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

Filippo Giuliani

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arXiv · 2009.04228

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

Filippo Giuliani

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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.

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Filippo Giuliani. 2021-03-12. Transfers of energy through fast diffusion channels in some resonant PDEs on the circle. https://arxiv.org/abs/2009.04228

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