4. Deterministic FSI with Linear Coupling

This chapter delves into the mathematical analysis of fluid-structure interaction (FSI) through linearly coupled models, which describe the interaction between a fluid and a structure with linear approximations. The text focuses on the application of these models in fields such as aerodynamics, bioengineering, and engineering, where the fluid's impact on the structure or the structure's deformation is relatively small. A prototypical model is presented, involving the interaction between a two-dimensional incompressible Newtonian fluid and a one-dimensional elastic structure, with the fluid flowing through a two-dimensional channel with elastic top and bottom walls. The chapter explores the partial differential equations governing the dynamics of the fluid and structure separately and discusses the conditions that describe their coupling at the interface. The Lie operator splitting approach is introduced as a constructive proof of the existence of weak solutions, highlighting its importance in analyzing models presented later in the text. The chapter also provides an a priori energy estimate and weak formulation of the problem, culminating in a main existence result that demonstrates the existence of a unique weak solution to the linearly coupled FSI problem. The text concludes with a discussion on the robustness of the Lie splitting strategy and its applicability to a large family of FSI problems.