ON SOME RESULTS FOR QUANTUM HYDRODYNAMICAL MODELS (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

In this paper we review some recent results on the existence of finite energy weak solutions to a class of quantum hydrodynamics (QHD) system. Our approach is based on a polar factorization method. This method allows to overcome the mathematical difficulty arising in the classical WKB approach, to define the velocity field inside the vacuum regions. Our methods to show existence of finite energy weak solutions fully exploit the dispersive and the local smoothing properties of the underlying nonlinear Schrödinger evolution in order to establish suitable "a priori" bounds for the hydrodynamical quantities. We finally sketch some new results towards a purely hydrodynamic theory in 1-D and recent developments of a low Mach number analysis of Quantum Vortices.