1. Introduction and Outline

This monograph delves into the mathematical theory of fluid-structure interaction (FSI) problems, focusing on systems where fluids and structures interact dynamically. It covers both deterministic and stochastic FSI, with a particular emphasis on finite energy (weak) solutions. The text systematically explores the well-posedness of these problems, proving existence, uniqueness, and stability. It introduces fundamental concepts for establishing the existence of weak solutions using time discretization via operator splitting, a method that allows for systematic analysis of these coupled FSI systems. The monograph is divided into four parts: Preliminaries, Deterministic FSI, Stochastic FSI, and FSI involving poroelastic media. Each part builds on the previous one, gradually increasing in complexity. The text concludes with a discussion of the well-posedness theory for fluid-poroelastic structure interaction problems, pushing the boundaries of applicability of weak solutions theory to studying FSI involving bulk poroelasticity.