Log-Lipschitz continuity of the vector field on the attractor of certain parabolic equations

Abstract

We discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that certain dissipative partial differential equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor A lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. Consequently, the global attractor A has zero Lipschitz deviation and, therefore, there are linear maps L into finite-dimensional spaces, whose inverses restricted to LA are Hölder continuous with an exponent arbitrarily close to one.

Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations

“…We remarked after the proof of Lemma 42 that one can obtain log-Lipschitz continuity of on the attractor under the assumption that is analytic, but with logarithmic exponent 2. Continuity of 1/2 on the attractor was shown by Kukavica [19] with logarithmic exponent 1/2 (see (272)), and it was shown by Pinto de Moura & Robinson [107] that a similar argument can be used to show that is log-Lipschitz with logarithmic exponent 1. If we could obtain an embedding with Lipschitz inverse then this would be sufficient; but the embedding Theorem 32 only guarantees that −1 is Hölder continuous.…”

“…One can deduce from this [81,107] that the Lipschitz deviation of the Navier-Stokes attractor is zero, even when we only have ∈ . Proposition 43.…”