Evolution problems with local/nonlocal coupling

• Autor(a) principal: Santos, Bruna Cassol dos
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45132/tde-05082021-085051/
• topic: Asymptotic behavior; Comportamento assintótico; Difusão não local; Domínios finos; Equação do calor; Heat equation; Métodos numéricos; Nonlocal diffusion; Numerical methods; Thin domains

Classical models, such as Partial Differential Equations (PDE), are widely used for making local approximations even if they have some limitations for capturing long-range effects. On the other side, the modeling of nonlocal effects is getting attention in many applied areas, like ecology, epidemiology, physics, and engineering. The development of a rigorous theoretical and computational framework for nonlocal models is far less developed than its local counterpart. In this work, we propose and study an evolution problem that couple local and nonlocal equations. The local part is classically represented by the Laplacian operator, while the nonlocal part is represented by a diffusion operator with an integrable kernel in convolution form, J(x y). As a first approximation, we study the properties of the model in the one-dimensional case. Results of existence, uniqueness, mass conservation, and asymptotic decay of solutions were verified. Next, we extend these results to higher dimensions. For the one-dimensional case, with the appropriate rescale of the nonlocal kernel, it is possible to recover the heat equation in the whole domain. Next, we continue our analysis of this coupled problem and, taking advantage of the particular coupling structure, we use the Splitting Operator method to provide a different proof of existence and uniqueness. We also develop some numerical experiments to illustrate the obtained theoretical results. Using classical numerical methods for PDE, we check that the solution of the discrete model converges to the mean value of the initial condition (when we assume Neumann type boundary conditions), as we have shown theoretically. Finally, we study the properties of the evolution problem in a thin domain. We consider the limit case when the nonlocal subdomain is narrowed in one direction, making the nonlocal domain concentrates in a set of smaller dimension. In this way, we obtain a model in which the local and nonlocal parts of the problem are defined in subdomains of different dimensions. We also show that the limit problem shares the same properties obtained in the one-dimensional case; existence and uniqueness, mass conservation, comparison, and asymptotic decay of solutions for large times.