  In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD)
system, by adopting an intrinsically hydrodynamic approach. The present work
continues the analysis initiated in [6] where the one dimensional case was
studied. Here we extend the analysis to the multi-dimensional problem, in
particular by considering two physically relevant classes of solutions. First
of all we consider two-dimensional initial data endowed with point vortices; by
assuming the continuity of the mass density and a quantization rule for the
vorticity we are able to study the Cauchy problem and provide global finite
energy weak solutions. The same result can be obtained also by considering
spherically symmetric initial data in the multi-dimensional setting. For rough
solutions with finite energy, we are able to provide suitable dispersive
estimates, which also apply to a more general class of Euler-Korteweg
equations. Moreover we are also able to show the sequential stability of weak
solutions with positive density. Analogously to the one dimensional case this
is achieved through the a priori bounds given by a new functional first
introduced in [6].
