Analysis of some diffusive and kinetic models in mathematical biology and physics

Descripció del recurs: el 29 de juny de 201

Optimal design problems governed by the nonlocal p -Laplacian equation

In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal p-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal p-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288– 310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed

Lp theory for the multidimensional aggregation equation

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A well-posedness theory in measures for some kinetic models of collective motion

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models

Asymptotic flocking dynamics for the kinetic Cucker-Smale model

In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution

Mis casos clínicos de especialidades odontológicas

Libro que muestra la atención de casos clínicos particulares referente a las diferentes especialidades odontológicasLibro que muestra la atención de casos clínicos particulares referente a las diferentes especialidades odontológicasUniversidad Autónoma de Campeche Universidad Autónoma del Estado de Hidalgo Universidad Autónoma del Estado de Méxic

Existence and approximation of nonlocal optimal design problems driven by parabolic equations

This work is a follow-up to a series of articles by the authors where the same topic for the elliptic case is analyzed. In this article, a class of nonlocal opti mal design problem driven by parabolic equations is examined. After a review of results concerning existence and uniqueness for the state equation, a detailed formulation of the nonlocal optimal design is given. The state equation is of non local parabolic type, and the associated cost functional belongs to a broad class of nonlocal integrals. In the first part of the work, a general result on the existence of nonlocal optimal design is proved. The second part is devoted to analyzing the convergence of nonlocal optimal design problems toward the corresponding classical problem of optimal design. After a slight modification of the problem, either on the cost functional or by considering a new set of admissibility, the G-convergence for the state equation and, consequently, the convergence of the nonlocal optimal design problem are proved

Collective behavior of animals: swarming and complex patterns

En esta nota repasamos algunos modelos basados en individuos para describir el movimiento colectivo de agentes, a lo que nos referimos usando la voz inglesa swarming. Estos modelos se basan en EDOs (ecuaciones diferenciales ordinarias) y muestran un comportamiento asintótico complejo y rico en patrones, que mostramos numéricamente. Además, comentamos cómo se conectan estos modelos de partículas con las ecuaciones en derivadas parciales para describir la evolución de densidades de individuos de forma continua. Las cuestiones matemáticas relacionadas con la estabilidad de de estos modelos de EDP's (ecuaciones en derivadas parciales) despiertan gran interés en la investigación en biología matemáticaIn this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs (ordinary differential equations) exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE (partial differential equations) models are questions of actual interest in mathematical biology researc

Lp theory for the multidimensional aggregation equation

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