Diffusion and wave behaviour in linear Voigt model

Monica De Angelis; Pasquale Renno

doi:10.1016/S1631-0721(02)01421-3

Diffusion and wave behaviour in linear Voigt model
[Diffusion et comportement ondulatoire dans le modèle linéaire de Voigt]

Monica De Angelis ¹ ; Pasquale Renno ¹

¹ Dipartimento di Matematica e Applicazioni, Facoltà di Ingegneria, via Claudio 21, 80125, Napoli, Italy

Comptes Rendus. Mécanique, Volume 330 (2002) no. 1, pp. 21-26.

Résumé
Abstract

On analyse un problème aux limites $𝒫_{ϵ}$ pour une équation parabolique du troisième ordre. Cette équation décrit l'évolution monodimensionnelle de beaucoup de materiaux dissipatifs comme les fluides ou les solides viscoélastiques, les gaz visqueux, les materiels superconducturs, les fluides incompressibles conducteurs de l'électricité. De plus l'opérateur parabolique du troisième ordre regularise divers equations non lineaires des ondes du deuxième ordre. On examine dans ce travail le comportment hyperbolique ou parabolique de la solution du problème $𝒫_{ϵ}$ à l'aide des temps lent et rapide. En conséquence, on donne une approximation asymptotique rigooreose de la solution du problème $𝒫_{ϵ}$ .

A boundary value problem $𝒫_{ϵ}$ related to a third order parabolic equation with a small parameter ε is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third order parabolic operator regularizes various nonlinear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution of $𝒫_{ϵ}$ is estimated by means of slow time τ=εt and fast time θ=t/ε. As consequence, a rigorous asymptotic approximation for the solution of $𝒫_{ϵ}$ is established.

Reçu le : 2001-09-01
Accepté le : 2001-10-26
Publié le : 2002-01-01

DOI : 10.1016/S1631-0721(02)01421-3

Keywords: continuum media, partial different equations, viscoelasticity, superconductivity, boundary layer
Mot clés : milieux continus, équations aux dérivées partielles, viscoéllasticité, supraconductivité

Affiliations des auteurs :

Monica De Angelis ¹ ; Pasquale Renno ¹

¹ Dipartimento di Matematica e Applicazioni, Facoltà di Ingegneria, via Claudio 21, 80125, Napoli, Italy

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Monica De Angelis; Pasquale Renno. Diffusion and wave behaviour in linear Voigt model. Comptes Rendus. Mécanique, Volume 330 (2002) no. 1, pp. 21-26. doi : 10.1016/S1631-0721(02)01421-3. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01421-3/

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