9. Stochastic FSI: A Fully Coupled Model with Nonlinear Coupling

The chapter focuses on the stochastic fluid-structure interaction (FSI) problem, particularly in the context of hemodynamics. It explores the stochastic fluctuations in the systolic and diastolic rhythm of the heart, which give rise to stochasticity in the FSI describing the flow of blood in coronary arteries. The text delves into the existence of weak martingale solutions to a nonlinearly coupled stochastic FSI problem, using a Lie-Trotter splitting scheme. It also discusses the application of Arbitrary Lagrangian-Eulerian (ALE) mappings to handle the geometric nonlinearity arising from the motion of the fluid domain. The chapter provides a detailed analysis of the stochastic framework and the definition of martingale solutions, along with the construction of approximate solutions and the passing to the limit as the time-step size goes to zero. Additionally, it covers the existence of solutions to the structure and fluid subproblems, and the uniform estimates on the expectation of the kinetic and elastic energy for the full, semidiscrete coupled problem. The chapter concludes with a discussion on the stopping time argument and the agreement of the structure and artificial structure displacements, summarizing the findings in the context of the stochastic FSI problem.