Optimal bounds for self-similar solutions to coagulation equations with product kernel

We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with multiplicative kernel of homogeneity $2l\lambda \in (0,1)$. We establish rigorously that such solutions exhibit a singular behavior of the form $x^{-(1+2\lambda)}$ as $x \to 0$. This property had been conjectured, but only weaker results had been available up to now

A class of dust-like self-similar solutions of the massless Einstein-Vlasov system

In this paper the existence of a class of self-similar solutions of the Einstein-Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein-Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein-Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point $P_0$ and converges to a stationary solution $P_1$ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes.Comment: 47 page

A Kinetic Model for Grain Growth

We provide a well-posedness analysis of a kinetic model for grain growth introduced by Fradkov which is based on the von Neumann-Mullins law. The model consists of an infinite number of transport equations with a tri-diagonal coupling modelling topological changes in the grain configuration. Self-consistency of this kinetic model is achieved by introducing a coupling weight which leads to a nonlinear and nonlocal system of equations. We prove existence of solutions by approximation with finite dimensional systems. Key ingredients in passing to the limit are suitable super-solutions, a bound from below on the total mass, and a tightness estimate which ensures that no mass is transported to infinity in finite time.Comment: 24 page

Interpolación de funciones en el marco del formalismo de los espacios de Hilbert con núcleo reproductor y ejemplos de aplicación

[ES] Los métodos de interpolación en el marco del formalismo de los espacios de Hilbert con núcleo reproductor (EHNR) se han aplicado en múltiples trabajos durante la pasada década para la obtención de superficies de energía potencial (SEP) de interacción de sistemas moleculares de pocos átomos. La interpolación usualmente se basa en las energías obtenidas de resolver la ecuación de Schrödinger para muchos cuerpos. La representación analítica de la SEP de interacción es esencial para el estudio de la dinámica clásica de los sistemas moleculares. Específicamente, la interpolación basada en RKHS presenta varias ventajas sobre otros métodos usualmente utilizados para el mismo propósito, por ejemplo, es un método genérico y no utiliza parámetros que deban ser ajustados. En el presente trabajo utilizaremos la función hypergeométrica de Gauss para representar la función kernel (núcleo). El comportamiento asintótico correspondiente según el sistema objetivo de estudio está incorporado desde la propia construcción de la función kernel. Los elementos fundamentales de la definición de este tipo de kernel se muestran en una primera parte, dando paso, en una segunda parte, a la descripción de ejemplos de aplicación tanto en el marco de los sistemas moleculares como en otros posibles contextos.[EN] The interpolation methods in the framework of the reproducing kernel Hilbert space (RKHS) formalism have been successfully applied to obtain the potential energy surface (PES) of small molecular systems over the last decade. The interpolation is usually based on high level ab initio calculations from solving the many-body Schrödinger equation. The right representation of the PES is a central issue when carrying out classical molecular dynamics simulations of molecular systems. Specifically, the interpolation methods in the framework of the RKHS show several advantages over other methods, for instance, they are generic and parameter-free. In this work, a kernel based on the Gauss hypergeometric function is used. One advantage of this type of kernel is that the asymptotic behavior of the PES goes to zero when the separation between any two atoms is taken to infinity. Examples of PES interpolations in the context of molecular systems and other contexts are described.Castro-Palacio, JC.; Cuador, J.; Velazquez, L.; Monsoriu Serra, JA. (2014). Interpolación de funciones en el marco del formalismo de los espacios de Hilbert con núcleo reproductor y ejemplos de aplicación. Nereis. Revista Iberoamericana Interdisciplinar de Métodos, Modelización y Simulación. 7:67-76. http://hdl.handle.net/10251/109830S6776