Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-31T16:27:03.711Z Has data issue: false hasContentIssue false
  • English
  • Français

(In-)stability of singular equivariant solutions to the Landau–Lifshitz–Gilbert equation

Published online by Cambridge University Press:  09 August 2013

JAN BOUWE VAN DEN BERG  and
J. F. WILLIAMS
JAN BOUWE VAN DEN BERG
Affiliation:
Department of Mathematics, VU University Amsterdam, de Boelelaan 1081, 1081 HV Amsterdam, the Netherlands email: janbouwe@math.vu.nl
J. F. WILLIAMS
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada email: jfw@math.sfu.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we use formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau–Lifshitz–Gilbert model for ferromagnets. We also analyse both the harmonic map heatflow and Schrödinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic.

Keywords

Landau–Lifshitz–GilbertHarmonic map heatflowSchrödinger map flowAsymptotic analysisBlowupNumerical simulationsAdaptive numerical methods

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 6 , December 2013 , pp. 921 - 948
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alouges, F. & Soyeur, A. (1992) On global weak solutions for Landau–Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal. 18, 1071–1084.Google Scholar
[2]Angenent, S. & Hulshof, J. (2005) Singularities at t=∞ in equivariant harmonic map flow. In: Chang, S., Chow, B., Chu, S. C.et al. (editors), Geometric Evolution Equations, Contemp. Math. vol. 367, Amer. Math. Soc., Providence, RI, pp. 1–15.Google Scholar
[3]Angenent, S. B., Hulshof, J. & Matano, H. (2009) The radius of vanishing bubbles in equivariant harmonic map flow from D 2 to S 2. SIAM J. Math. Anal. 41, 1121–1137.Google Scholar
[4]Bartels, S., Ko, J. & Prohl, A. (2008) Numerical analysis of an explicit approximation scheme for the Landau–Lifshitz–Gilbert equation. Math. Comp. 77, 773–788.Google Scholar
[5]Bejenaru, I. & Tataru, D. (2010) Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions, preprint (arXiv:1009.1608).Google Scholar
[6]Bertsch, M., Passo, R. D. & van der Hout, R. (2002) Nonuniqueness for the heat flow of harmonic maps on the disk. Arch. Ration. Mech. Anal. 161, 93–112.Google Scholar
[7]Bertsch, M., Podio Guidugli, P. & Valente, V. (2001) On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest. Ann. Mat. Pura Appl. 179 (4), 331–360.Google Scholar
[8]Bertsch, M., van der Hout, R. & Hulshof, J. (2011) Energy concentration for 2-dimensional radially symmetric equivariant harmonic map heat flows. Commun. Contemp. Math. 13, 675–695.Google Scholar
[9]Bethuel, F., Brezis, H., Coleman, B. D. & Hélein, F. (1992) Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders. Arch. Ration. Mech. Anal. 118, 149–168.CrossRefGoogle Scholar
[10]Bizoń, P., Chmaj, T. & Tabor, Z. (2000) Dispersion and collapse of wave maps. Nonlinearity 13, 1411–1423.CrossRefGoogle Scholar
[11]Budd, C. J. & Williams, J. F. (2009) Moving mesh generation using the parabolic Monge-Ampère equation. SIAM J. Sci. Comp. 31, 3438–3465.Google Scholar
[12]Budd, C. J. & Williams, J. F. (2010) How to adaptively resolve evolutionary singularities in differential equations with symmetry. J. Eng. Math. 66, 217–236.Google Scholar
[13]Chang, K.-C., Ding, W. Y. & Ye, R. (1992) Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36, 507–515.Google Scholar
[14]Chang, N.-H., Shatah, J. & Uhlenbeck, K. (2000) Schrödinger maps. Commun. Pure Appl. Math. 53, 590–602.Google Scholar
[15]Coron, J.-M. (1990) Nonuniqueness for the heat flow of harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 335–344.Google Scholar
[16]E, W. & Wang, X.-P. (2001) Numerical methods for the Landau–Lifshitz equation. SIAM J. Numer. Anal. 38, 1647–1665.CrossRefGoogle Scholar
[17]Freire, A. (1995) Uniqueness for the harmonic map flow in two dimensions. Calc. Var. Partial Differ. Equ. 3, 95–105.Google Scholar
[18]Guan, M., Gustafson, S. & Tsai, T.-P. (2009) Global existence and blow-up for harmonic map heat flow. J. Differ. Equ. 246, 1–20.Google Scholar
[19]Guo, B. L. & Hong, M. C. (1993) The Landau–Lifshitz equation of the ferromagnetic spin chain and harmonic maps. Calc. Var. Partial Differ. Equ. 1, 311–334.Google Scholar
[20]Gustafson, S., Kang, K. & Tsai, T.-P. (2007) Schrödinger flow near harmonic maps. Commun. Pure Appl. Math. 60, 463–499.Google Scholar
[21]Gustafson, S., Nakanishi, K. & Tsai, T.-P. (2010) Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau–Lifshitz, and Schrödinger maps on $\Bbb R^2$ Commun. Math. Phys. 300, 205–242.Google Scholar
[22]Hatcher, A. (2002) Algebraic Topology, Cambridge University Press, Cambridge.Google Scholar
[23]Huang, W., Ren, Y. & Russell, R. D. (1994) Moving mesh partial differential equations (MMPDES) based on the equidistribution principle SIAM J. Numer. Anal. 31, 709–730.CrossRefGoogle Scholar
[24]Hubert, A. & Schäfer, R. (1998) Magnetic Domains: The Analysis of Magnetic Microstructures, Springer, Berlin/Heidelberg/New York.Google Scholar
[25]Krieger, J., Schlag, W. & Tataru, D. (2008) Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171, 543–615.Google Scholar
[26]Kružík, M. & Prohl, A. (2006) Recent developments in the modeling, analysis, and numerics of ferromagnetism. SIAM Rev. 48, 439–483 (electronic).CrossRefGoogle Scholar
[27]Lemaire, L. (1978) Applications harmoniques de surfaces riemanniennes. J. Differ. Geom. 13, 51–78.Google Scholar
[28]Merle, F., Raphaël, P. & Rodnianski, I. (2011) Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. C. R. Math. Acad. Sci. Paris 349, 279–283.Google Scholar
[29]Ovchinnikov, Y. N. & Sigal, I. M. (2011) On collapse of wave maps. Phys. D 240, 1311–1324.Google Scholar
[30]Raphaël, P. & Rodnianski, I. (2012) Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. Publ. Math. Inst. Hautes Études Sci. 1–122.Google Scholar
[31]Raphaël, P. & Schweyer, R. (2013) Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow. Commun. Pure Appl. Math. 66, 414–480.CrossRefGoogle Scholar
[32]Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. & Mikhailov, A. P. (1995) Blow-up in quasilinear parabolic equations, de Gruyter Expositions in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin (Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors).Google Scholar
[33]Struwe, M. (1985) On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558–581.Google Scholar
[34]Struwe, M. (1996) Geometric evolution problems In: Hardt, R. & Wolf, M. (editors), Nonlinear Partial Differential Equations in Differential Geometry (Park City, UT, 1992), IAS/Park City Math. Ser. vol. 2, Amer. Math. Soc., Providence, RI, pp. 257–339.Google Scholar
[35]Topping, P. (2002) Reverse bubbling and nonuniqueness in the harmonic map flow. Int. Math. Res. Not., 505–520.Google Scholar
[36]van den Berg, J. B., Hulshof, J. & King, J. R. (2003) Formal asymptotics of bubbling in the harmonic map heat flow. SIAM J. Appl. Math. 63, 1682–1717 (electronic).CrossRefGoogle Scholar