Abstract
We prove via a Galerkin approximation the time local existence of regular solutions to a Landau–Lifshitz–Bloch equation with applied current in a bounded domain. The uniqueness of the solution is also established. Moreover, we show the global in time existence of a regular solution in dimension two.
Acknowledgements
The authors would like to thank the referee and the editor for their valuable comments and suggestions.
References
[1] U. Atxitia, D. Hinzke and U. Nowak, Fundamentals and applications of the Landau–Lifshitz–Bloch equation, J. Phys. D 50 (2017), no. 3, Article ID 033003. 10.1088/1361-6463/50/3/033003Search in Google Scholar
[2] U. Atxitia, P. Nieves and O. Chubykalo-Fesenko, Landau–Lifshitz–Bloch equation for ferrimagnetic materials, Phys. Rev. B. 86 (2012), Article ID 104414. 10.1103/PhysRevB.86.104414Search in Google Scholar
[3] C. Ayouch, El-H. Essoufi and M. Tilioua, Global weak solutions to a spatio-temporal fractional Landau–Lifshitz–Bloch equation, Comput. Math. Appl. 77 (2019), no. 5, 1347–1357. 10.1016/j.camwa.2018.11.016Search in Google Scholar
[4] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B. 54 (1996), Article ID 9353. 10.1103/PhysRevB.54.9353Search in Google Scholar
[5] A. Berti and C. Giorgi, Derivation of the Landau–Lifshitz–Bloch equation from continuum thermodynamics, Phys. B. 500 (2016), 142–153. 10.1016/j.physb.2016.07.035Search in Google Scholar
[6] G. Carbou and P. Fabrie, Regular Solutions for Landau–Lifschitz equations in a bounded domain, Differential Integral Equations 14 (2001), 213–22. 10.57262/die/1356123353Search in Google Scholar
[7] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell and D. Garanin, Dynamic approach for micromagnetics close to the Curie temperature, Phys. Rev. B. 74 (2006), Article ID 094436. 10.1103/PhysRevB.74.094436Search in Google Scholar
[8] A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Search in Google Scholar
[9] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer, Berlin, 2000. Search in Google Scholar
[10] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1997. Search in Google Scholar
[11] G. Foias and R. Temam, Remarques sur les équations de Navier–Stokes stationnaires et les phénomènes successifs de bifurcation, Ann. Sc. Norm. Super. Pisa Cl. Sci. IV 5 (1978), 29–63. Search in Google Scholar
[12] D. A. Garanin, Fokker–Planck and Landau–Lifshitz–Bloch equations for classical ferromagnets, Phys. Rev. B. 55 (1997), 3050–3057. 10.1103/PhysRevB.55.3050Search in Google Scholar
[13] K. Hamdache and D. Hamroun, Solutions to the Landau–Lifshitz–Bloch Equation, preprint (2019), https://hal.archives-ouvertes.fr/hal-01879023/document. Search in Google Scholar
[14] Z. Jia, Local strong solution to general Landau–Lifshitz–Bloch equation, preprint (2019), https://arxiv.org/abs/1802.00144. Search in Google Scholar
[15] R. Jizzini, Étude mathématique d’un modèle de fil ferromagnétique en présence d’un courant électrique, Ph.D. thesis, Université Bordeaux I, 2013. Search in Google Scholar
[16] K. N. Le, Weak solutions of the Landau–Lifshitz–Bloch equation, J. Differential Equations. 261 (2016), no. 12, 6699–6717. 10.1016/j.jde.2016.09.002Search in Google Scholar
[17] C. Melcher and M. Ptashnyk, Landau–Lifshitz–Slonczewski equations: Global weak and classical solutions, SIAM J. Math. Anal. 45 (2013), 407–429. 10.1137/120878847Search in Google Scholar
[18] C. Schieback, D. Hinzke, M. Kläui, U. Nowak and P. Nielaba, Temperature dependence of the current-induced domain wall motion from a modified Landau–Lifshitz–Bloch equation, Phys. Rev. B. 80 (2009), Article ID 214403. 10.1103/PhysRevB.80.214403Search in Google Scholar
[19] J. C. Slonczewski, Current-driven excitation of magnetic multilayer, J. Magn. Mater. 159 (1996), no. 1–2, L1–L7. 10.1016/0304-8853(96)00062-5Search in Google Scholar
[20] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Grad. Stud. Math. 140, American Mathematical Society, Providence, 2012. 10.1090/gsm/140Search in Google Scholar
[21] M. Tilioua, Current-induced magnetization dynamics. Global existence of weak solutions, J. Math. Anal. Appl. 373 (2011), no. 2, 635–642. 10.1016/j.jmaa.2010.08.024Search in Google Scholar
[22] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970. 10.1007/978-3-642-86405-6Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston