Gabriel J Lord, Jacques Rougemont
Topological and $\epsilon$-entropy for Large Volume
Limits of Discretised Parabolic Equations
(254K, Postscript)
ABSTRACT. We consider semi-discrete and fully discrete
approximations of nonlinear parabolic equations in the limit of
unbounded domains, which by a scaling argument is equivalent to the
limit of small viscosity. We define the spatial density of $\epsilon
$-entropy, topological entropy and dimension for the attractors and
show that these quantities are bounded. We also provide practical
means of computing lower bounds on them. The proof uses the property
that solutions lie in
Gevrey classes of analyticity, which we
define in a way that does not depend on the size of the spatial
domain. As a specific example we discuss the complex Ginzburg-Landau
equation.