The Center for Nonlinear Phenomena and Magnetic Materials

Abstract

Abstract : We have proved existence and obtained estimates for the (finite) Hausdorff and fractal dimensions of global (maximal compact) attractors for the Landau-Lifschitz equations. These are the fundamental equations of the classical theory ferromagnetism. In order to obtain more detailed information about these attractors, we are currently developing approximation methods based on the theory of inertial manifolds. Inertial manifolds are finite dimensional manifolds which attract all solutions at an exponential rate. They contain the global attractor and have the advantage that they are manifolds whereas the attractors generally are not (they can be complicated fractal sets). The equations reduce to a finite-dimensional system of O.D.E.'s on the inertial manifolds. There is a class of calculational methods that have been developed in recent years, called nonlinear or modified Galerkin methods, which are closely related to the concept of inertial manifold and which are especially useful for the long-time integration of nonlinear differential equations. In the usual Galerkin approach, solutions of the nonlinear equation are sought in linear manifolds PuH which are spanned by the eigenfunctions of a linear operator which occurs in the problem. In the case of the Landau-Lifschitz equations, this is the Laplacian.