Title:

Global existence and stability for some models in quantum fluid dynamics
Abstract:
A quantum fluid consists of a system of interacting particles characterized by the fact that the effects of quantum statistics remain non-negligible also on a macroscopic scale. Typical examples comprise superfluid Helium II or atomic Bose-Einstein condensates. Nevertheless, the expression “quantum fluid” is also used in a broader sense, describing similar phenomena, not necessarily occurring at ultracold temperatures: this is the case for instance of quasiparticle condensates, electron transport in semiconductor devices, or quantum plasmas.
From the mathematical point of view, the prototypical example of such fluid dynamical systems is the so called quantum hydrodynamics (QHD) system, describing a compressible, inviscid fluid, subject to a stress tensor depending on the particle density and its derivatives.

In this talk I will review some analytical results about global in time weak solutions to the QHD system, obtained in collaboration with Pierangelo Marcati and Hao Zheng. In particular, I will address the issue of existence of such solutions and their stability property. By introducing the polar factorization and the wave function lifting methods, it is possible to rigorously establish an analogy between the wave function dynamics, given by a nonlinear Schrödinger equation, and the QHD system, describing the dynamics of the physical observables, associated to the wave function and formally defined through the Madelung transform. Moreover, we define a functional, formally controlling the L^2 norm of the chemical potential with respect to the probability measure related to the mass density. By exploiting the a priori estimates given by the total energy and the new functional, it is then possible to show that any sequence of uniformly bounded weak solutions has a strongly converging subsequence and that the limit is indeed a weak solution to the QHD system.

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