An Intrinsically Hydrodynamic Approach to Multidimensional QHD Systems

In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present continues analysis initiated in Antonelli et al. (Commun Math Phys 383:2113–2161, 2021) where one dimensional case was investigated. Here extend to problem, particular considering two physically relevant classes of solutions. First all two-dimensional initial data endowed with point vortices; assuming continuity mass density and a quantization rule for vorticity, are able study Cauchy problem provide global finite energy weak same result can be obtained also spherically symmetric setting. For rough solutions energy, suitable dispersive estimates, which apply more general class Euler–Korteweg equations. Moreover show sequential stability positive density. Analogously case, is achieved through priori bounds given new functional, first introduced (2021).