Well-posedness and regularity in thermodynamics of compressible fluid-structure interactions

Abstract

We consider the interaction of a general viscous compressible and heat-conducting fluid with an elastic shell located at the boundary of the fluid's domain. The fluid is described by the compressible Navier-Stokes-Fourier equations and the shell evolves in accordance with a viscoelastic beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. Our first result is the local-in-time well-posedness of the underlying coupled system of nonlinear PDEs in smooth function spaces. Eventually, we prove a conditional regularity criterion which is in the spirit of the Beale-Kato-Majda condition for the incompressible Euler equations combined with the boundedness of density and temperature. It rests upon a weak-strong uniqueness result for the underlying system.