Comptes Rendus
no. 1
Diffusion and wave behaviour in linear Voigt model
[Diffusion et comportement ondulatoire dans le modèle linéaire de Voigt]
Monica De Angelis1  ; Pasquale Renno1
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.

On analyse un problème aux limites 𝒫ϵ 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 𝒫ϵ à l'aide des temps lent et rapide. En conséquence, on donne une approximation asymptotique rigooreose de la solution du problème 𝒫ϵ.

A boundary value problem 𝒫ϵ 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 𝒫ϵ 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 𝒫ϵ is established.

Reçu le :
Accepté le :
Publié le :
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
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/

[1] D.D. Joseph; M. Renardy; J.C. Saut Hyperbolicity and change of type in the flow of viscoestic fluids, Arch. Rational Mech. Anal., Volume 87 (1985), pp. 213-251

[2] J.A. Morrison Wave propagations in rods of Voigt material and viscoelastic materials with three-parameters models, Quart. Appl. Math., Volume 14 (1956), pp. 153-173

[3] P. Renno On some viscoelastic models, Atti Acad. Naz. Lincei Rend. Fis., Volume 75 (1983) no. 6, pp. 1-10

[4] A. Morro; L.E. Payne; B. Straughan Decay, growth, continuous dependence and uniqueness results of generalized heat theories, Appl. Anal., Volume 38 (1990), pp. 231-243

[5] N. Flavin; S. Rionero Qualitative Estimates for Partial Differential Equations, CRC Press, 1996 (p. 368)

[6] H. Lamb Hydrodynamics, Dover, 1932 (p. 708)

[7] R. Nardini Soluzione di un problema al contorno della magneto idrodinamica, Ann. Mat. Pura Appl., Volume 35 (1953), pp. 269-290

[8] A. Barone; G. Paterno Physics and Application of the Josephson Effect, Wiley, New York, 1982 (p. 530)

[9] J.E.M. Rivera; L.H. Fatori Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., Volume 206 (1997), pp. 397-427

[10] V.P. Maslov; P.P. Mosolov Nonlinear Wave Equations Perturbed by Viscous Terms, Walter de Gruyher, Berlin, 2000 (p. 329)

[11] A.I. Kozhanov; N.A. Lar'kin; N.N. Yanenko A mixed problem for a class of equation of third order, Siberian Math. J., Volume 22 (1981) no. 6, pp. 867-872

[12] S. Kawashima; Y. Shibata Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys. (1992), pp. 189-208

[13] G.I. Barenblatt; M. Bertsch; R. Del Passo; M. Ughi A degenerate pseudoparabolic regularization of a nonlinear forward–backward heat equation arising in the theory of heat and mass exchange in stably stratified turbolent shear flow, SIAM J. Math. Anal., Volume 24 (1993) no. 6, pp. 1414-1439

[14] B. D'Acunto; M. De Angelis; P. Renno Fundamental solution of a dissipative operator, Rend. Acad. Sci. Fis. Mat. Napoli, Volume LXIV (1997), pp. 295-314

[15] B. D'Acunto; M. De Angelis; P. Renno Estimates for the perturbed sine-Gordon equation, Rend. Circ. Mat. Palermo (2) Suppl., Volume 57 (1998), pp. 199-204

[16] A.T. Cousin; C.L. Frota; N.A. Lar'kin Regular solution and energy decay for the equation of viscoelasticity with nonlinear damping on the boundary, J. Math. Anal. Appl., Volume 224 (1998), pp. 273-296

[17] A.T. Cousin; N.A. Lar'kin On the nonlinear initial boundary value problem for the equation of viscoelasticity, Nonlinear Anal., Volume 31 (1998) no. 1/2, pp. 229-242

[18] Y. Shibata On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci., Volume 23 (2000), pp. 203-226

[19] A.I. Kozhanov; N.A. Lar'kin Wave equation with nonlinear dissipation in noncylindrical domains, Dokl. Math., Volume 62 (2000) no. 2, pp. 17-19

[20] A. Nayfey A comparison of perturbation methods for nonlinear hyperbolic waves, Proc. Adv. Sem. Wisconsin, Volume 45 (1980), pp. 223-276

[21] M. De Angelis Asymptotic analysis for the strip problem related to a parabolic third order operator, Appl. Math. Lett., Volume 14 (2001) no. 4, pp. 425-430

[22] D.S. Mitrinovic Analytic Inequalities, Springer, 1970 (p. 404)

[23] P. Renno On a wave theory for the operator ε∂t(∂t2−c12Δn)+∂t2−c02Δn, Ann. Mat. Pura Appl., Volume 136 (1984) no. 4, pp. 355-389

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