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 paper continues the analysis initiated in Antonelli et al. (Commun Math Phys 383:2113–2161, 2021) where the one dimensional case was investigated. 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 Antonelli et al. (2021).