Quantum hydrodynamics, Wigner transforms and the classical limit

We analyse the classical limit of the quantum hydrodynamic equations as the Planck constant tends to zero. The equations have the form of an Euler system with a constant pressure and a dispersive regularisation term, which (formally) tends to zero in the classical limit. The main tool of the analysis is the exploitation of a kinetic equation, which lies behind the quantum hydrodynamic system. The presented analysis can also be interpreted as an alternative approach to the geometrical optics (WKB)-analysis of the Schrödinger equation.