Partially regular and singular solutions to the Landau-Lifshitz (Gilbert) equations

The Landau-Lifshitz and Landau-Lifshitz Gilbert equations are the basic evolution equations in micromagnetics, a continuum model for magnetic behavior in ferromagnetic materials. In the setting where the magnetic behavior is determined by the Dirichlet energy, these equations are a hybrid Schrodinger map flow and harmonic map heat flow into the unit sphere S2. In this thesis, more general geometries involving targets that are compact smooth hypersurfaces are considered. We address the questions of whether global, partially regular solutions exist in the space of finite energy for these equations and whether singularities form from smooth finite energy data when evolved by these equations. The domain is always taken to be flat and two-dimensional. A general framework is given to construct partially regular solutions to the LLG. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution into the discrete setting. The question of singularity formation for these equations is motivated by the fact that under a suitable transformation, these equations are reminiscent of the cubic nonlinear Schrodinger equation for which singular solutions abound. Analytical attempts have met impasses but our numerical investigations here demonstrate the existence of singular solutions for the LLG. Issues related to the numerical computation of the LL, which only involves the Schrodinger term, are also discussed.