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Inertial manifold and large deviations approach to reduced PDE dynamics. (English) Zbl 1373.82063

Summary: In this paper a certain type of reaction-diffusion equation – similar to the Allen-Cahn equation – is the starting point for setting up a genuine thermodynamic reduction i.e. involving a finite number of parameters or collective variables of the initial system. We firstly operate a finite Lyapunov-Schmidt reduction of the cited reaction-diffusion equation when reformulated as a variational problem. In this way we gain a finite-dimensional ODE description of the initial system which preserves the gradient structure of the original one and that is exact for the static case and only approximate for the dynamic case. Our main concern is how to deal with this approximate reduced description of the initial PDE. To start with, we note that our approximate reduced ODE is similar to the approximate inertial manifold introduced by Temam and coworkers for Navier-Stokes equations. As a second approach, we take into account the uncertainty (loss of information) introduced with the above mentioned approximate reduction by considering the stochastic version of the ODE. We study this reduced stochastic system using classical tools from large deviations, viscosity solutions and weak KAM Hamilton-Jacobi theory. In the last part we suggest a possible use of a result of our approach in the comprehensive treatment non equilibrium thermodynamics given by Macroscopic Fluctuation Theory.

MSC:

82C35 Irreversible thermodynamics, including Onsager-Machlup theory
35K57 Reaction-diffusion equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35Q84 Fokker-Planck equations
60F10 Large deviations
82C27 Dynamic critical phenomena in statistical mechanics
35D40 Viscosity solutions to PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35B42 Inertial manifolds
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