A fluid–structure model coupling the Navier–Stokes equations and the Lamé system

Abstract

We study a fluid–structure system describing the motion of an elastic solid inside an incompressible viscous fluid in three dimensions. The motion of the solid is described by the Lamé system of linear elasticity and the fluid obeys the incompressible Navier–Stokes equations in a time-dependent domain. At the fluid–solid interface, natural conditions are imposed, continuity of the velocities and of the Cauchy stress forces. The fluid and the solid are coupled through these conditions. By this interaction, the fluid deforms the boundary of the solid which in turn influences the fluid motion. We prove the existence and uniqueness of local strong solutions by rewriting the coupled system in Lagrangian variables and by using the method of successive approximations. Our analysis relies on new regularity results for the linearized coupled system. In particular, if u˜ is the Lagrangian velocity of our system, we have to estimate separately Pu˜ and (I−P)u˜, where P is the Leray projector for the fluid model. Moreover, we prove new hidden regularity results for the Lamé system, that we are able to establish only in the case when the interface, in the reference configuration, between the fluid and the solid is flat. Due to this restriction, the model is completed by periodic boundary conditions in two directions. The other results may be transposed to more general configurations.