2. Deterministic Preliminaries

This chapter provides a comprehensive overview of the fundamental concepts from real and functional analysis that are essential for the study of coupled fluid-structure dynamics. It begins with an introduction to abstract function spaces, including Banach and Hilbert spaces, and their key properties. The chapter then explores specific examples of these spaces that play a crucial role in the analysis of fluid-structure interaction (FSI), with a focus on spaces and Sobolev spaces. The properties of the Fourier transform are also discussed, highlighting its relevance for studying reduced fluid-structure interaction models. Additionally, the chapter introduces function spaces that account for both spatial and temporal dependence, specifically Bochner spaces, which are essential for analyzing evolutionary partial differential equations (PDEs). The analysis extends to function spaces adapted to moving domains, including extensions of Bochner spaces to accommodate time-dependent geometries and generalizations of classical inequalities. The chapter also discusses compact embeddings and presents several classical results that are crucial for the analysis of fluid-structure interaction. Furthermore, it provides an overview of the fundamental equations governing fluid dynamics and elasticity, discussing the concepts of a priori estimates and finite energy weak solutions. The chapter concludes by introducing two key constructive methods—Rothe’s method and Lie operator splitting—at an abstract level, discussing their role in constructing approximate solutions to partial differential equations and their application to fluid-structure interaction problems.