论文标题
在基尔chhoff方程的正常形式下
On the normal form of the Kirchhoff equation
论文作者
论文摘要
考虑Kirchhoff方程$$ \ partial_ {tt} u-ΔU\ big（1 + \ int _ {\ Mathbb {t}^d} | \ nabla u |^2 \ big）= 0 $ big）= 0 $ $在上一篇论文中，我们证明，在准线性正常形式的第一步之后，谐振的立方术语显示了可整合的行为，即对能量估计没有任何贡献。这导致了一个问题，同一结构是否也在正常形式的下一步处出现。在本文中，我们执行第二步，并对上一个问题给出负面答案：五重共振术语对能量估计给出了非零的贡献。这不仅是正式计算，因为我们证明了正常形式的转换在索博尔夫空间之间。
Consider the Kirchhoff equation $$ \partial_{tt} u - Δu \Big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \Big) = 0 $$ 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.
