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

An existence theorem for the magneto-viscoelastic problem

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.
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]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[2]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[3]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

[4]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[5]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[6]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[7]

Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122

[8]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[9]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322

[10]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[11]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083

[12]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021

[13]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[14]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[15]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[16]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[17]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039

[18]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[20]

Aisling McGlinchey, Oliver Mason. Observations on the bias of nonnegative mechanisms for differential privacy. Foundations of Data Science, 2020, 2 (4) : 429-442. doi: 10.3934/fods.2020020

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (5)

[Back to Top]