Abstract: I will discuss the existence of turbulent solutions to a quantum hydrodynamic (QHD) system, with periodic boundary conditions. A suitable nonlinear change of variables (the Madelung transform) formally connects the QHD system to a non-linear Schrödinger (NLS) equation, for which we can construct smooth solutions displaying arbitrarily large growth of Sobolev norms above the energy regularity level. This amounts to a cascade in time of the energy to higher Fourier modes. In addition, these solutions can be designed to be small amplitude perturbations of stationary states, which implies in particular absence of quantum vortices. This allows to exploit an equivalence between high regularity QHD- and NLS- norms, which eventually yields the existence of smooth, turbulent solutions to the quantum hydrodynamic system.
Based on joint work with F. Giuliani.

[Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022CHELC7 "Singular Interactions and Effective Models in Mathematical Physics" finanziato dall’Unione europea – Next Generation EU.]

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