An existence theorem for the magneto-viscoelastic problem

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June 2012, 5(3): 435-447. doi: 10.3934/dcdss.2012.5.435

An existence theorem for the magneto-viscoelastic problem

Sandra Carillo ^1, , Vanda Valente ^2, and Giorgio Vergara Caffarelli ^1,

1.	Dipartimento di Scienze di Base e Applicate per l’Ingegneria, sez. matematica – 16, Via A. Scarpa, SAPIENZA Università di Roma, Rome, 00161, Italy, Italy
2.	Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, Rome, 00185, Italy

Received August 2010 Revised September 2010 Published October 2011

Abstract
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The dynamics of magneto-viscoelastic materials is described by a nonlinear system which couples the equation of the magnetization, given in Gibert form, and the viscoelastic integro-differential equation for the displacements. We study the general three-dimensional case and establish a theorem for the existence of weak solutions. The existence is proved by compactness of the approximated penalty problem.

Keywords: nonlinear integro-differential problem, Magneto-viscoelastic materials, materials with memory..

Mathematics Subject Classification: Primary: 74H20, 35Q74; Secondary: 45K0.

Citation: Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. An existence theorem for the magneto-viscoelastic problem. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 435-447. doi: 10.3934/dcdss.2012.5.435

References:

[1]	W. F. Brown, "Magnetoelastic Interactions,", Springer Tracts in Natural Philosophy, 9 (1966).
[2]	S. Carillo, V. Valente and G. Vergara, Caffarelli, Existence and uniqueness in magneto-viscoelasticity,, Applicable Analysis, (2010).
[3]	M. Chipot and G. Vergara-Caffarelli, Viscoelasticity without initial conditions,, in, 190 (1989), 52.
[4]	M. Chipot and G. Vergara Caffarelli, Some results in viscoelasticity theory via a simple perturbation argument,, Rend. Sem. Mat. Univ. Padova, 84 (1990), 223.
[5]	M. Chipot, I. Shafrir, V. Valente and G. Vergara-Caffarelli, A nonlocal problem arising in the study of magneto-elastic interactions,, Boll. UMI (9), 1 (2008), 197.
[6]	M. Chipot, I. Shafrir, V. Valente and G. Vergara-Caffarelli, On a hyperbolic-parabolic system arising in magnetoelasticity,, J. Math. Anal. Appl., 352 (2009), 120. doi: 10.1016/j.jmaa.2008.04.013.
[7]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity,, J. Diff. Equations, 7 (1970), 554. doi: 10.1016/0022-0396(70)90101-4.
[8]	C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rat. Mech. Anal., 37 (1970), 297. doi: 10.1007/BF00251609.
[9]	T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetization field,, Phys. Rev., 100 (1955).
[10]	S. He, "Modélisation et Simulation Numérique de Matériaux Magnétostrictifs,", Ph.D thesis, (1999).
[11]	D. Kinderlehrer, Magnetoelastic interactions,, in, 25 (1996), 177.
[12]	L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,, Phys. Z. Sowjet., 8 (1935).
[13]	D. Sforza and G. Vergara-Caffarelli, A Volterra integro-differential equation "without initial conditions,", Adv. Math. Sci. Appl., 11 (2001), 153.
[14]	V. Valente and G. Vergara-Caffarelli, On the dynamics of magneto-elastic interactions: Existence of solutions and limit behaviors,, Asymptotic Analysis, 51 (2007), 319.
[15]	G. Vergara-Caffarelli, Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare,, Atti Accad. Naz. Lincei, 82 (1988), 483.
[16]	G. Vergara-Caffarelli, Dissipatività ed esistenza per il problema dinamico unidimensionale della viscoelasticità lineare,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 489.

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References:

[1]	W. F. Brown, "Magnetoelastic Interactions,", Springer Tracts in Natural Philosophy, 9 (1966).
[2]	S. Carillo, V. Valente and G. Vergara, Caffarelli, Existence and uniqueness in magneto-viscoelasticity,, Applicable Analysis, (2010).
[3]	M. Chipot and G. Vergara-Caffarelli, Viscoelasticity without initial conditions,, in, 190 (1989), 52.
[4]	M. Chipot and G. Vergara Caffarelli, Some results in viscoelasticity theory via a simple perturbation argument,, Rend. Sem. Mat. Univ. Padova, 84 (1990), 223.
[5]	M. Chipot, I. Shafrir, V. Valente and G. Vergara-Caffarelli, A nonlocal problem arising in the study of magneto-elastic interactions,, Boll. UMI (9), 1 (2008), 197.
[6]	M. Chipot, I. Shafrir, V. Valente and G. Vergara-Caffarelli, On a hyperbolic-parabolic system arising in magnetoelasticity,, J. Math. Anal. Appl., 352 (2009), 120. doi: 10.1016/j.jmaa.2008.04.013.
[7]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity,, J. Diff. Equations, 7 (1970), 554. doi: 10.1016/0022-0396(70)90101-4.
[8]	C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rat. Mech. Anal., 37 (1970), 297. doi: 10.1007/BF00251609.
[9]	T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetization field,, Phys. Rev., 100 (1955).
[10]	S. He, "Modélisation et Simulation Numérique de Matériaux Magnétostrictifs,", Ph.D thesis, (1999).
[11]	D. Kinderlehrer, Magnetoelastic interactions,, in, 25 (1996), 177.
[12]	L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,, Phys. Z. Sowjet., 8 (1935).
[13]	D. Sforza and G. Vergara-Caffarelli, A Volterra integro-differential equation "without initial conditions,", Adv. Math. Sci. Appl., 11 (2001), 153.
[14]	V. Valente and G. Vergara-Caffarelli, On the dynamics of magneto-elastic interactions: Existence of solutions and limit behaviors,, Asymptotic Analysis, 51 (2007), 319.
[15]	G. Vergara-Caffarelli, Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare,, Atti Accad. Naz. Lincei, 82 (1988), 483.
[16]	G. Vergara-Caffarelli, Dissipatività ed esistenza per il problema dinamico unidimensionale della viscoelasticità lineare,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 489.

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