ASYMPTOTIC LIMITS FOR QUANTUM TRAJECTORY MODELS

The semi-classical and the inviscid limit in quantum trajectory models given by a one-dimensional steady-state hydrodynamic system for quantum fluids are rigorously performed. The model consists of the momentum equation for the particle density in a bounded domain, with prescribed current density, and the Poisson equation for the electrostatic potential. The momentum equation can be written as a dispersive third-order differential equation which may include viscous terms. It is shown that the semi-classical and inviscid limit commute for sufficiently small data (i.e. current density) corresponding to subsonic states, where the inviscid non-dispersive solution is regular. In addition, we show that these limits do not commute in general. The proofs are based on a reformulation of the problem as a singular second-order elliptic system and on elliptic and W ^1,1 estimates.

Keywords

• Semi-classical limit,
• Inviscid limit,
• Quantum hydrodynamic equations,
• Semiconductors