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

The author is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 757802

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  • 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.

    Mathematics Subject Classification: Primary: 37K45, 35B34; Secondary: 35B35.

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