Mathematical Analysis Seminars @ DISIM

Title: Growth of Sobolev norms for dispersive hydrodynamic models

Abstract: I discuss the existence of turbulent solutions for a quantum hydrodynamic (QHD) system, with periodic boundary conditions. A suitable nonlinear change of variables formally connects the QHD system to a semilinear 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, a typical phenomenon predicted by the weak turbulence theory. The unstable 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. Finally, I discuss partial extensions to the Euler-Korteweg system, which is formally connected to a quasilinear NLS.

Based on joint works with Y. Cacchiò, F. Iandoli and F. Giuliani.