Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2020-10-22 , DOI: 10.1007/s10884-020-09904-w Pietro Baldi , Emanuele Haus
Consider the Kirchhoff equation
$$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$on the d-dimensional torus \(\mathbb {T}^d\). In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.
中文翻译：
关于基尔霍夫方程的范式
考虑基尔霍夫方程
$$ \ begin {aligned} \ partial _ {tt} u-\ Delta u \ Big（1 + \ int _ {\ mathbb {T} ^ d} | \ nabla u | ^ 2 \ Big）= 0 \ end {已对齐} $$在d维环面\（\ mathbb {T} ^ d \）上。在先前的论文中，我们证明了，在拟线性法线形式的第一步之后，共振立方项表现出可积性，即它们对能量估计没有任何贡献。这就提出了一个问题，即在正常形式的后续步骤中是否还会出现相同的结构。在本文中，我们执行第二步，并对先前的问题给出否定的答案：五次共振项对能量估计值贡献非零。这不仅是形式上的计算，因为我们证明了正规形式变换在Sobolev空间之间是有界的。
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