Transfers of energy through fast diffusion channels in some resonant PDEs on the circle

TL;DR

This paper demonstrates how non-convex resonant Hamiltonian PDEs on the circle can exhibit energy transfer through fast diffusion channels, leading to significant Sobolev norm growth by constructing solutions with prescribed energy transfer.

Contribution

It introduces a method to create solutions with controlled Sobolev norm growth in resonant PDEs using linear potentials and analyzes the role of non-convexity in enabling fast energy diffusion.

Findings

Existence of solutions with prescribed Sobolev norm growth.

Identification of fast diffusion channels in non-convex Hamiltonian PDEs.

Energy transfer among Fourier modes causes high-order Sobolev norm increase.

Abstract

In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of fast diffusion channels along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions which in turn is responsible for the growth of higher order Sobolev norms.