Breakdown of smooth solutions in one-dimensional magnetostriction
Author:
Harumi Hattori
Journal:
Quart. Appl. Math. 58 (2000), 1-16
MSC:
Primary 35Q72; Secondary 35B65, 74F15, 74H99, 78A55
DOI:
https://doi.org/10.1090/qam/1738555
MathSciNet review:
MR1738555
Full-text PDF Free Access
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Abstract: We discuss a hyperbolic aspect of magnetostriction. The equations governing the longitudinal motion consist of the nonlinear wave equations and the rate equations for the motion of spin for the magnetic moment. We show that the breakdown of smooth solutions will take place in finite time even if the initial data are smooth.
W.F. Brown, Micromagnetics, John Wiley and Sons, New York, 1963
W. F. Brown, Magnetoelastic Interactions, Springer Tracts in Natural Philosophy, Vol. 9, Springer-Verlag, New York, 1966
- C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal. 91 (1985), no. 3, 193–205. MR 806001, DOI https://doi.org/10.1007/BF00250741
- C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44 (1986), no. 3, 463–474. MR 860899, DOI https://doi.org/10.1090/S0033-569X-1986-0860899-8
- Harumi Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Quart. Appl. Math. 40 (1982/83), no. 2, 113–127. MR 666668, DOI https://doi.org/10.1090/S0033-569X-1982-0666668-X
- William J. Hrusa and Salim A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 (1990), no. 2, 135–151. MR 1057652, DOI https://doi.org/10.1007/BF00375405
R. D. James and D. Kinderlehrer, A theory of magnetostriction with application to TbDyFe$_{2}$, Phil. Mag., to appear
- D. Kinderlehrer and L. Ma, The hysteretic event in the computation of magnetization, J. Nonlinear Sci. 7 (1997), no. 2, 101–128. MR 1437985, DOI https://doi.org/10.1007/BF02677975
L. Landau and E. Lifshitz, Electrodynamics of continuous media, Pergamon Press, New York, 1960
- Mitchell Luskin and Ling Ma, Analysis of the finite element approximation of microstructure in micromagnetics, SIAM J. Numer. Anal. 29 (1992), no. 2, 320–331. MR 1154269, DOI https://doi.org/10.1137/0729021
- W. L. Miranker and B. E. Willner, Global analysis of magnetic domains, Quart. Appl. Math. 37 (1979/80), no. 3, 219–238. MR 548986, DOI https://doi.org/10.1090/S0033-569X-1979-0548986-4
- Robert C. Rogers, A nonlocal model for the exchange energy in ferromagnetic materials, J. Integral Equations Appl. 3 (1991), no. 1, 85–127. MR 1094932, DOI https://doi.org/10.1216/jiea/1181075602
- M. Slemrod, Instability of steady shearing flows in a nonlinear viscoelastic fluid, Arch. Rational Mech. Anal. 68 (1978), no. 3, 211–225. MR 509225, DOI https://doi.org/10.1007/BF00247740
- H. F. Tiersten, Coupled magnetomechanical equations for magnetically saturated insulators, J. Mathematical Phys. 5 (1964), 1298–1318. MR 181224, DOI https://doi.org/10.1063/1.1704239
- H. F. Tiersten, Variational principle for saturated magnetoelastic insulators, J. Mathematical Phys. 6 (1965), 779–787. MR 177766, DOI https://doi.org/10.1063/1.1704334
- A. Visintin, On Landau-Lifshitz’ equations for ferromagnetism, Japan J. Appl. Math. 2 (1985), no. 1, 69–84. MR 839320, DOI https://doi.org/10.1007/BF03167039
W.F. Brown, Micromagnetics, John Wiley and Sons, New York, 1963
W. F. Brown, Magnetoelastic Interactions, Springer Tracts in Natural Philosophy, Vol. 9, Springer-Verlag, New York, 1966
C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal. 91, 193–205 (1986)
C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44, 463–474 (1986)
H. Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Quart. Appl. Math. 40, 113–127 (1982)
W. J. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity 111, 135–151 (1990)
R. D. James and D. Kinderlehrer, A theory of magnetostriction with application to TbDyFe$_{2}$, Phil. Mag., to appear
D. Kinderlehrer and L. Ma, The hysteretic event in the computation of magnetization and magnetostriction, preprint
L. Landau and E. Lifshitz, Electrodynamics of continuous media, Pergamon Press, New York, 1960
M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics, SIAM J. Numer. Anal. 29, 320–331 (1992)
M. W. Miranker and B. E. Willner, Global analysis of magnetic domain, Quart. Appl. Math. 37, 221–238 (1987)
R. C. Rogers, A nonlocal model for the exchange energy in ferromagnetic materials, J. Integral Equations Appl. 3, 85–127 (1991)
M. Slemrod, Instability of steady shearing flows in a non-linear viscoelastic fluid, Arch. Rational Mech. Anal. 68, 211–225 (1978)
H. F. Tiersten, Coupled magnetomechanical equations for magnetically saturated insulators, J. Math. Phys. 5, 1298–1318 (1964)
H. F. Tiersten, Variational principle for saturated magnetoelastic insulators, J. Math. Phys. 6, 779–787 (1995)
A. Visintin, On Landau-Lifshitz’ equations for ferromagnetism, Japan J. Appl. Math. 2, 69–84 (1985)
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