A variational approach to hyperbolic evolutions and fluid-structure interactions

Abstract

We show the existence of a weak solution for a system of partial differential equations describing the motion of a flexible solid inside a fluid: A nonlinear, viscoelastic, $n$-dimensional bulk solid governed by a PDE including inertia is interacting with an incompressible fluid governed by the ($n$-dimensional) Navier–Stokes equation for $n\ge 2$. The result is the first allowing for large bulk deformations in the regime of long time existence for fluid-structure interactions. The existence is achieved by introducing a novel variational scheme involving two time-scales that allows us to extend the method of minimizing movements to hyperbolic problems involving nonconvex and degenerate energies.