Moving boundary problems

. Moving boundary problems are ubiquitous in nature, technology, and engineering. Examples include the human heart and heart valves inter- acting with blood flow, biodegradable microbeads swimming in water to clean up water pollution, a micro camera in the human intestine used for early colon cancer detection, and the design of next-generation vascular stents to prop open clogged arteries and to prevent heart attacks. These are time-dependent, dynamic processes, which involve the interaction between fluids and various structures. Analysis and numerical simulation of fluid-structure interaction (FSI) problems can provide insight into the “invisible” properties of flows and structures, and can be used to advance design of novel technologies and im-prove the understanding of many physical and biological phenomena. Math- ematical analysis of FSI models is at the core of this understanding. In this paper we give a brief survey of recent progress in the area of mathematical well-posedness for moving boundary problems describing fluid-structure interaction between incompressible, viscous fluids and elastic, viscoelastic, and rigid solids.