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On the continuous limit for a system of classical spins. (English) Zbl 0614.35087
A classical model for an isotropic ferromagnet is provided by a collection of three-dimensional spin vectors with unit length and arbitrary directions, located at the nodes of a d-dimensional cubic lattice. A simple hypothesis consists in assuming that each spin processes in the local magnetic field created by the closest neighbours. When interested in the phenomena at scales large compared to the lattices mesh size, we are led to consider the equation \[ \partial S/\partial t=S\wedge \Delta S \] where \(S=S(x,t)\) is a three-dimensional continuous vector field.
The main results established in the paper concern the existence in the large of weak solutions, obtained as (weak) limits of solutions of the discrete problem when the mesh size goes to zero. For smooth initial conditions, there exists, locally in time, a unique smooth solution. This solution is global in time if initially the spins do display small deviations. In this case, the long time dynamics is essentially linear.

35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI
[1] Fogedby, H.C.: Solitons and magnons in the classical Heisenberg chain. J. Phys. A.13, 1467-1499 (1980)
[2] Friedman, A.: Partial differential equations. New York: Holt-Rinehart Winston 1969 · Zbl 0224.35002
[3] Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys.49, 435-480 (1977)
[4] Klainerman, S.: Commun. Pure Appl. Math.33, 43-101 (1980) · Zbl 0414.35054
[5] Klainerman, S., Majda, A.: Commun. Pure Appl. Math.34, 481-524 (1981) · Zbl 0476.76068
[6] Klainerman, S., Ponce, G.: Commun. Pure Appl. Math.37, 133-141 (1983) · Zbl 0509.35009
[7] Ladysenskaya, O.A.: The boundary value problems of Mathematical Physics. Moscow: Nauka 1973 (in Russian); English translation: Applied Mathematical Sciences, Vol.49. Berlin, Heidelberg, New York: Springer 1985
[8] Lakshmanan, M., Ruijgrok, T.W., Thompson, C.J.: On the dynamics of a continuum spin system. Physica84, 577-590 (1976)
[9] Lions, J.L.: Quelques methodes de r?solution de probl?mes aux limites non lin?aires. Paris: Dunod, Gauthier Villars 1969 · Zbl 0189.40603
[10] Ma, S.K.: Modern theory of critical phenomena. New York: W. A. Benjamin 1975
[11] Ma, S.K., Mazenko, F.: Critical dynamics of ferromagnets in 6?? dimensions: General discussion and detailed calculation. Phys. Rev. B11, 4077 (1975)
[12] Moser, J.: Ann. Sc. Norm. Super. Pisa (3)20, 265-315 and 499-535 (1966)
[13] Nirenberg, L.: Ann. Sc. Norm. Super. Pisa13, 116-162 (1959)
[14] Shatah, J.: Differ. Equations46, 409-625 (1982) · Zbl 0518.35046
[15] Takhtajan, L.A.: Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett.64, 235-237 (1977)
[16] Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP34, 62-69 (1972)
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