Abstract
In this paper, we prove the existence of global weak solutions to the periodic fractional Landau–Lifshitz–Gilbert equation through the Ginzburg–Landau approximation and the Galerkin approximation. Since the nonlinear term is nonlocal and of full order of the equation, some special structures of the equation, the commutator estimate and some calculus inequalities of fractional order are exploited to get the convergence of the approximating solutions. The equation considered in this paper can also be regarded as a generalization of the heat flow of harmonic maps to the fractional order.
Similar content being viewed by others
References
Alouges F., Soyeur A.: On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonl. Anal. TMA 19(11), 1071–1084 (1992)
Carbou G., Fabrie P.: Time average in micromagnetism. J. Differ. Equ. 147(2), 383–409 (1998)
Chang N.-H., Uhlenbeck K.: Schrödinger maps. Commun. Pure Appl. Math. 53(5), 0590–0602 (2000)
Chen Y.: The weak solutions to the evolution problems of harmonic maps. Math. Z. 201, 69–74 (1989)
Chen Y., Struwe M.: Existence and partial regularity for the het flow for harmonic maps. Math. Z. 201, 83–103 (1989)
Coifman, R., Meyer, Y.: Nonlinear harmonic analysis, operator theory and P.D.E. In: Beijing Lectures in Harmonic Analysis, pp. 3–45. Princeton University Press (1986)
Constantin P.: Energy spectrum of quasigeostrophic turbulence. Phys. Rev. Lett. 89, 184501 (2002)
Constantin P.: On the Euler equations of incompressible fluids. Bull. Am. Math. Soc. 44(4), 603–621 (2007)
Constantin P., Cordoba D., Wu J.: On the critical dissipative quasi-geostrophic equations. Indiana Univ. Math. J. 50, 97–107 (2001)
Constantin P., Majda A., Tabak E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)
Constantin P., Wu J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999)
Ding W., Wang Y.: Schrödinger flow of maps into symplectic manifolds. Sci. China Ser. A 41(7), 746–755 (1998)
Ding, S., Wang, C.: Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int. Math. Res. Not. 2007, 1–25 (2007)
Ding W., Tang H., Zeng C.: Self-similar solutions of Schrodinger flows. Calc. Var. 34(2), 267–277 (2009)
Eells J., Sampson H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
Gilbert T.L.: A Lagrangian formulation of gyromagnetic equation of the magnetization field. Phys. Rev. 100, 1243–1255 (1955)
Guan M., Gustafson S., Tsai T.P.: Global existence and blow-up for harmonic map heat flow. J. Differ. Equ. 246, 1–20 (2009)
Guo B., Hong M.: The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps. Calc. Var. 1, 311–334 (1993)
Gustafson, S., Nakanishi, K., Tsai, T.P.: Asymptotic stability, concentration, and oscillation in harmonic map heat-flow. Landau-Lifshitz, and Schrodinger maps on R 2. Commun. Math. Phys. 300, 205–242 (2010)
Ju N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251, 365–376 (2004)
Ju N.: The Maximum principle and the global attractor for the dissipative 2d quasi-geostrophic equations. Commun. Math. Phys. 155, 161–181 (2005)
Kato T.: Liapunov Functions and Monotonicity in the Navier-Stokes Equations, Lecture Notes in Mathematics, vol. 1450. Springer-Verlag, Berlin (1990)
Kato T., Ponce G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)
Kenig C., Ponce G., Vega L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)
Kenig C., Ponce G., Vega L.: Well-posedness and scattering results for the generalized korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46(4), 453–620 (1993)
Landau, L.D., Lifshitz, E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowj. 8, 153–169 (1935); Reproduced in: Collected papers of L.D. Landau, pp. 101–114. Pergamon Press, New York (1965)
Lin F., Wang C.: The analysis of harmonic maps and their heat flows. World Scientific, River Edge (2008)
Lions, J.L.: Quelques methodes de resolution des problemes aux limits non lineeaires. Dunod, Paris (1969)
Liu X.: Partial regularity for the Landau-Lifshitz systems. Calc. var. 20(2), 153–173 (2004)
Miller K.S., Ross B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
Nahmod A., Stefanov A., Uhlenbeck K.: On Schrödinger maps. Commun. Pure Appl. Math. 56(1), 114–151 (2003)
Podlubny I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)
Shatah J., Zeng C.: Schrödinger maps and anti-ferromagnetic chains. Commun. Math. Phys. 262, 299–315 (2006)
Sulem P.L., Sulem C., Bardos C.: On the continuous limit for a system of classical spins. Commun. Math. Phys. 107, 431–454 (1986)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comm. Math. Helv. 60, 558–581 (1985)
Tarasov V.E.: Fractional Heisenberg equation. Phys. Lett. A 372, 2984–2988 (2008)
Temam R.: Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Struwe.
Rights and permissions
About this article
Cite this article
Pu, X., Guo, B. The fractional Landau–Lifshitz–Gilbert equation and the heat flow of harmonic maps. Calc. Var. 42, 1–19 (2011). https://doi.org/10.1007/s00526-010-0377-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-010-0377-4