In this three-part lecture series, we will discuss the fundamental role that the mathematics of free- and moving-boundary problems plays in the life sciences. We will begin with classical examples, such as blood flow in elastic and viscoelastic arteries, and conclude with a cutting-edge application in the design of bioartificial organs.

To describe these phenomena, we will develop a mathematical framework that couples incompressible viscous flows (governed by the Stokes and Navier–Stokes equations) with various structural models, including elastic shells, membranes, and poroelastic solids described by the Biot equations.

The resulting systems are highly nonlinear, multiphysics PDE problems posed on domains that are not known a priori, presenting significant mathematical and computational challenges. Throughout the lectures, I will present benchmark models and survey well-posedness results within both deterministic and stochastic frameworks. We will consider both linearly and nonlinearly coupled problems, which typically give rise to mixed hyperbolic–parabolic systems defined on moving domains.

In addition, we will discuss a classical simplified one-dimensional averaged fluid–structure interaction model—a hyperbolic system analogous to the shallow water equations or the p-system of isentropic gas dynamics. 

We will conclude with a survey of some of the most pressing open problems in the field.