论文标题
通过快速扩散通道的某些共振PDES的快速扩散通道传输
Transfers of energy through fast diffusion channels in some resonant PDEs on the circle
论文作者
论文摘要
在本文中，我们在圆圈上考虑了两类的共鸣汉密尔顿PDE，它具有非凸（针对动作）一阶共振汉密尔顿人。我们表明，对于非线性的适当选择，我们可以找到与时间无关的线性电势，从而可以构建解决方案的解决方案，从而在Sobolev规范中经历规定的增长。我们提供的解决方案非常遵循非线性谐振模型的轨道，这是完整方程式的良好近似值。共振汉密尔顿共振的非体积允许存在快速扩散通道，谐振模型的轨道在最佳时间内会在动作中经历大量漂移。这种现象引起了能量在溶液的傅立叶模式之间的转移，而溶液的傅立叶模式又导致了高阶Sobolev规范的增长。
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.
