# Stochastic and geometric aspects of reduced reaction–diffusion dynamics

## Abstract

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.

## Keywords

Non-equilibrium thermodynamics Lyapunov–Schmidt reduction Inertial manifolds Collective variables Fokker–Planck equation Hamilton–Jacobi equation Large deviations## Mathematics Subject Classification

82C05 60F10 37L25 35Q84 70H20## References

- 1.Abbondandolo, A., Mayer, P.: A Morse complex for infinite-dimensional manifolds, I. Adv. Math.
**197**(2), 321–410 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 2.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
- 3.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)MathSciNetzbMATHGoogle Scholar - 4.Amann, H.: Saddle points and multiple solutions of differential equations. Math. Z.
**169**, 127–166 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser Boston Inc., Boston (1997)CrossRefzbMATHGoogle Scholar
- 6.Cardin, F.: Elementary symplectic topology and mechanics. Lecture Notes of the Unione Matemaica Italiana, 16. Springer, xviii+222 pp. (2015)Google Scholar
- 7.Cardin, F., De Marco, G., Sfondrini, A.: Finite reduction and Morse index estimates for mechanical systems. Nonlinear Differ. Equ. Appl.
**18**(5), 557–569 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Cardin, F., Favretti, M., Lovison, A.: Inertial manifold and large deviations approach to reduced pde dynamics. J. Stat. Phys.
**5**(168), 1000–1015 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Cardin, F., Lovison, A.: Finite mechanical proxies for a class of reducible continuum systems. Netw. Heterog. Media
**9**, 417–432 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Cardin, F., Lovison, A., Putti, M.: Implementation of an exact finite reduction scheme for steady-state reaction–diffusion equations. Int. J. Numer. Methods Eng.
**69**(9), 1804–1818 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Cardin,F., Masci,L.: A Morse Index invariant reduction of non equilibrium thermodynamics, Rendiconti Lincei, Matematica e Applicazioni.
**29**(1) (2018)Google Scholar - 12.Cardin, F., Tebaldi, C.: Finite reductions for dissipative systems and viscous fluid-dynamic models on \(\mathbb{T}^2\). J. Math. Anal. Appl.
**345**, 213–222 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Conley, C.C., Zehnder, E.: The Birkhoff–Lewis fixed point theorem and a conjecture of V. I. Arnol’d. Invent. Math.
**73**(1), 33–49 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Cottle, R.W.: On manifestations of the Schur complement. Rend. Sem. Mat. Fis. Milano 45 (1975)Google Scholar
- 15.Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, vol. 88. Cambridge university Press, Cambridge (2016)Google Scholar
- 16.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)CrossRefzbMATHGoogle Scholar - 17.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)MathSciNetzbMATHGoogle Scholar - 18.Freidlin, M.I., Ventzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1998)CrossRefGoogle Scholar
- 19.Marion, M.: Approximate inertial manifolds for reaction–diffusion equations in high space dimension. J. Dyn. Differ. Equ.
**1**, 245–267 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Moro, G.J.: Kinetic equations for site populations from the Fokker–Planck equation. J. Chem. Phys.
**103**, 7514–7528 (1995)CrossRefGoogle Scholar - 21.Moro, G.J., Cardin, F.: Variational layer expansion for kinetic processes. Phys. Rev. E
**55**(5), 4918–4934 (1997)CrossRefGoogle Scholar - 22.Temam, R.: Inertial manifolds. Math. Intell.
**12**(4), 68–74 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
- 24.Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep.
**478**, 1–69 (2009)MathSciNetCrossRefGoogle Scholar