Long-term Dynamics for Living Fluids and Heterogeneous Catalysis and an Approach to the Stokes Equations via Duality Scales

In this thesis we consider (in)stability and long-term behavior of a living fluids model, stability of a model for the heterogenous catalysis process and as a last topic the use of duality scales on complemented subspaces with regard to partial differential equations. The first model to be considered, a living fluids model, is given as generalized Navier-Stokes equations and describes dense bacterial suspensions at low Reynolds number. We establish a complete analysis of linear and nonlinear stability and instability in the periodic L²-setting about the two relevant types of equilibria and find parameter sets corresponding to stability and instability. Afterwards, we show that the living fluids model possesses a global attractor of finite dimension and arbitrary high regularity. The second model considered in this thesis stems from chemical engineering and describes the process of heterogeneous catalysis in a cylinder-shaped domain. Since the catalysis considered is heterogeneous, we assume the catalyzer to be on the lateral boundary of the cylinder, which results in a coupled system of equations in the bulk and on the lateral boundary. We show stability and instability for the heterogeneous catalysis model in the Lp-setting dependent on the chemical reaction which is chosen on the lateral boundary. In the last part of this thesis we consider the concept of duality scales of Banach spaces, which gives a more precise meaning to the concept of duality. We show that under certain assumptions regarding a consistent projection P on these scales, the property of being a duality scale is preserved if we consider the complemented subspaces and apply the result to the Stokes operator.