On the motion of an elastic solid inside of an incompressible viscous fluid
The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a coupled PDE system between parabolic and hyperbolic phases, the latter inducing a loss of regularity. In this talk, I will outline the proof of the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. The functional framework employed is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.