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.

中文翻译：

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能量通过圆上某些共振偏微分方程中的快速扩散通道传递

在本文中，我们考虑具有非凸（关于动作）一阶共振哈密顿量的圆上的两类共振哈密顿偏微分方程。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. 我们提供的解决方案密切遵循非线性谐振模型的轨道，这是完整方程的良好近似。共振哈密顿量的非凸性允许存在快速扩散通道共振模型的轨道在最佳时间内经历了大的动作漂移。这种现象会引起解的傅立叶模式之间的能量转移，这反过来又是导致高阶 Sobolev 范数增长的原因。