We present some considerations on time-oscillatory phenomena in hydrodynamical models for semiconductors and study the existence of periodic solutions. For the one-dimensional, viscous, isentropic model, written in Lagrangian mass coordinates, we state a first existence result and give a sketch of the proof.

The existence of global smooth solutions to the multi-dimensional hydrodynamic model for plasmas of electrons and positively charged ions is shown under the assumption that the initial densities are close to a constant. The model consists of the conservation laws for the particle densities and the current densities, coupled to the Poisson equation for the electrostatic potential. Furthermore, it is proved that the particle densities converge exponentially fast to the (constant) steady state. The proof uses a higher-order energy method inspired from extended thermodynamics.

We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal.

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multiphysics approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that combines the differential-algebraic network equations of the circuit with elliptic boundary value problems modeling the diodes. For this mixed initial-boundary value problem of partial differential-algebraic equations a first existence result is given.

The global existence of smooth solutions of the Cauchy problem for the $N$-dimensional Euler-Poisson model for semiconductors is established, under the assumption that the initial data is a perturbation of a stationary solution of the drift-diffusion equations with zero electron velocity, which is proved to be unique. The resulting evolutionary solutions converge asymptotically in time to the unperturbed state. The singular relaxation limit is also discussed.

We perform a multiple time scale, single space scale analysis of a compressible fluid in a time-dependent domain, when the time variations of the boundary are small with respect to the acoustic velocity. We introduce an average operator with respect to the fast time. The averaged leading order variables satisfy modified incompressible equations, which are coupled to linear acoustic equations with respect to the fast time. We discuss possible initial-boundary data for the asymptotic equations inherited from the initial-boundary data for the compressible equations.