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
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