Global weak solutions to Landau-Lifshitz equations into compact Lie algebras

Abstract

We consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra g; which can be viewed as the extension of Landau-Lifshitz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such LL equations from an n-dimensional closed Riemannian manifold T or a bounded domain in {ℝ}^n into a unit sphere {{\displaystyle S}}_g(\mathrm{1}) in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of {ℝ}^{\mathrm{3}} and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.

Keywords

Landau-Lifshitz (LL) equations / Lie algebra / test functions

References

[1] Alouges F, Soyeur A. On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal, 1992, 18(11): 1071–1084

[2] Arnold V I, Khesin B A. Topological Methods in Hydrodynamics. Appl Math Sci, Vol 125. New York: Springer-Verlag, 1998

[3] Balakrishnan R. On the inhomogeneous Heisenberg chain. J Phys C, 1982, 15(36): 1305–1308

[4] Bejenaru I, Ionescu A D, Kenig C E, Tataru D. Global Schrodinger maps in dimensions d= 2: small data in the critical Sobolev spaces. Ann Math, 2011, 173(3): 1443–1506

[5] Carbou G, Fabrie P. Regular solutions for Landau-Lifschitz equation in a bounded domain. Differential Integral Equations, 2001, 14(2): 213–229

[6] Carbou G, Rida J. Very regular solutions for the Landau-Lifschitz equation with electric current. Chin Ann Math Ser B, 2018, 39(5): 889–916

[7] Chen X, Jiang R, Wang Y. A class of periodic solutions of one-dimensional Landau-Lifshitz equations. J Math Study, 2017, 50(3): 199–214

[8] Chen Y. The weak solutions to the evolution problems of harmonic maps. Math Z, 1989, 201(1): 69–74

[9] Daniel M, Porsezian K, Lakshmanan M. On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions. J Math Phys, 1994, 35(12): 6498–6510

[10] Ding S, Guo B. Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations. J Differential Equations, 2008, 244(10): 2448–2472

[11] Ding S, Liu X, Wang C. The Landau-Lifshitz-Maxwell equation in dimension three. Pacific J Math, 2009, 243(2): 243–276

[12] Ding W, Wang H, Wang Y. Schrödinger flows on compact Hermitian symmetric spaces and related problems. Acta Math Sin (Engl Ser), 2003, 19(2): 303–312

[13] Ding W, Wang Y. Schrödinger flow of maps into symplectic manifolds. Sci China Ser A, 1998, 41(7): 746–755

[14] Ding W, Wang Y. Local Schrödinger flow into Kähler manifolds. Sci China Ser A, 2001, 44(11): 1446–1464

[15] Garcia-Cervera C J, Wang X. Spin-polarized transport: existence of weak solutions. Discrete Contin Dyn Syst Ser B, 2007, 7(1): 87–100

[16] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001

[17] Gilbert T L. A Lagrangian formulation of gyromagnetic equation of the magnetization field. Phys Rev, 1955, 100: 1243–1255

[18] Harpes P. Uniqueness and bubbling of the 2-dimensional Landau-Lifshitz flow. Calc Var Partial Differential Equations, 2004, 20: 213–229

[19] Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press, 1978

[20] Hsiang W, Hou Z, Meng D. Lectures of Lie Group. Beijing: Higher Education Press, 2014 (in Chinese)

[21] Jia Z, Wang Y. Local nonautonomous Schödinger flows on Kähler manifolds. Acta Math Sin (Engl Ser), 2019, 35(8): 1251–1299

[22] Jia Z, Wang Y. Global weak solutions to Landau-Lifshitz systems with spin-polarized transport. Preprint

[23] Kosevich A M, Ivanov B A, Kovalev A S. Magnetic solitons. Phys Rep, 1990, 194(3-4): 117–238

[24] Ladyzhenskaya O A. The Boundary Value Problem of Mathematical Physics. Appl Math Sci, Vol 49. New York: Springer-Verlag, 1985

[25] Landau L D, Lifshitz E M. On the theory of dispersion of magnetic permeability in ferromagnetic bodies, Phys Z Soviet, 1935, 8: 153–169

[26] Li Z, Zhao L F. Asymptotic behaviors of Landau-Lifshitz flows from ℝ2 to Kahler manifolds. Calc Var Partial Differential Equations, 2017, 56(4): Art 96 (35 pp)

[27] Melcher C. Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions. Indiana Univ Math J, 2012, 61(3): 1175–1200

[28] Pu X, Wang M, Wang W. The Landau-Lifshitz equation of the ferromagnetic spin chain and Oseen-Frank flow. SIAM J Math Anal, 2017, 49(6): 5134–5157

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