Kirchhoff方程的许多解的更长寿命
Longer lifespan for many solutions of the Kirchhoff equation
摘要：
本文研究了$d$维环面$\mathbb T^d$上的Kirchhoff方程$$ \partial_{tt} u - Δu \Big( 1 + \int_{\mathbb T^d} |\nabla u|^2 \Big) = 0 $$及其初值为$u(0,x)$，$\partial_t u(0,x)$，大小为$\varepsilon$的Sobolev类Cauchy问题.在五阶动力学的有效方程，在以前的论文中获得的准线性规范形式，包含共振对应的非平凡项的能量估计。这种共振不能通过调整外部参数来避免（简单地说，因为基尔霍夫方程不包含参数）。本文在初值上引入了非共振条件，并证明了相应解的寿命的一个下界$\varepsilon^{-6}$（标准局部理论给出$\varepsilon^{-2}$，三次项的规范形式给出$\varepsilon^{-4}$）.证明依赖于这样一个事实，即在这些非共振条件下，在大的时间间隔上的有效方程的“叠加”的增长率小于（由一个因子$\varepsilon^2$）比其先验估计的基础上的正常形式的三次条款。满足这样的非共振条件的初始数据集包含几个非平凡的例子，在本文中讨论。
英文摘要：
We consider the Kirchhoff equation $$ \partial_{tt} u - Δu \Big( 1 + \int_{\mathbb T^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb T^d$, and its Cauchy problem with initial data $u(0,x)$, $\partial_t u(0,x)$ of size $\varepsilon$ in Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound $\varepsilon^{-6}$ for the lifespan of the corresponding solutions (the standard local theory gives $\varepsilon^{-2}$, and the normal form for the cubic terms gives $\varepsilon^{-4}$). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the "superactions" of the effective equations on large time intervals is smaller (by a factor $\varepsilon^2$) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper.
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