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Finite-dimensional reduction, Inertial Manifolds, and Homoclinic structures in dissipative PDEs

耗散偏微分方程中的有限维约简、惯性流形和同宿结构

基本信息

批准号：
EP/P024920/1
负责人：
Sergey Zelik
金额：
$ 49.01万
依托单位：
University of Surrey
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP024920%2F1
关键词：
Finite dimensional reduction Inertial Manifolds

项目摘要

There is a common belief that the dissipative dynamics generated by partial differential equations in bounded domains is effectively finite-dimensional. In other words, despite the fact that the initial phase space is infinite-dimensional, there is a possibility that after an "unimportant" transient behaviour is discarded, the dynamics can be described by finitely many parameters which satisfying a system of ordinary differential equations (ODEs) - the so-called inertial form (IF). This idea is widely used, in particular, in modern theories of turbulence, in order to justify various scales (inertial, dissipative, etc.), energy cascades, and Kolmogorov laws. However, the precise and clear meaning of the above mentioned finite-dimensional reduction remains a mystery, despite the fundamental significance of the problem and permanent interest to it by the experts from various fields of science. Rigorously, the finite-dimensional reduction is justified in the case where the so-called inertial manifold (IM) exists. By definition, it is a finite-dimensional smooth invariant manifold in the phase space which attracts exponentially all other trajectories. Then, the IF is generated just by restricting the system to this invariant manifold. However, the existence of an IM requires the so called spectral gap conditions (SG) which are not satisfied in many interesting examples. In particular, the existence or non-existence of the IM for the 2D Navier-Stokes system is one of the major open problems in the field. In such cases, only non-smooth (Holder continuous) IFs are constructed in general, and it was unclear for a long time whether this loss of smoothness is just a drawback of the method or it has a principal significance. The situation has changed now due to our recent counterexamples which show that, in the absence of an IM, the dynamics may demonstrate features which cannot be observed in smooth ODEs, i.e. the dynamics may only "pretend" to be finite-dimensional (due to the existence of a non-smooth IF), while the "true nature" of the dynamics is infinite-dimensional. In the project we intend to give a precise meaning to this idea and investigate the effect in detail.This will potentially lead to an essential shift in the paradigm which would requires a comprehensive study of the new infinite-dimensional phenomena arising in problems which were previously thought to be finite-dimensional; this is the ultimate aim of the proposed project. We intend to act in two directions: on one hand, we develop new methods of verifying the existence of IMs and, on the other hand, by combining the methods of the Analysis of PDEs and Functional Analysis with Dynamical Systems theory and, in particular, normal hyperbolicity theory and the theory of homolcinic bifurcations, we will describe a wide class of PDEs which may demonstrate the new type of infinite-dimensional dynamics. As a result, we intend to show that for systems of PDEs with a non-trivial (recurrent) dynamics there is an almost sharp dichotomy between the existence of IM and the irreducibility to a finite-dimensional system of ODEs in practically every reasonable sense. The obtained results will be applied to various classes of physically important dissipative PDEs, such as 1D Burger's type equations and systems, complex Ginzburg-Landau equations, and (as an ultimate goal) tthe 2D Navier-Stokes system on a torus or on a sphere.

人们普遍认为，由偏微分方程在有界域中产生的耗散动力学实际上是有限维的。换句话说，尽管初始相空间是无限维的，但在丢弃“不重要”的瞬态行为之后，动力学可以用有限多个参数来描述，这些参数满足一个常微分方程（ode）系统——即所谓的惯性形式（IF）。这个想法被广泛使用，特别是在现代湍流理论中，以证明各种尺度（惯性，耗散等），能量级联和柯尔莫哥洛夫定律。然而，上述有限维约简的精确和清晰的含义仍然是一个谜，尽管这个问题具有根本意义，并且各个科学领域的专家都对它感兴趣。严格地说，在所谓的惯性流形（IM）存在的情况下，有限维缩减是合理的。根据定义，它是相空间中的有限维光滑不变流形，它以指数方式吸引所有其他轨迹。然后，IF是通过将系统限制为这个不变流形来生成的。然而，谱隙的存在需要所谓的谱隙条件（SG），而在许多有趣的例子中，这一条件并不满足。特别是二维Navier-Stokes系统的IM是否存在是该领域的主要开放性问题之一。在这种情况下，通常只构造非光滑的（Holder连续的）if，并且很长一段时间都不清楚这种平滑性的损失是该方法的缺点还是具有主要意义。由于我们最近的反例表明，在没有IM的情况下，动力学可能表现出在光滑的ode中无法观察到的特征，即动力学可能只是“假装”是有限维的（由于非光滑IF的存在），而动力学的“真实性质”是无限维的。在这个项目中，我们打算给这个想法一个精确的含义，并详细研究其效果。这可能会导致范式的本质转变，这将需要对以前被认为是有限维的问题中出现的新的无限维现象进行全面的研究；这是拟建项目的最终目的。我们打算在两个方向上采取行动：一方面，我们开发验证IMs存在的新方法；另一方面，通过将偏微分方程分析和泛函分析的方法与动力系统理论相结合，特别是正规双曲理论和齐次分岔理论，我们将描述一类可能展示新型无限维动力学的偏微分方程。因此，我们打算证明，对于具有非平凡（循环）动力学的偏微分方程系统，在几乎所有合理的意义上，在IM的存在性和对有限维偏微分方程系统的不可约性之间存在着几乎尖锐的二分法。所获得的结果将应用于各种物理上重要的耗散微分方程，如1D Burger型方程和系统，复杂的金兹堡-朗道方程，以及（作为最终目标）环面或球面上的2D Navier-Stokes系统。