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

人们普遍认为，有界域上偏微分方程解的耗散动力学实际上是有限维的。换言之，尽管初始相空间是无限维的，但在丢弃“不重要的”暂态行为后，动力学可以用满足常微分方程组(ODE)的有限多个参数来描述--所谓的惯性形式(IF)。这一思想被广泛应用于现代湍流理论中，以证明各种尺度(惯性、耗散等)、能量级联和柯尔莫戈洛夫定律的合理性。然而，上述有限维约化的精确而清晰的含义仍然是一个谜，尽管这个问题具有根本意义，而且来自不同科学领域的专家对它有着永久的兴趣。严格地说，在存在所谓的惯性流形(IM)的情况下，有限维约化是合理的。根据定义，它是相空间中的有限维光滑不变流形，它指数地吸引所有其他轨迹。然后，通过将系统限制在这个不变流形上，就产生了IF。然而，IM的存在需要所谓的谱间隙条件(SG)，而这些条件在许多有趣的例子中并不满足。尤其是二维Navier-Stokes系统的IM的存在与否是该领域的一个主要开放问题。在这种情况下，通常只构造非光滑(Holder连续)IF，很长一段时间都不清楚这种光滑度的损失是该方法的缺点还是具有主要意义。这种情况现在已经改变了，因为我们最近的反例表明，在没有IM的情况下，动力学可能展示出在光滑的颂歌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确切的含义，并详细研究它的影响。这可能会导致范式的根本转变，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；这是拟议项目的最终目标。我们打算朝两个方向行动：一方面，我们发展了新的方法来验证IMS的存在性；另一方面，通过将偏微分方程组的分析和泛函分析的方法与动力系统理论，特别是正规双曲性理论和同伦分叉理论相结合，我们将描述一大类可能证明新型无限维动力学的偏微分方程组。因此，我们打算证明，对于具有非平凡(递归)动力学的偏微分方程组，IM的存在和有限维常微分方程组的不可约性在几乎所有合理的意义上几乎是一分为二的。所得结果将应用于各类物理上重要的耗散偏微分方程组，如一维Burger型方程和系统、复杂的Ginzburg-Landau方程，以及(作为最终目标)环面或球面上的二维Navier-Stokes系统。

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.