Finite mechanical proxies for a class of reducible continuum systems

Cardin, Franco; Lovison, Alberto

doi:10.3934/nhm.2014.9.417

Cited by 4 publications

(1 citation statement)

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“…Namely, a static PDE variational problem becomes perfectly equivalent to a finite algebraic system, see Section 2.1. This method has been employed in conjunction with topological techniques for proving results of existence and multiplicity of solutions for nonlinear differential equations [9,10,29]. Notably for our purposes, we have that the ACZ philosophy can be applied also to dissipative dynamical equations, giving rise to the notion of inertial manifolds [26].…”

Section: Introductionmentioning

confidence: 99%

Approximate Inertial Manifold Approach to Non-Equilibrium Thermodynamics

Cardin¹,

Favretti²,

Lovison³

2016

Preprint

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In this paper a reaction-diffusion type 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. This program is carried over by firstly operating a finite Lyapunov-Schmidt reduction of the cited reaction-diffusion equation when reformulated as a variational problem. In this way we gain an approximate finite-dimensional o.d.e. description of the initial system which preserves the gradient structure of the original one and that is similar to the approximate inertial manifold description of a p.d.e. introduced by Temam and coworkers. Secondly, we resort to the stochastic version of the o.d.e., taking into account in this way the uncertainty (loss of information) introduced with the above mentioned reduction. 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 highlight some essential similarities existing between our approach and the comprehensive treatment non equilibrium thermodynamics given by Jona-Lasinio and coworkers. The starting point of their axiomatic theory -motivated by large deviations description of lattice gas modelsof systems in a stationary non equilibrium state is precisely a conservation/balance law which is akin to our simple model of reaction-diffusion equation. Large Deviations 113.2. The gradient vector field case 13 Viscosity solutions 132010 Mathematics Subject Classification. 82C05 Classical dynamic and nonequilibrium statistical mechanics (general) 60F10 Large deviations 37L25 Inertial manifolds and other invariant attracting sets 35Q84 Fokker-Planck equations 70H20 Hamilton-Jacobi equations .

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Section: Introductionmentioning

confidence: 99%

Approximate Inertial Manifold Approach to Non-Equilibrium Thermodynamics

Cardin¹,

Favretti²,

Lovison³

2016

Preprint

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Erratum to: Discrete structures equivalent to nonlinear Dirichlet and wave equations

Lovison

Cardin

Bobbo

2016

Continuum Mech. Thermodyn.

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A perturbed wave equation amenable to exact reductionIn [2] we have considered the problem of finding T -periodic solutions∂ x 2 is the D'Alembert wave operator while F is a Nemitski Lipschitz operator. To produce an exact finite reduction in the wave equation we needed the inverse operator of the D'Alembertian. To arrive to this scope we first expanded v ∈ H in Fourier series and then we laid down the formal explicit component-wise inverse of :where k ∈ Z n , k = 0. However, this formula is not valid in general, i.e., the D'Alembert operator does not admit a bounded inverse in H both with rational and irrational periods T . This could be overcome by considering numbers of constant typeas in [4]; this, however, is not compatible with the contractivity needed for the reduction procedure. We have instead adopted the standard approach by Rabinowitz [5], i.e., we considered the vibrating string (n = 1), weThe online version of the original article can be found under

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