: The first part of the talk will provide a self-contained introduction to the formal deduction of the quantum hydrodynamics (QHD) in the form of an Euler Dispersive Irrotational compressible fluid system.
We will then introduce in a possible simple way the a macroscopic approach to various related to superfluids due to the Russian school by Landau, Khalaktikov and other important models in superconductivity and semiconductor devices.
Then we show important mathematical difficulties related to the presence of vacuum and explain their physical counterpart, even in relation with quantum vortices and Bose Einstein condensation. We also will relate our analysis to the Gross-Pitaevskii model.
We will show how to develop a mathematical theory in 2-D and 3-D, consistent with the physics for the problem of large data weak solutions, in the energy norm. All the theory, including irrotationality, is formulated by using observable quantities (density and linear momentum), avoiding the need of defining the velocity fields. The methods are based on a ad hoc" polar factorization, dispersive analysis and local smoothing. The initial data are restricted to be momenta of a wave funtiona.
It has been recently developed a large data 1-D theory purely hydrodynamical of weal solutions with strong stability.
In conclusion we will just mention several other related problems connected to quantum vortices, dispersive shocks and the presence of magnetic fields.