There is a common belief that the dissipative dynamics generated by partial differential equations in bounded domains is effectively finitedimensional. In other words, despite the fact that the initial phase space is infinitedimensional, 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 socalled 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 finitedimensional 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 finitedimensional reduction is justified in the case where the socalled inertial manifold (IM) exists. By definition, it is a finitedimensional 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 nonexistence of the IM for the 2D NavierStokes system is one of the major open problems in the field. In such cases, only nonsmooth (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 finitedimensional (due to the existence of a nonsmooth IF), while the "true nature" of the dynamics is infinitedimensional. 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 infinitedimensional phenomena arising in problems which were previously thought to be
finitedimensional; 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 infinitedimensional dynamics. As a result, we intend to show that for systems of PDEs with a nontrivial (recurrent) dynamics there is an almost sharp dichotomy between the existence of IM and the irreducibility to a finitedimensional 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 GinzburgLandau equations, and (as an ultimate goal) tthe 2D NavierStokes system on a torus or on a sphere.
