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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

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, Springer-Verlag, 1966. Google Scholar

[2]

S. Carillo, V. Valente and G. Vergara, Caffarelli, Existence and uniqueness in magneto-viscoelasticity, Applicable Analysis, 2010, in press. Google Scholar

[3]

M. Chipot and G. Vergara-Caffarelli, Viscoelasticity without initial conditions, in "Volterra Integrodifferential Equations in Banach Spaces and Applications" (eds. G. Da Prato and M. Iannelli), Pitman Res. Notes Math. Ser., 190, Longman Sci. Tech., Harlow, (1989), 52-66.  Google Scholar

[4]

M. Chipot and G. Vergara Caffarelli, Some results in viscoelasticity theory via a simple perturbation argument, Rend. Sem. Mat. Univ. Padova, 84 (1990), 223-239.  Google Scholar

[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), 197-221.  Google Scholar

[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), 120-131. doi: 10.1016/j.jmaa.2008.04.013.  Google Scholar

[7]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Diff. Equations, 7 (1970), 554-569. doi: 10.1016/0022-0396(70)90101-4.  Google Scholar

[8]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609.  Google Scholar

[9]

T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243. Google Scholar

[10]

S. He, "Modélisation et Simulation Numérique de Matériaux Magnétostrictifs," Ph.D thesis, Université Pierre et Marie Currie, 1999. Google Scholar

[11]

D. Kinderlehrer, Magnetoelastic interactions, in "Variational Methods for Discontinuous Structures" (Como, 1994), Prog. Nonlinear Differential Equations Appl., 25, Birkhäuser, Basel, (1996), 177-189.  Google Scholar

[12]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjet., 8 (1935), 153. Google Scholar

[13]

D. Sforza and G. Vergara-Caffarelli, A Volterra integro-differential equation "without initial conditions," Adv. Math. Sci. Appl., 11 (2001), 153-160.  Google Scholar

[14]

V. Valente and G. Vergara-Caffarelli, On the dynamics of magneto-elastic interactions: Existence of solutions and limit behaviors, Asymptotic Analysis, 51 (2007), 319-333.  Google Scholar

[15]

G. Vergara-Caffarelli, Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare, Atti Accad. Naz. Lincei, 82 (1988), 483-488. Google Scholar

[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), 489-496.  Google Scholar

show all references

References:
[1]

W. F. Brown, "Magnetoelastic Interactions," Springer Tracts in Natural Philosophy, 9, Springer-Verlag, 1966. Google Scholar

[2]

S. Carillo, V. Valente and G. Vergara, Caffarelli, Existence and uniqueness in magneto-viscoelasticity, Applicable Analysis, 2010, in press. Google Scholar

[3]

M. Chipot and G. Vergara-Caffarelli, Viscoelasticity without initial conditions, in "Volterra Integrodifferential Equations in Banach Spaces and Applications" (eds. G. Da Prato and M. Iannelli), Pitman Res. Notes Math. Ser., 190, Longman Sci. Tech., Harlow, (1989), 52-66.  Google Scholar

[4]

M. Chipot and G. Vergara Caffarelli, Some results in viscoelasticity theory via a simple perturbation argument, Rend. Sem. Mat. Univ. Padova, 84 (1990), 223-239.  Google Scholar

[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), 197-221.  Google Scholar

[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), 120-131. doi: 10.1016/j.jmaa.2008.04.013.  Google Scholar

[7]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Diff. Equations, 7 (1970), 554-569. doi: 10.1016/0022-0396(70)90101-4.  Google Scholar

[8]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609.  Google Scholar

[9]

T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243. Google Scholar

[10]

S. He, "Modélisation et Simulation Numérique de Matériaux Magnétostrictifs," Ph.D thesis, Université Pierre et Marie Currie, 1999. Google Scholar

[11]

D. Kinderlehrer, Magnetoelastic interactions, in "Variational Methods for Discontinuous Structures" (Como, 1994), Prog. Nonlinear Differential Equations Appl., 25, Birkhäuser, Basel, (1996), 177-189.  Google Scholar

[12]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjet., 8 (1935), 153. Google Scholar

[13]

D. Sforza and G. Vergara-Caffarelli, A Volterra integro-differential equation "without initial conditions," Adv. Math. Sci. Appl., 11 (2001), 153-160.  Google Scholar

[14]

V. Valente and G. Vergara-Caffarelli, On the dynamics of magneto-elastic interactions: Existence of solutions and limit behaviors, Asymptotic Analysis, 51 (2007), 319-333.  Google Scholar

[15]

G. Vergara-Caffarelli, Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare, Atti Accad. Naz. Lincei, 82 (1988), 483-488. Google Scholar

[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), 489-496.  Google Scholar

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