Non-resonant Solutions in Hyperbolic-Parabolic Systems with Periodic Forcing in Bloomington, MN

This thesis is a mathematical investigation of damping effects in hyperbolic systems. In the first part two models from nonlinear acoustics are studied. Existence of time-periodic solutions to the Blackstock-Crighton equation and the Kuznetsov equation are established for time-periodic data sufficiently restricted in size. This leads to the conclusion that the dissipative effects in these models are sufficient to avoid resonance. In the second part the interaction of a viscous fluid with an elastic structure is studied. A periodic cell structure filled with a viscous fluid interacting with a deformable boundary of the cell is considered under time-periodic forcing. The motion of the fluid is governed by the Navier-Stokes equations and the deformable boundary is governed by the plate equation. It is shown that the damping mechanism induced by the viscous fluid is sufficient to avoid resonance in the elastic structure.