We study the gradient flow associated with the functional F-phi(u) := 1/2 integral(I) phi(u(x)) dx, where phi is non convex, and with its singular perturbation F-phi(epsilon)(u) := 1/2 integral(I) (epsilon(2)(u(xx))(2) + phi(u(x)))dx. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions u(epsilon) of the singularly perturbed equation u(t) = - epsilon(2)u(xxxx) + 1/2 phi''(u(x)) u(xx) for small values of epsilon > 0. Our analysis leads to a reinterpretation of the unperturbed equation u(t) = 1/2 (phi'(u(x)))(x), and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of u(epsilon) as epsilon --> 0(+).

Bellettini, G., Fusco, G., Guglielmi, N. (2006). A concept of solution and numerical experiments for forward-backward diffusion equations. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 16(4), 783-842.

A concept of solution and numerical experiments for forward-backward diffusion equations

BELLETTINI, GIOVANNI;
2006-01-01

Abstract

We study the gradient flow associated with the functional F-phi(u) := 1/2 integral(I) phi(u(x)) dx, where phi is non convex, and with its singular perturbation F-phi(epsilon)(u) := 1/2 integral(I) (epsilon(2)(u(xx))(2) + phi(u(x)))dx. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions u(epsilon) of the singularly perturbed equation u(t) = - epsilon(2)u(xxxx) + 1/2 phi''(u(x)) u(xx) for small values of epsilon > 0. Our analysis leads to a reinterpretation of the unperturbed equation u(t) = 1/2 (phi'(u(x)))(x), and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of u(epsilon) as epsilon --> 0(+).
2006
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
Forward-backward parabolic equations; Fourth order regularization; Microstructures; Nonconvex functional; Singular perturbations; Stiff problems
60
http://matematica.univaq.it/~guglielm/PAPERS/bfg.pdf
Bellettini, G., Fusco, G., Guglielmi, N. (2006). A concept of solution and numerical experiments for forward-backward diffusion equations. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 16(4), 783-842.
Bellettini, G; Fusco, G; Guglielmi, N
Articolo su rivista
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/37216
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