Arbitrarily large growth of Sobolev norms for a quantum Euler system

In this talk we present a result of existence of solutions to the quantum hydrodynamic (QHD) system, under periodic boundary conditions, which undergo an arbitrarily large growth of higher order Sobolev norms in polynomial times.
The proof is based on the connection between the QHD system and the cubic NLS equation, provided by the Madelung transform. We show that the cubic NLS equation on the two dimensional torus possesses solutions which starts close to plane waves and undergo an arbitrarily large growth of higher order Sobolev norms in polynomial times. This is an improvement of the result by Guardia-Hani-Haus-Maserp-Procesi (JEMS 2023) and it is achieved by a refined normal form approach.
Then we show that the existence of such solutions to NLS implies the existence of solutions to the QHD system exhibiting a large growth in Sobolev norms.