Counterexamples to regularity of Mañé projections in the theory of attractors

Abstract

We study the global attractors of abstract semilinear parabolic equations and their projections to finite-dimensional planes. It is well-known that the attractor can be embedded into the finite-dimensional inertial manifold if the so-called spectral gap condition is satisfied. We show that in the case when the spectral gap condition is violated, it is possible to construct the nonlinearity in such way that the corresponding attractor cannot be embedded into any finite-dimensional Log-Lipschitz manifold and, therefore, does not possess any Mane projections with Log-Lipschitz inverse. In addition, we give an example of finitely smooth nonlinearity such that the attractor has finite Hausdorff but infinite fractal dimension.1991 Mathematics Subject Classification. 35B40, 35B45.

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

“…But, unfortunately, as recently shown in [14], the conjecture is actually wrong (the corresponding counterexample is discussed in the next section). Nevertheless, we discuss below some results obtained under the attempts to prove this conjecture and introduce a number of interesting quantities which occurred to be helpful, in particular, in order to disprove the conjecture.…”

“…The proof of this proposition is an elementary exercise and is given in [14], so we leave it to the reader. Remark 3.35.…”

“…In addition, there are examples with very regular non-linearities F when the Bouligand dimension of the attractor is infinite (see [14] and next section) and, finally, no reasons and no mechanisms which can guarantee the finiteness of this dimension without the inertial manifold existence are known. Thus, it seems natural to make a conclusion that the concept of Bouligand dimension and related technical tools are not sufficient to solve the uniqueness problem.…”

“…To conclude this paragraph, we introduce following [14] one more quantity which will be used in order to disprove the log-Lipschitz conjecture.…”

“…So, the drastic and in a sense unacceptable loss of smoothness under that reduction looks unavoidable. Moreover, as recent counterexamples of [14] show even the log-Lipschitz regularity may be lost.…”