Semiclassical Limit for One-dimensional Viscous Quantum Hydrodynamic Model

The paper deals with a viscous quantum hydrodynamic model in one dimension, which has been recently investigated by Gamba, Jungel and Vasseur (cfr. ref [2]). It is shown that the semiclassical limit of the underlying system can be performed, giving in the limit a system in which the quantum term is no more present. The evaluation of the semiclassical limit requires a new estimate on the square root of the solution, apparently not available in paper [2], which is proved in Lemma 2.2. This gives a uniform control on the quantum term which allows the passage to the limit. This limiting process describes the relation from quantum mechanics to the classical Newtonian mechanics.