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On the stability of steady-states of a two-dimensional system of ferromagnetic nanowires. (English) Zbl 1382.35286

The paper concerns the stability of steady states of the two-dimensional finite system of straight ferromagnetic nanowires. The magnetization vector is assumed to be independent of the space variable and the Landau-Lifshitz equation becomes a classical system of coupled ODEs. The authors defined a stray field for an entire system with the interaction among the wires, and established the steady results when the wires are sufficiently far from each other.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
82D40 Statistical mechanics of magnetic materials
82D77 Quantum waveguides, quantum wires
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[1] C. Abert, L. Exl, G. Selke, A. Drews and T. Schrefl, Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, J. Magnetism Magnetic Mater. 326 (2013), 176-185.; Abert, C.; Exl, L.; Selke, G.; Drews, A.; Schrefl, T., Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, J. Magnetism Magnetic Mater., 326, 176-185 (2013)
[2] S. Agarwal, G. Carbou, S. Labbé and C. Prieur, Control of a network of magnetic ellipsoidal samples, Math. Control Relat. Fields 1 (2011), no. 2, 129-147.; Agarwal, S.; Carbou, G.; Labbé, S.; Prieur, C., Control of a network of magnetic ellipsoidal samples, Math. Control Relat. Fields, 1, 2, 129-147 (2011) · Zbl 1231.93013
[3] A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford University Press, Oxford, 2000.; Aharoni, A., Introduction to the Theory of Ferromagnetism (2000)
[4] F. Alouges, A new finite element scheme for Landau-Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, 187-196.; Alouges, F., A new finite element scheme for Landau-Lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S, 1, 2, 187-196 (2008) · Zbl 1152.35304
[5] F. Alouges, Mathematical models in micromagnetism: An introduction, ESAIM Proc. 22 (2008), 114-117.; Alouges, F., Mathematical models in micromagnetism: An introduction, ESAIM Proc., 22, 114-117 (2008) · Zbl 1166.35378
[6] F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal. 18 (1992), no. 11, 1071-1084.; Alouges, F.; Soyeur, A., On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal., 18, 11, 1071-1084 (1992) · Zbl 0788.35065
[7] L. Baňas, Stochastic Ferromagnetism: Analysis and Numerics, De Gruyter Stud. Math. 58, Walter de Gruyter, Berlin, 2014.; Baňas, L., Stochastic Ferromagnetism: Analysis and Numerics (2014)
[8] L. Baňas, S. Bartels and A. Prohl, A convergent implicit finite element discretization of the Maxwell-Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 46 (2008), no. 3, 1399-1422.; Baňas, L.; Bartels, S.; Prohl, A., A convergent implicit finite element discretization of the Maxwell-Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 46, 3, 1399-1422 (2008) · Zbl 1173.35321
[9] S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44 (2006), no. 4, 1405-1419.; Bartels, S.; Prohl, A., Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44, 4, 1405-1419 (2006) · Zbl 1124.65088
[10] G. Boling and S. Fengqiu, Global weak solution for the Landau-Lifshitz-Maxwell equation in three space dimensions, J. Math. Anal. Appl. 211 (1997), no. 1, 326-346.; Boling, G.; Fengqiu, S., Global weak solution for the Landau-Lifshitz-Maxwell equation in three space dimensions, J. Math. Anal. Appl., 211, 1, 326-346 (1997) · Zbl 0877.35122
[11] W. F. Brown, Micromagnetics, Wiley, New York, 1963.; Brown, W. F., Micromagnetics (1963)
[12] F. Bruckner, D. Suess, M. Feischl, T. Führer, P. Goldenits, M. Page, D. Praetorius and M. Ruggeri, Multiscale modeling in micromagnetics: Existence of solutions and numerical integration, Math. Models Methods Appl. Sci. 24 (2014), no. 13, 2627-2662.; Bruckner, F.; Suess, D.; Feischl, M.; Führer, T.; Goldenits, P.; Page, M.; Praetorius, D.; Ruggeri, M., Multiscale modeling in micromagnetics: Existence of solutions and numerical integration, Math. Models Methods Appl. Sci., 24, 13, 2627-2662 (2014) · Zbl 1320.35336
[13] G. Carbou, Domain walls dynamics for one-dimensional models of ferromagnetic nanowires, Differential Integral Equations 26 (2013), no. 3-4, 201-236.; Carbou, G., Domain walls dynamics for one-dimensional models of ferromagnetic nanowires, Differential Integral Equations, 26, 3-4, 201-236 (2013) · Zbl 1289.35151
[14] G. Carbou and P. Fabrie, Time average in micromagnetism, J. Differential Equations 147 (1998), no. 2, 383-409.; Carbou, G.; Fabrie, P., Time average in micromagnetism, J. Differential Equations, 147, 2, 383-409 (1998) · Zbl 0931.35170
[15] G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential Integral Equations 14 (2001), no. 2, 213-229.; Carbou, G.; Fabrie, P., Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential Integral Equations, 14, 2, 213-229 (2001) · Zbl 1161.35421
[16] G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in \mathbb{R}^3, Commun. Appl. Anal. 5 (2001), no. 1, 17-30.; Carbou, G.; Fabrie, P., Regular solutions for Landau-Lifschitz equation in \mathbb{R}^3, Commun. Appl. Anal., 5, 1, 17-30 (2001) · Zbl 1084.35519
[17] G. Carbou and S. Labbé, Stability for static walls in ferromagnetic nanowires, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 2, 273-290.; Carbou, G.; Labbé, S., Stability for static walls in ferromagnetic nanowires, Discrete Contin. Dyn. Syst. Ser. B, 6, 2, 273-290 (2006) · Zbl 1220.82163
[18] G. Carbou and S. Labbé, Stabilization of walls for nano-wires of finite length, ESAIM Control Optim. Calc. Var. 18 (2012), no. 1, 1-21.; Carbou, G.; Labbé, S., Stabilization of walls for nano-wires of finite length, ESAIM Control Optim. Calc. Var., 18, 1, 1-21 (2012) · Zbl 1235.35029
[19] G. Carbou, S. Labbé and E. Trélat, Control of travelling walls in a ferromagnetic nanowire, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 1, 51-59.; Carbou, G.; Labbé, S.; Trélat, E., Control of travelling walls in a ferromagnetic nanowire, Discrete Contin. Dyn. Syst. Ser. S, 1, 1, 51-59 (2008) · Zbl 1310.82059
[20] I. Cimrák, A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism, Arch. Comput. Methods Eng. 15 (2008), no. 3, 277-309.; Cimrák, I., A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism, Arch. Comput. Methods Eng., 15, 3, 277-309 (2008) · Zbl 1206.78008
[21] D. J. Griffiths, Introduction to Electrodynamics, 3rd ed., Pearson Benjamin Cummings, San Francisco, 2008.; Griffiths, D. J., Introduction to Electrodynamics (2008) · Zbl 1377.78001
[22] A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures, Springer, Berlin, 1998.; Hubert, A.; Schäfer, R., Magnetic Domains: The Analysis of Magnetic Microstructures (1998)
[23] M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev. 48 (2006), no. 3, 439-483.; Kružík, M.; Prohl, A., Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev., 48, 3, 439-483 (2006) · Zbl 1126.49040
[24] S. Labbé, Fast computation for large magnetostatic systems adapted for micromagnetism, SIAM J. Sci. Comput. 26 (2005), no. 6, 2160-2175.; Labbé, S., Fast computation for large magnetostatic systems adapted for micromagnetism, SIAM J. Sci. Comput., 26, 6, 2160-2175 (2005) · Zbl 1107.78016
[25] S. Labbé, Y. Privat and E. Trélat, Stability properties of steady-states for a network of ferromagnetic nanowires, J. Differential Equations 253 (2012), no. 6, 1709-1728.; Labbé, S.; Privat, Y.; Trélat, E., Stability properties of steady-states for a network of ferromagnetic nanowires, J. Differential Equations, 253, 6, 1709-1728 (2012) · Zbl 1247.35173
[26] L. Landau and E. Lifschitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8 (1935), no. 153, 101-114.; Landau, L.; Lifschitz, E., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8, 153, 101-114 (1935)
[27] L. Landau and E. Lifschitz, Électrodynamique des Milieux Continus, Cours de Physique Théorique. Tome 8, Mir, Moscow, 1969.; Landau, L.; Lifschitz, E., Électrodynamique des Milieux Continus, Cours de Physique Théorique. Tome 8 (1969) · Zbl 0146.23803
[28] C. B. Muratov and V. V. Osipov, Bit storage by \(360^{\circ}\) domain walls in ferromagnetic nanorings, IEEE Trans. Magnetics 45 (2009), no. 8, 3207-3209.; Muratov, C. B.; Osipov, V. V., Bit storage by \(360^{\circ}\) domain walls in ferromagnetic nanorings, IEEE Trans. Magnetics, 45, 8, 3207-3209 (2009)
[29] S. S. P. Parkin, M. Hayashi and L. Thomos, Magnetic domain-wall racetrack memory, Science 320 (2008), 190-194.; Parkin, S. S. P.; Hayashi, M.; Thomos, L., Magnetic domain-wall racetrack memory, Science, 320, 190-194 (2008)
[30] A. Prohl, Computational Micromagnetism, B. G. Teubner, Stuttgart, 2001.; Prohl, A., Computational Micromagnetism (2001)
[31] D. Sanchez, Behaviour of the Landau-Lifschitz equation in a ferromagnetic wire, Math. Methods Appl. Sci. 32 (2009), no. 2, 167-205.; Sanchez, D., Behaviour of the Landau-Lifschitz equation in a ferromagnetic wire, Math. Methods Appl. Sci., 32, 2, 167-205 (2009) · Zbl 1152.35504
[32] A. Visintin, On Landau-Lifschitz’ equations for ferromagnetism, Japan J. Appl. Math. 2 (1985), no. 1, 69-84.; Visintin, A., On Landau-Lifschitz’ equations for ferromagnetism, Japan J. Appl. Math., 2, 1, 69-84 (1985) · Zbl 0613.35018
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