中心流形形式级数展开的改进 Gevrey-1 估计
Improved Gevrey-1 estimates of formal series expansions of center manifolds
AI总结：
本文结合既有方法与 Borel–Laplace 方法并改进 Banach 范数，证明平面解析鞍结中心流形形式级数系数具有不可改进的 Γ(n+a) 型 Gevrey-1 增长。
摘要：
本文证明，对于平面解析鞍结的中心流形，其形式级数展开 y=∑_{n=1}^∞ ϕ_n x^n∈x C[[x]] 的系数 ϕ_n 在重新标度 x 后，当 n→∞ 时按 Γ(n+a) 增长。其中，量 a 是与该鞍结相关的形式解析不变量（遵循 J. Martinet 与 J.-P. Ramis 的工作）。ϕ_n 的这一增长性质在中心流形非解析时是不可改进的；作者最近（2024 年）曾与 P. Szmolyan 合作，针对一类受限的非线性项描述了该性质。这项合作工作反过来又受到 Merle、Raphaël、Rodnianski 和 Szeftel（2022 年）工作的启发，后者描述了一个与可压缩 Euler 方程自相似解相关系统的系数增长。本文将以往方法与 Borel–Laplace 方法相结合。具体而言，我们调整了 Bonckaert 与 De Maesschalck（2008 年）的 Banach 范数，以捕捉复平面中的奇性。
英文摘要：
In this paper, we show that the coefficients $ϕ_n$ of the formal series expansions $y=\sum_{n=1}^\infty ϕ_n x^n\in x\mathbb C[[x]]$ of center manifolds of planar analytic saddle-nodes grow like $Γ(n+a)$ (after rescaling $x$) as $n\rightarrow \infty$. Here the quantity $a$ is the formal analytic invariant associated with the saddle node (following the work of J. Martinet and J.-P. Ramis). This growth property of $ϕ_n$, which cannot be improved when the center manifold is nonanalytic, was recently (2024) described for a restricted class of nonlinearities by the present author in collaboration with P. Szmolyan. This joint work was in turn inspired by the work of Merle, Raphaël, Rodnianski, and Szeftel (2022), which described the growth of the coefficients for a system related to self-similar solutions of the compressible Euler. In the present paper, we combine the previous approaches with a Borel-Laplace approach. Specifically, we adapt the Banach norm of Bonckaert and De Maesschalck (2008) in order to capture the singularity in the complex plane.
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