Title:

On embeddings and dimensions of global attractors associated with dissipative partial differential equations

Hunt and Kaloshin (1999) proved that it is possible to embed a compact subset X of a Hilbert space with upper boxcounting dimension d into RN for any N > 2d+1, using a linear map L whose inverse is Hölder continuous with exponent α < (N  2d)/N(1 + τ(X)/2), where τ(X) is the 'thickness exponent' of X. More recently, Ott et al. (2006) conjectured that "many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero". In Chapter 2 we study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c0 of null sequences. These examples are used to show that Hunt and Kaloshin's result, and a related result due to Robinson (2009) for subsets of Banach spaces, are asymptotically sharp. An analogous argument shows that the embedding theorems proved by Robinson (2010), in terms of the Assouad dimension, for the Hilbert and Banach space case are asymptotically sharp. In Chapter 3 we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X). We show that Hunt and Kaloshin's result and Corollary 3.9 in Ott et al. (2006) remain true with the thickness replaced by the Lipschitz deviation. We then prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt, & Kaloshin. In Chapter 4 we study the regularity of the vector field on the global attractor associated with parabolic equations. We show that certain dissipative equations possess a linear term that is logLipschitz 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. This provides an alternative proof that the global attractor A has zero Lipschitz deviation. In Chapter 5 we use shape theory and the concept of cellularity to show that if A is the global attractor associated with a dissipative partial differential equation in a real Hilbert space H and the set A  A has finite Assouad dimension d, then there is an ordinary differential equation in Rm+1, with m > d, that has unique solutions and reproduces the dynamics on A. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor X arbitrarily close to LA, where L is a homeomorphism from A into Rm+1.
