Análisis y control de problemas de tipo Navier-Stokes

Abstract

This work is framed within the field of EDPs and Numerical Analysis, aiming to continue the study of the Navier-Stokes equations which was already started on the subjects “EDPyA”, “ANED” and “MySN”. It is meant for deepening our knowledge on the regularity and control of the Navier-Stokes equations. Additionally, an analogous study of the Boussinesq system is carried out. Throughout this text we broach the existence, uniqueness and regularity of the Navier-Stokes equations as well as the Boussinesq system. The article of Caffarelli, Kohn and Nirenberg plays a central role in the research of the regularity of Navier-Stokes equations. Besides the above-mentioned aspects, various numerical methods to approximate the solutions to the Navier-Stokes equations and Boussinesq system are wide developed. The applicability of the results is exemplified by studying the simulations of these methods. The text concludes with a theoretical analysis of the control of the Navier-Stokes equations.