Theoretical and numerical analysis and control of some PDEs

Abstract

This dissertation presents theoretical and numerical results for some Partial Differential Equations (PDEs). For the distinction of the topics addressed during the development of this work, we have divided its goals into four parts. Precisely, in the rst two chapters, we focus on the development of some results associated with the k{" model, and in the last two chapters, some numerical results related to the heat and wave equations. The k{" turbulence model is one of the most used models in Computational Fluid Dynamics (CFD) to simulate medium ow characteristics for turbulent ow conditions. It is a model formed by equations of the type Navier-Stokes (called Reynolds equations) coupled to two equations that in general describe the evolution in time, the transport, the di usion and the generation of turbulence. In Chapter 1, we prove a renormalized weak solution result for a simpli ed model from the k{" model. In the following chapters, we present some controllability results for some models. More speci cally, we proof the existence of controls which drive the average eld of velocity from a prescribed initial data to a desired nal state in a positive time given. Furthermore, we present in Chapter 2 a result of the local null partial control for a simpli ed model from the k{" model. On the other hand, aiming the numerical development of hierarchical control problems, Chapters 3 and 4 are presented. These two chapters show the theoretical and numerical developments that were performed, as well as Freefem++ which is a software that was used to present some graphical results of experiments associated with the problems. Particularly in Chapter 3, we develop numerical results in optimal controls for heat equations (linear and semi-linear cases) in combination with the notions of cooperative and non-cooperative games, according to Pareto and Nash strategies, respectively. Motivated by the results found in Chapter 3, we developed Chapter 4, where we obtained similar numerical results of optimal control in bi-objective problems associated with the notions of Nash and Pareto equilibrium for wave equations.