On embeddings and dimensions of global attractors associated with dissipative partial differential equations
Pinto de Moura, Eleonora (2010) On embeddings and dimensions of global attractors associated with dissipative partial differential equations. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2341202~S15
Hunt and Kaloshin (1999) proved that it is possible to embed a compact
subset X of a Hilbert space with upper box-counting 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,
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 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. 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.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Embeddings (Mathematics), Attractors (Mathematics), Differential equations, Partial, Hilbert space|
|Official Date:||September 2010|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Robinson, James C. (James Cooper), 1969-|
|Sponsors:||Brazil. Coordenação do Aperfeiçoamento de Pessoal de Nível Superior (CAPES)|
|Extent:||v, 94 leaves|
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