Programa – XV ENAMA

A extensão de técnicas lineares para uma configuração não-linear, além de seu interesse matemático intrínseco, é uma tarefa importante para potenciais aplicações e bastante desafiadora, uma vez que os argumentos lineares são comumente ineficazes em um cenário mais geral.
Nesta palestra, vamos apresentar algumas técnicas em análise não-linear que consistem em levar alguns resultados da teoria linear dos operadores absolutamente somantes para um contexto completamente abstrato, com pouca ou nenhuma estrutura algébrica, no intuito de unificar e mostrar a validade desses resultados para várias classes de operadores (lineares e não-lineares) somantes conhecidas ou que venham a ser definidas, apenas fazendo escolhas adequadas de parâmetros.

In this talk we shed new light on the mathematical studies of nonlinear dispersive evolution equations on metric graphs. This trend has been mainly motivated by the demand of reliable mathematical models for different phenomena in branched systems which, in meso- or nano-scales, resemble a thin neighborhood of a graph, such as Josephson junction networks, electric circuits, or nerve impulses in complex arrays of neurons, just to mention a few examples. Our dynamic problems are related to the (in)stability of solitons-profiles. Models associated to Y−junctions or tadpole graphs will be considered.
The arguments presented in this talk have prospects for the study of the (in)stability of soliton-profiles solutions of other nonlinear evolution equations on branched systems.

We consider the α−Navier-Stokes equations coupled with a Vlasov type equation to model the flow of an incompressible fluid containing small particles. We prove the existence of global weak solutions to the coupled system subject to periodic boundary conditions. Moreover, we investigate the regularity of weak solutions and the uniqueness of regular solutions. The convergence of its solutions to that of the Navier-Stokes-Vlasov equations when α tends to zero is also established.

Results are extended to the model with the diffusion of spray, i.e., to the α−Navier-Stokes-Vlasov-Fokker-Planck equations.

Brief introduction to boundary value problems. Variational methods vs. topological methods.
Shooting method and applications. Poincaré map and Brouwer fixed point theorem. Fixed points in infinite dimension, the theorems of Banach and Schauder. The method of upper and lower solutions. Basic notions of topological degree theory, from Brouwer to Leray-Schauder. Application to ordinary and functional differential equations.

Linear dynamics is a relatively recent area of mathematics which lies at the intersection of operator theory and dynamical systems.
It arose from the studies of operators on Banach spaces which have the property of being cyclic, supercyclic, hypercyclic, etc.
In the first half of this talk, we will give an overview and general background of the area. In the second half of this talk, we will study composition operators on L^p(μ) spaces where μ is a σ-finite measure. Some open problems involving measures and and operators on L^p(μ) will be stated.

The linear Boltzmann equation is a fundamental model in the study of particle transport. Numerous and relevant applications and the high complexity of such a model motivate continued research in this area.
In this course, we will focus on the analysis and applications of numerical techniques in developing benchmark solutions to the linear Boltzmann equation. In the first part, we introduce the application of extrapolation techniques to analyze the asymptotic convergence of the spatial and angular discretization for two-dimensional discrete ordinates solutions. In addition, we will discuss convergence acceleration schemes. In the second part, we will pay particular attention to the convergence analysis of iterative methods to the solution of a linear system relevant to the same class of applications.