Non linear Schrödinger equations and quantum fluids non vanishing at infinity: incompressible limits and quantum vortices

This thesis concerns the mathematical analysis of some hydrodynamic models describing quantum fluids, namely fluids whose macroscopic behavior still exhibits quantum effects. The prototype model for such fluids is the quantum hydrodynamic (QHD) system arising as model in the description of phenomena like superfluidity, Bose-Einstein condensation (BEC), superconductivity, quantum plasmas and semi conductor devices. From a mathematical point of view, the QHD system is given by a compressible Euler system augmented by a stress tensor accounting for the quantum features in the fluid and which depends on the density and its derivatives. Stress tensors of this kind also appear in the theory of capillarity developed by Korteweg, one refers to these systems as Euler–Korteweg and Navier–Stokes-Korteweg systems when inviscid or viscous respectively. Motivated by the analysis of some physically relevant solutions like quantized vortices, we consider the system on the whole space complemented with non-zero boundary conditions at infinity. The Cauchy problem for finite and infinite energy weak solutions (including vortex solutions) is investigated. Our method relies on the equivalence between QHD systems and NLS type equations through the Madelung transforms and the polar factorization method that renders the equivalence rigorous for rough solutions. We are thus led to study the well-posedness theory in the energy space for a class of nonlinear Schrödinger equations with non-zero boundary conditions at infinity that we develop ad-hoc. Moreover, we consider the asymptotic behavior of weak solutions to the QHD system in a suitable scaling regime that is linked to the study of quantized vortices and can be interpreted as quantum counterpart of the low Mach number limit of classical fluid dynamics. The dispersion relation of acoustic waves turns out to be characterized by the Bogoliubov dispersion relation. In the scaling regime, the dynamics of vortex solutions can be asymptotically described by the Kirchhoff-Onsager ODE system. Secondly, we study the quantum Navier-Stokes (QNS) equations that can be understood as a viscous regularization of the QHD system with density dependent viscosity tensor. Physically, it is motivated by applications in the modeling of dissipative quantum fluids and as a showcase model for capillary fluids. We introduce global existence of finite energy weak solutions of the quantum Navier-Stokes system with non-trivial far-field behavior. In contrast to the results for the QHD system, the analysis of the QNS system is entirely based on techniques from fluid dynamics. Finally, we investigate the low Mach number limit and prove strong convergence to weak solutions of the incompressible Navier-Stokes equations for general ill-prepared data.