On the Initial-Boundary Value Problem for One-Dimensional Quantum Hydrodynamics

Vortrag von Paolo Antonelli

Datum: 21.05.26  Zeit: 16.15 - 18.00  Raum: Y27H35/36

The Quantum Hydrodynamics (QHD) system is a fundamental model for quantum fluids, such as Bose-Einstein condensates and superfluid Helium. Mathematically, QHD is described by a compressible Euler system augmented by a dispersive quantum stress tensor. While the formal connection to nonlinear Schrödinger (NLS) equations via the Madelung transformation is well-established, the rigorous, general analytical framework relies on the polar factorization method.
In the first part of this talk, I will review existing results for the Cauchy problem. Focusing on the one-dimensional case, I will discuss a wave-function lifting method that, under physically consistent assumptions, reconstructs a wave function directly from hydrodynamic data. This allows for a "genuine" hydrodynamic treatment of the theory without a priori reliance on the Schrödinger picture. Furthermore, I will introduce a novel entropy related to the system’s chemical potential, which provides the necessary a priori estimates to determine a compactness class for 1D weak solutions.
Finally, I will address the initial-boundary value problem (IBVP). I will motivate our choice of boundary conditions and their correspondence to the IBVP for the 1D NLS equation, then present a global existence result for finite-energy weak solutions. I will conclude by discussing potential applications and future directions for this research.