Existence of Regular Solutions of the Landau-Lifshitz-Gilbert Equation in 3D with Natural Boundary Conditions

We prove that the Landau--Lifshitz--Gilbert equation in three space dimensions with homogeneous Neumann boundary conditions admits arbitrarily regular solutions, given that the initial data is sufficiently close to a constant function. This validates the assumptions needed for strong convergence of Alouges's numerical integrator as proved in a recent work by the authors.