论文标题
光束方程的亚指数稳定性
Sub-exponential stability for the Beam equation
论文作者
论文摘要
我们考虑在周期性边界条件下，在一个空间维度下具有哈密顿非线性的一部分梁方程家族。在一个统一的功能框架中，我们研究了两类可不同性的初始数据的长时间演变：（i）Sobolev规则性的子空间，（ii）无限多个可微分函数的子空间严格包含在Sobolev空间中，但严格包含Gevrey。在这两种情况下，我们都证明了指数类型的稳定时间。结果几乎适用于所有质量参数，并且可以通过将正常形式技术与合适的二氧化苯胺条件相结合的弱（比波尔加因提出的弱苯胺条件弱）获得。对于退化方程式，这是Sobolev规则性的第一个结果，其中仅使用一个参数来调整振荡的线性频率。
We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two categories of differentiability: (i) a subspace of Sobolev regularity, (ii) a subspace of infinitely many differentiable functions which is strictly contained in the Sobolev space but which strictly contains the Gevrey one. In both cases we prove exponential type times of stability. The result holds for almost all mass parameters and it is obtained by combining normal form techniques with a suitable Diophantine condition weaker than the one proposed by Bourgain. This is the first result of this kind in Sobolev regularity for a degenerate equation, where only one parameter is used to tune the linear frequencies of oscillations.
