An A Posteriori KAM Theorem for Whiskered Tori in Hamiltonian Partial Differential Equations with Applications to some Ill-Posed Equations
An A Posteriori KAM Theorem for Whiskered Tori in Hamiltonian Partial Differential Equations with Applications to some Ill-Posed Equations
复制标题
哈密顿偏微分方程中晶须托里的后验KAM定理及其在某些病态方程中的应用
DOI：
10.1007/s00205-018-1293-6
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发表时间：
2019
影响因子：
2.5
通讯作者：
Sire, Yannick
中科院分区：
文献类型：
--
作者：
de la Llave, Rafael;Sire, Yannick
本文的目标是发展具有双曲方向的环面的 KAM 理论，该理论适用于哈密顿偏微分方程，甚至适用于一些不适定方程。主要结果具有后验格式，也就是说，如果不变性方程存在一个近似解并且满足一些非简并条件，那么附近就有一个真解。除了处理准可积的情况之外，这还允许验证数值计算或形式微扰展开以及在简并情况下获得准周期解。后验格式还具有其他自动后果（对参数的平滑依赖、正则性引导等）。我们强调，所需的非简并条件只是在近似解上评估的量（没有对系统的全局假设，例如扭曲）。因此，它们很容易在微扰展开中得到验证。我们将关注近似值大小与非简并条件之间的定量关系。这将使我们能够证明专家所说的小扭曲定理（当扰动变为零时，非简并条件消失，但比近似误差慢得多）。证明方法基于迭代方法来求解环面参数化的函数方程，表示参数化的范围允许演化并且是不变的。我们还求解束的函数方程，这意味着在线性化下保持不变。迭代方法不使用变换理论或作用角变量。主要结果并不假设系统接近可积。更令人惊讶的是，我们不需要我们研究的方程定义所有初始条件的演化并且是适定的。即使我们研究的系统不承认所有初始条件的解，我们也表明存在一种系统的方法来选择初始条件，在该初始条件上人们可以定义准周期性的演化。我们首先提出一个抽象定理。然后，我们展示这个抽象结果如何应用于一些具体例子。本文考虑的例子是标量 Boussinesq 方程和 Boussinesq 系统（两者都是旨在描述长波极限下的水波的偏微分方程模型）。对于这些方程，我们构造小幅度时间准周期解，其在空间变量中均匀。抽象定理的策略受到 Fontich 等人的启发。 （Electron Res Announc Math Sci 16:9–22, 2009；J Differ Equ 246(8):3136–3213, 2009）。本文的主要部分是研究二分法的无限维类似物，它甚至适用于不适定方程，并且在添加无界扰动的情况下是稳定的。这要求我们假设平滑特性。我们还提出了扰动下分裂变化的非常详细的界限。
The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has ana-posterioriformat, that is, we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. This allows, besides dealing with the quasi-integrable case, for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi-periodic solutions in degenerate situations. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). We emphasize that the non-degeneracy conditions required are just quantities evaluated on the approximate solution (no global assumptions on the system such as twist). Hence, they are readily verifiable in perturbation expansions. We will pay attention to the quantitative relations between the sizes of the approximation and the non-degeneracy conditions. This will allow us to prove what experts callsmall twist theorems(the non-degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation). The method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant. We also solve functional equations for bundles which imply that are invariant under the linearization. The iterative method does not use transformation theory nor action-angle variables. The main result does not assume that the system is close to integrable. More surprisingly, we do not need that the equations we study define an evolution for all initial conditions and are well posed. Even if the systems we study do not admit solutions for all initial conditions, we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi-periodic. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples. The examples considered in this paper are the scalar Boussinesq equation and the Boussinesq system (both are PDE models that aim to describe water waves in the long wave limit). For these equations we construct small amplitude time quasi-periodic solutions which are even in the spatial variable. The strategy for the abstract theorem is inspired by that in Fontich et al. (Electron Res Announc Math Sci 16:9–22, 2009; J Differ Equ 246(8):3136–3213, 2009). The main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill-posed equations and which is stable under addition of unbounded perturbations. This requires that we assume smoothing properties. We also present very detailed bounds on the change of the splittings under perturbations.
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DOI：
10.1007/978-1-4684-0313-8
发表时间：
1993-09
期刊：
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影响因子：
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作者：
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DOI：
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发表时间：
2017
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P. Polácik;Darío A. Valdebenito
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影响因子：
1.4
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DOI：
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发表时间：
2002
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作者：
J. Vano
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J. Vano
DOI：
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发表时间：
1979
期刊：
影响因子：
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作者：
K. Kirchgässner;J. Scheurle
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