Stable manifolds to bounded solutions in possibly ill-posed PDEs
Stable manifolds to bounded solutions in possibly ill-posed PDEs
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DOI：
10.1016/j.jde.2019.10.042
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发表时间：
2020-04
影响因子：
2.4
通讯作者：
Hon-Wing Cheng;Rafael de la Llave
中科院分区：
文献类型：
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作者：
Hon-Wing Cheng;Rafael de la Llave
我们证明了几个结果建立稳定流形的存在性和规律性的不同类别的特殊解决方案的发展方程（这些方程可能是不适定的）：一个单一的特定的解决方案，一个不变的环面充满拟周期轨道或更一般的流形的解决方案。在后一种情况下，其中包括几个轨道，我们也建立不变流形的轨道光滑依赖于轨道（解析的情况下，准周期轨道和可微的情况下，更一般的家庭）。我们首先建立一个一般的抽象定理，在适当的（谱，非退化，解析）假设下的线性方程，建立所需的流形的存在性。相关的结果出现在文献中，但我们的结果允许的非线性是无界的，我们得到的不变流形的光滑性。这使得本文的结果适用于一些当前感兴趣的模型，不能以其他方式处理。详细讨论了水波的Boussinesq方程（在其它长波近似中也有类似的现象）和复Ginzburg-Landau方程。最近，我们发现[11]我们的结果也适用于平均场博弈。由于我们考虑的方程可能是不适定的，对稳定流形的部分要求是人们可以定义它们上的（前向）动力学。还要注意的是，基于动力学存在的方法（如图变换）不适用于不适定方程。我们使用基于积分方程的方法（Perron方法）与部分动力学，但我们需要利用部分动力学的光滑性质。请注意，即使我们开始时的解族是有限维的，稳定流形也可能是无限维的。
We prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations (these equations may be ill-posed): a single specific solution, an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions. In the later cases, which include several orbits, we also establish the invariant manifolds of an orbit depend smoothly on the orbit (analytically in the case of quasi-periodic orbits and finitely differentiably in the case of more general families).We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, analyticity) assumptions on the linearized equation, establishes the existence of the desired manifold. Related results appear in the literature, but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds. This makes the results in this paper applicable to some several models of current interest that could not be treated otherwise. We discuss in detail the Boussinesq equation of water waves (similar phenomena happen in other long wave approximations) and complex Ginzburg-Landau equation. More recently, we observed [11] that our results also apply to Mean Field Games.Since the equations we consider may be ill-posed, part of the requirements for the stable manifold is that one can define the (forward) dynamics on them. Note also that the methods that are based in the existence of dynamics (such as graph transform) do not apply to ill-posed equation. We use the methods based on integral equations (Perron method) associated with the partial dynamics, but we need to take advantage of smoothing properties of the partial dynamics. Note that, even if the families of solutions we started with are finite dimensional, the stable manifolds may be infinite dimensional.