Qualitative properties of localized dissipative systems

• Ano de defesa: 2022
• Autor(a) principal: Sozzo, Bruna Thaís Silva
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Laboratório Nacional de Computação Científica; Coordenação de Pós-Graduação e Aperfeiçoamento (COPGA); Brasil; LNCC; Programa de Pós-Graduação em Modelagem Computacional
• Palavras-chave em Português: Diferenciabilidade; Dissipação localizada; Classe de Gevrey; Teoria de semigrupos; Equação diferencial parcial; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE
• Link de acesso: https://tede.lncc.br/handle/tede/360

The elimination of noise and vibration effects in elastic structures is an important topic in materials science. The way these structures are built has a direct impact on minimizing the vibration effects. Then, it is important to study structures with more than one type of material, i.e., models in which the dissipative mechanism differs or acts only on a part of the material. Models describing dissipative systems acting on the whole domain are well-studied and their qualitative properties (asymptotic behavior) are well-known. On the other hand, models that exhibit localized dissipative mechanisms demand further studies. Regarding to differentiability and analyticity properties, we have not found works that have heterogeneous domains. For this reason, in this Ph.D. dissertation, we focus on working with models that have heterogeneous domains and investigate qualitative properties of the model solution. In this case, we are concerned with concepts of differentiability and analyticity. First, we assume that the Euler-Bernoulli beam model is composed of two materials, one of which is an elastic material and the other is a Kelvin-Voigt type viscoelastic material. In the second model, we use the Euler-Bernoulli model to study the vibrations of a beam composed of two components, one of which is a thermoelastic material and the other is an elastic material that does not generate dissipation. In the third model, we study qualitative properties for a plate with a thermoelastic dissipative mechanism only in a part of the domain. Using semigroup theory and the functional analysis tools, we prove that the models are defined by an immediately differentiable semigroup belonging to the Gevrey class. In particular, the models are exponentially stable, have the linear stability property, and have a smoothing effect property on the initial data. The results for the Euler-Bernoulli beam models are new results that have been accepted in journals with great impact factors. The result found for the plate model is also new and is in the submission process. According to our research, these are the first papers in the literature showing this type of qualitative results for models with localized dissipative mechanisms. The qualitative properties found not only represent important results for materials science, but are also significant for working with numerical approximations.