5. Deterministic FSI with Nonlinear Coupling

This chapter focuses on the existence of weak solutions for nonlinearly coupled deterministic fluid-structure interaction (FSI) problems. The analysis centers around the challenges posed by moving fluid domains and the geometric nonlinearity of coupling conditions. The chapter employs the Lie operator splitting strategy to construct approximate solutions and provides a rigorous proof of the existence of weak solutions in finite energy spaces. Key topics include the prototypical nonlinearly coupled FSI problem, the existence of weak solutions, and the use of Lie operator splitting to handle the moving boundary. The chapter concludes with a discussion on the well-posedness of the problem and the conditions under which the solution exists. The detailed exploration of the geometric nonlinearity and the rigorous proof of existence make this chapter a valuable resource for professionals in the field of applied mathematics and fluid dynamics.