Landau-Lifshitz-Slonczewski Equations: Global Weak and Classical Solutions

We consider magnetization dynamics under the influence of a spin-polarized current, given in terms of a spin-velocity field v, governed by the following modification of the Landau– Lifshitz–Gilbert equation ∂m ∂t + v · ∇m = m × (α ∂m ∂t + β v · ∇m − Δm), called the Landau– Lifshitz–Slonczewski equation. We focus on the situation of magnetizations defined on the entire Euclidean space m(t) : R3 → S2. Our construction of global weak solutions relies on a discrete lattice approximation in the spirit of Slonczevski’s spin-transfer-torque model and provides a rigorous justification of the continuous model. Using the method of moving frames, we show global existence and uniqueness of classical solutions under smallness conditions on the initial data in terms of the Ẇ 1,3 norm and on the spin velocity v in terms of weighted-in-time Lebesgue norms, both optimal with respect to the natural scaling of the equation.