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

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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.