different densities. This problem is mathematically analogous to the dynamics interface for convection in porous media, where the free boundary evolves between fluids with different temperature. We find a new formula for the evolution equation of the free boundary parameterized as a function in the periodic case. In this formula there is no a principal value in the non-local integral operator involved in the equation, giving a simpler system. Using this formulation, we perform numerical simulations in the stable case (denser fluid below) which show a strong regularity effect in the periodic interface.Ministerio de Educación y CienciaJunta de Castilla-La ManchaMinisterio de Educación y Ciencia (España) MTM2005-0071

Asymptotics for the spectrum of a floquet-parametric family of homogenization problems associated with a Dirichlet waveguide

In this chapter, we address the asymptotic behavior of the eigenvalues and eigenfunctions of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane. The perforations are periodically placed along the ordinate axis. The two parameters of the problem are the period, which converges toward zero, and the so-called Floquet parameter. The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Dirichlet over the rest. In particular, we provide bounds for convergence rates of the eigenvalues which are uniform in both parameter

Matemáticas del planeta Tierra : unidad didáctica

Con: Matemáticas del planeta Tierra : [cuaderno de actividades] / Fernando Alcaide, Miguel NietoResumen basado en el de la publicaciónRevisor didáctico: Luis RicoEsta unidad didáctica nace dentro de una iniciativa internacional de gran relevancia, la proclamación de 2013 como Año de las Matemáticas del Planeta Tierra (Mathematics Planet Earth, MPE 2013). Esta declaración ha tenido su origen en las sociedades matemáticas e institutos de investigación de Estados Unidos y Canadá, y posteriormente ha recibido el apoyo de la Unión Matemática Internacional (IMU) y la UNESCO. El objetivo del MPE 2013 es señalar la importancia de las matemáticas para conocer y gestionar mejor el funcionamiento de nuestro planeta –su propia estructura, la vida que alberga, los fenómenos en su corteza, en su atmósfera y en sus océanos, la influencia de la actividad humana, nuestro entorno astronómico– y también para estar mejor preparados ante catástrofes que nos alcanzan a veces de una manera terrible. Se ha dividido la obra en 16 capítulos, desarrollado cada uno de ellos por expertos en el tema y con materiales complementarios (libros, películas, series televisivas, portales de Internet) que pueden resultar de utilidad en las clases para amenizar e ilustrar los textos.ES