Topological and Epsilon-Entropy for Large Volume Limits of Discretized Parabolic Equations

We consider semidiscrete 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 vanishing viscosity. We define the spatial density of $\varepsilon $-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.

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