Energy cascades

We prove growth of Sobolev norms for time dependently perturbed quantum harmonic oscillators in higher dimensions.

We prove growth of Sobolev norms for a quasilinear NLS with fractional dispersion, introducing a novel paradigm to obtain energy cascades.

One of the most important problem in physics, mathematical analysis and engineering is to study how energy is transferred and redistributed from macro to micro scales in a deterministic system. This question is central to understand the insurgence of turbulent dynamics. In fluids, it is characterized by a nonlinear transfer of energy from low to high Fourier modes, in a process called cascade, which is the result of non-linear resonant interactions between different wave components. The goal of this project is to study cascades formation in dispersive Hamiltonian fluids.