Compressible fluids and elastic plates in 2D: a conditional no-contact theorem

We consider the interaction of a compressible fluid with a flexible plate in two space dimensions. The fluid is described by the Navier–Stokes equations in a domain that is evolving in accordance with the motion of the structure. The displacement of the latter evolves according to a beam equation. The two systems are coupled through kinematic boundary conditions and balance of forces. We prove that for any weak solution to the coupled system that additionally satisfies certain regularity conditions, contact between the elastic wall and the bottom of the fluid cavity cannot occur. This conditional no-contact result applies to both isentropic and heat-conducting fluids.