In this paper we consider a nonlinear reaction–diffusion equation in which the nonlinear term is described by a potential energy function. For this class of pde, equilibria admit a variational formulation and they can be determined by a suitable finite dimensional reduction technique called Amann–Conley–Zendher (ACZ) reduction. By extension, the ACZ reduction applies also to the pde dynamics, leading to a finite dimensional description in terms of an ode. It turns out that this ode is a gradient dynamics defined by a potential. While the static case is recovered perfectly in the reduced representation, the reduced dynamics appears to be only a good approximation of the original pde dynamics. The main aim of this paper is to present a number of tools that allow to deal with this uncertainty. First of all we show that the reduced potential W is derived by the original variational functional for the equilibria, and that it contains all the relevant information about the reduced dynamics. In particular we show that the Morse Index of the equilibra of the reduced potential W is the same of the original one, therefore the reduced dynamics give a faithful representation of the stability properties of the original equilibria. Then we highlight the strict analogy between the ACZ reduced dynamics and the approximate inertial manifold description by Temam. This allows to control the discrepancy between the exact and the reduced dynamics, by showing that the orbits of the exact dynamics enter in a thin neighborhood of our approximate inertial manifold after a certain transient time. Finally, the lack of information due to the above approximation may be modelled by adding a stochastic noise term to the reduced ode, and by considering a description of this randomly perturbed system using Freidlin–Wentzell theory, we show that the Large Deviations rate function is given by the reduced potential.
Abbondandolo, A., Mayer, P.: Lectures on the Morse complex for infinite-dimensional manifolds In: Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology, pp. 1–74, Springer (2006)Google Scholar
Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7, 539–603 (1980)MathSciNetMATHGoogle Scholar
Cottle, R.W.: On manifestations of the Schur complement. Rend. Sem. Mat. Fis. Milano 45 (1975)Google Scholar
Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, vol. 88. Cambridge university Press, Cambridge (2016)Google Scholar
Foias, C., Manley, O., Temam, R.: Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal. Theory Models Appl. 11(8), 939–967 (1987)CrossRefMATHGoogle Scholar
Foias, C., Manley, O., Temam, R.: Modelling of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO - Modelisation mathematique et analyse numerique 22, 93–118 (1988)MathSciNetMATHGoogle Scholar
Freidlin, M.I., Ventzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1998)CrossRefGoogle Scholar