In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.
F.Alouges and A.Soyeur, On global weak solutions for Landau–Lifshitz equations: Existence and non uniqueness, Nonlinear Anal.18 (1992), 1071–1084. doi:10.1016/0362-546X(92)90196-L.
2.
C.Ayouch, M.Benmouane and E.H.Essoufi, Regular solution for the compressible Landau–Lifshitz–Bloch equation in a bounded domain of , J Elliptic Parabol Equ.8 (2022), 419–441. doi:10.1007/s41808-022-00160-1.
3.
C.Ayouch, E.H.Essoufi and M.Tilioua, On a model of magnetization dynamics with vertical spin stiffness, Boundary Value Problems2016 (2016), 110. doi:10.1186/s13661-016-0618-3.
4.
C.Ayouch, E.H.Essoufi and M.Tilioua, A finite difference scheme for the time-fractional Landau–Lifshitz–Bloch equation, Research in Applied Mathematics1 (2017), Article ID 101264.
5.
C.Ayouch, E.H.Essoufi and M.Tilioua, On a non scalar damping model in micromagnetism, Int. J. Dynamical Systems and Differential Equations8(1–2) (2018), 6–18. doi:10.1504/IJDSDE.2018.089091.
6.
C.Ayouch, K.S.Nisar, M.Tilioua and M.Zakarya, On the Landau–Lifshitz–Bloch equation with spin torque effects, Alexandria Engineering Journal.60(5) (2021), 4433–4439. doi:10.1016/j.aej.2021.03.025.
7.
C.Ayouch and M.Tilioua, Local existence and uniqueness of regular solutions to a Landau–Lifshitz–Bloch equation with applied current, J. Appl. Anal.29(1) (2023), 113–122. doi:10.1515/jaa-2022-2003.
8.
V.Baryakhtar, In Front. Magn. Reduc. Dimens. Syst., pp. 63–94 (1998).
9.
Z.Brzeźniak, B.Goldys and K.N.Le, Existence of a unique solution and invariant measures for the stochastic Landau–Lifshitz–Bloch equation, J. Differ. Equ.269(11) (2020), 9471–9507. doi:10.1016/j.jde.2020.06.061.
10.
G.Carbou and R.Jizzini, Very regular solutions for the Landau–Lifschitz equation with electric current, Chinese Annals of Mathematics – Series B39(5) (2018), 889–916. doi:10.1007/s11401-018-0103-7.
11.
R.Dautray and J.-L.Lions, Mathematical Analysis and Numerical Methods, Sciences and Technology, Springer-Verlag, 2000.
12.
G.Di Fratta, M.Innerberger and D.Praetorius, Weak-strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics, Nonlinear Analysis: Real World Applications55 (2020), 103122.
13.
G.Di Fratta, C.-M.Pfeiler, D.Praetorius and M.Ruggeri, The mass-lumped midpoint scheme for computational micromagnetics: Newton linearization and application to magnetic skyrmion dynamics, Comput. Methods Appl. Math.1(23) (2023), 145–175.
14.
G.Di Fratta, C.-M.Pfeiler, D.Praetorius, M.Ruggeri and B.Stiftner, Linear second-order IMEX-type integrator for the (Eddy current) Landau–Lifshitz–Gilbert equation, IMA J. Numer. Anal.40(4) (2020), 2802–2838. doi:10.1093/imanum/drz046.
15.
S.Ding, B.Guo, J.Lin and M.Zeng, Global existence of weak solutions for Landau–Lifshitz–Maxwell equations, Discrete & Continuous Dynamical Systems A17(4) (2007), 867–890. doi:10.3934/dcds.2007.17.867.
16.
M.Dvornik, A.Vansteenkiste and B.Van Waeyenberge, Micromagnetic modeling of anisotropic damping in magnetic nanoelements, Phys. Rev. B88 (2013), 054427. doi:10.1103/PhysRevB.88.054427.
17.
G.Foias and R.Temam, Remarques sur les équations de Navier–Stokes stationnaires et les phénomènes successifs de bifurcation, An. Sc. Norm. Super. Pisa IV5 (1978), 29–63.
18.
S.Gokhale and U.Manna, Wong–Zakai approximations for the stochastic Landau–Lifshitz–Bloch equations, J. Math. Phys.63 (2022), 091512. doi:10.1063/5.0088961.
19.
M.Hadda and M.Tilioua, On magnetization dynamics with inertial effects, J. Engineering Mathematics88 (2014), 197–206. doi:10.1007/s10665-014-9691-8.
20.
K.Hamdache and D.Hamroun, Asymptotic behaviours for the Landau–Lifshitz–Bloch equation, Adv. Theory Nonlinear Anal. Appl.4(3) (2019), 174–191.
21.
K.N.Le, Weak solutions of the Landau–Lifshitz–Bloch equation, J. Differ. Equ.261(12) (2016), 6699–6717. doi:10.1016/j.jde.2016.09.002.
22.
Q.Lia, B.Guo, F.Liu and W.Liu, Weak and strong solutions to Landau–Lifshitz–Bloch–Maxwell equations with polarization, J. Differ. Equ.286 (2021), 47–83. doi:10.1016/j.jde.2021.02.042.
23.
J.L.Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod & Gauthier-Villars, Paris, 1969.
24.
N.J.Mauser, C.-M.Pfeiler, D.Praetorius and M.Ruggeri, Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics, Appl. Numer. Math.180 (2022), 33–54. doi:10.1016/j.apnum.2022.05.008.
25.
P.Podio-Guidugli and V.Valente, Existence of global-in-time weak solutions to a modified Gilbert equation, Nonlinear Anal.47 (2001), 147–158. doi:10.1016/S0362-546X(01)00164-X.
26.
D.Praetorius, M.Ruggeri and B.Stiftner, Convergence of an implicit–explicit midpoint scheme for computational micromagnetics, Comput. Math. Appl.75(5) (2018), 1719–1738. doi:10.1016/j.camwa.2017.11.028.
27.
M.Ruggeri, Numerical analysis of the Landau–Lifshitz–Gilbert equation with inertial effects, ESAIM: Mathematical Modelling and Numerical Analysis.56(4) (2022), 1199–1222. doi:10.1051/m2an/2022043.
28.
M.Tilioua, Current-induced magnetization dynamics. Global existence of weak solutions, J. Math. Anal. Appl.373(2) (2011), 635–642. doi:10.1016/j.jmaa.2010.08.024.
29.
W.Wanget al., Phenomenological description of the nonlocal magnetization relaxation in magnonics, spintronics, and domain-wall dynamics, Physical Review B92 (2015), 054430. doi:10.1103/PhysRevB.92.054430.