Some contributions to the study of incompressible micropolar fluids in two dimensional domains - CUNHA, Cleyton Natanael Lopes de Carvalho (2020)
This thesis deals with some theoretical aspects related to the two-dimensional incompressible micropolar fluids model. In particular, two problems were addressed. The first of them is known in the literature as the initialization problem. The fundamental idea of this type of problem is to recover information from the initial data based on observations of the state of the system. For this purpose, a bounded and smooth domain of R2 with Dirichlet boundary conditions was considered. Optimal control theory techniques ensure the existence of at least one and at most a finite number of solutions for the problem. We also provide sufficient conditions to guarantee uniqueness of the solution. The second problem was the study of well posedness, in Hadamard’s sense, for the micropolar model with partial viscosity and singular initial data, including the possibility of measures as initial data in Morrey spaces. Through integral techniques and compactness arguments, the existence of a weak solution was established. Uniqueness and stability of these solutions were also analyzed, showing the model to be well posed.