An existence theorem for the magneto-viscoelastic problem

Previous Article
Some identities on weighted Sobolev spaces
DCDS-S Home
This Issue
Next Article
Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $

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
Related Papers

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). Google Scholar
[2]	S. Carillo, V. Valente and G. Vergara, Caffarelli, Existence and uniqueness in magneto-viscoelasticity,, Applicable Analysis, (2010). Google Scholar
[3]	M. Chipot and G. Vergara-Caffarelli, Viscoelasticity without initial conditions,, in, 190 (1989), 52. 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. 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. 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. 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. 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. 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). Google Scholar
[10]	S. He, "Modélisation et Simulation Numérique de Matériaux Magnétostrictifs,", Ph.D thesis, (1999). Google Scholar
[11]	D. Kinderlehrer, Magnetoelastic interactions,, in, 25 (1996), 177. 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). Google Scholar
[13]	D. Sforza and G. Vergara-Caffarelli, A Volterra integro-differential equation "without initial conditions,", Adv. Math. Sci. Appl., 11 (2001), 153. 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. Google Scholar
[15]	G. Vergara-Caffarelli, Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare,, Atti Accad. Naz. Lincei, 82 (1988), 483. 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. Google Scholar

show all references

References:

[1]	W. F. Brown, "Magnetoelastic Interactions,", Springer Tracts in Natural Philosophy, 9 (1966). Google Scholar
[2]	S. Carillo, V. Valente and G. Vergara, Caffarelli, Existence and uniqueness in magneto-viscoelasticity,, Applicable Analysis, (2010). Google Scholar
[3]	M. Chipot and G. Vergara-Caffarelli, Viscoelasticity without initial conditions,, in, 190 (1989), 52. 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. 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. 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. 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. 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. 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). Google Scholar
[10]	S. He, "Modélisation et Simulation Numérique de Matériaux Magnétostrictifs,", Ph.D thesis, (1999). Google Scholar
[11]	D. Kinderlehrer, Magnetoelastic interactions,, in, 25 (1996), 177. 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). Google Scholar
[13]	D. Sforza and G. Vergara-Caffarelli, A Volterra integro-differential equation "without initial conditions,", Adv. Math. Sci. Appl., 11 (2001), 153. 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. Google Scholar
[15]	G. Vergara-Caffarelli, Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare,, Atti Accad. Naz. Lincei, 82 (1988), 483. 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. Google Scholar

[1]	Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
[2]	Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[3]	John Murrough Golden. Constructing free energies for materials with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 447-483. doi: 10.3934/eect.2014.3.447
[4]	Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[5]	Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[6]	Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
[7]	Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525
[8]	Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235
[9]	Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249
[10]	Mariano Giaquinta, Paolo Maria Mariano, Giuseppe Modica. A variational problem in the mechanics of complex materials. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 519-537. doi: 10.3934/dcds.2010.28.519
[11]	Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
[12]	Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
[13]	Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[14]	Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026
[15]	Dariusz Idczak, Stanisław Walczak. Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2281-2292. doi: 10.3934/dcdsb.2019095
[16]	Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885
[17]	Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
[18]	Tomás Caraballo, I. D. Chueshov, Pedro Marín-Rubio, José Real. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 253-270. doi: 10.3934/dcds.2007.18.253
[19]	Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379
[20]	Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences