Moving boundary problems are ubiquitous in nature, technology, and engineering. Examples include the human heart and heart valves interacting 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 improve the understanding of many physical and biological phenomena. Mathematical 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.
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