重整化不变流形

中文摘要 AI 辅助

偏微分方程（PDE）的不变流形理论在技术上具有挑战性，因为我们常常缺乏谱间隙。受数学各领域近期利用重整化取得进展的启发，我们针对半线性抛物方程引入了一种谱重整化方法，该方法在高频处人为地制造一个谱间隙，同时保持低模态不变。对于由此产生的重整化方程族，经典惯性流形理论给出了有限维不变流形，其维数随着重整化参数趋于零而发散。我们证明，这些流形上的约化动力学通过近似半共轭逼近原始半流，在有界有限时间区间上的误差为$O(\epsilon^2)$。在全局增量耗散条件下，半共轭在时间上是一致的，并给出全局吸引子（若存在）的$O(\epsilon^2)$豪斯多夫估计。我们还建立了谱重整化的进一步动力学性质，如双曲平衡点的持久性及其莫尔斯指标。我们在有和没有自然谱间隙的情况下说明了该构造，并将其嵌入一个双参数快慢PDE框架中，其中人工间隙使我们能够为有界域上任意维数的系统构造慢流形。因此，我们的新方法即使在原始方程不存在精确惯性流形的区域也提供了有限维动力学近似。特别是，这在完全不变流形和直接估计之间提供了一种实用的平衡。

英文摘要

Invariant manifold theory for partial differential equations (PDEs) is technically challenging as we often lack spectral gaps. Motivated by recent progress using renormalization in various areas of mathematics, we introduce a spectral renormalization method for semilinear parabolic equations that creates an artificial spectral gap at high frequencies while leaving the low modes unchanged. For the resulting family of renormalized equations, classical inertial-manifold theory yields finite-dimensional invariant manifolds whose dimension diverges as the renormalization parameter tends to zero. We show that the reduced dynamics on these manifolds approximate the original semiflow through an approximate semi-conjugacy, with an $O(ε^2)$ error on bounded finite-time intervals. Under global incremental dissipativity, the semi-conjugacy is uniform in time and yields an $O(ε^2)$ Hausdorff estimate for the global attractors, if they exist. We also establish further dynamical properties of the spectral renormalization such as persistence of hyperbolic equilibria and their Morse indices. We illustrate the construction in settings with and without a natural spectral gap and embed it into a two-parameter fast-slow PDE framework, where the artificial gap enables us to construct slow manifolds for systems on bounded domains with arbitrary dimension. Thus our new approach provides a finite-dimensional dynamical approximation even in regimes where an exact inertial manifold for the original equation is not available. In particular, this provides a practical balance between fully invariant manifolds and direct estimates.

发表机构

• Technical University of Munich, School of Computation, Information and Technology（慕尼黑工业大学计算、信息与技术学院）
• Leiden University, Mathematical Institute（莱顿大学数学研究所）

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