BZ Seminar in Analysis: Paolo Antonelli (GSSI L'Aquila)

The Cauchy problem for the quantum hydrodynamics system and the stability of weak solutions

The quantum hydrodynamics (QHD) system is a model used to describe macroscopic objects exhibiting quantum features, as for instance in superfluidity, Bose-Einstein condensation or semiconductor devices. They are characterized by a compressible, inviscid flow subject to a stress tensor depending on the particle density and its derivatives, that adds dispersive effects in the system. The Cauchy theory for finite energy weak solutions exploits the analogy between the QHD system and nonlinear Schrödinger (NLS) dynamics, through the Madelung transform. By using a polar factorization approach it is possible to make this analogy rigorous in the space of finite energy weak solutions. Furthermore the hydrodynamic formulation allows to include some dissipative effects which cannot be treated with the NLS dynamics. The polar decomposition can also be exploited to study the problem with non-trivial conditions at infinity. This analysis is motivated by future applications to quantized vortex dynamics. The main drawback of this approach is that the initial data for the hydrodynamical system need to be consistent with a wave function by means of the Madelung transform; at present it is not clear what is the degree of generality of such initial data. We show that in the one-dimensional case we can eliminate this assumption and we prove global existence of finite energy weak solutions to the QHD system with general initial data. Moreover, by introducing a novel functional we are also able to determine a class of solutions for which we can prove stability, namely given a sequence of solutions satisfying some uniform bounds there is a subsequence converging to a weak solution to the QHD system. This talk is based on a series of works done in collaboration with P. Marcati and our PhD students L. E. Hientzsch and H. Zheng.

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