On 3D domain walls for the Landau Lifshitz equations
Olivier Guès, Franck Sueur · Dynamics of Partial Differential Equations · 2007
We show that the Landau-Lifshitz equations of micromagnetics admit solutions with large variations as the exchange coefficient ε 2 tends to zero, corresponding to a large gradient (∼ ε -1 ) of the magnetic moment.These solutions are described by an asymptotic expansion involving internal layers of width O(ε) and amplitude O(1) located in a neighborhood of a smooth fixed hypersurface contained in the domain.The magnetic moment varies fastly accross this hypersurface, called a wall in micromagnetism.The evolution of the transition layer is governed by a nonlinear PDE and our results apply for interval of times of lenght O(1).As ε → 0 the solution converges to a discontinuous solution of the "hyperbolic" model, with no exchange term.