Strong solvability of regularized stochastic Landau–Lifshitz–Gilbert equation

Abstract

We examine a stochastic Landau-Lifshitz-Gilbert equation based on an exchange energy functional containing second-order derivatives of the unknown field. Such regularizations are featured in advanced micromagnetic models recently introduced in connection with nanoscale topological solitons. We show that, in contrast to the classical stochastic Landau-Lifshitz-Gilbert equation based on the Dirichlet energy alone, the regularized equation is solvable in the stochastically strong sense. As a consequence it preserves the topology of the initial data, almost surely.

Nonlocal stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution

“…2 of [15]). Equation ( 5) is in contrast to regularizing the LLG equation by changing the energy functional to include a term to control the modulus of continuity [16]. It also compliments other works that derive equations to preserve the equilibrium distribution, such as in the case of inhomogeneous magnitude of magnetic spins [17], for temporally-colored noise but for finitely many spins [18], and for the stochastic Landau-Lifshitz-Bloch equation [19].…”

Global weak solutions for the Landau–Lifshitz–Gilbert–Vlasov–Maxwell system coupled via emergent electromagnetic fields

“…3. Due to the strong regularizing effect of the governing micromagnetic energy (5), the free LLG equation (7) for j = 0 allows for regular solutions m up to any order. In the coupled system, however, the regularity of m is limited by the regularity of j arising from f .…”

“…where we use the following convenient notations where ∇m : ∇m, ∇ 2 (v • m) stands for i, j ∂ i m • ∂ j m, ∂ 2 i j (v • m) . Details can be found in [5,7]. We now assume there exist two distinct solutions to the LLG equation (56l)…”

Global weak solutions for the Landau-Lifshitz-Gilbert-Vlasov-Maxwell system coupled via emergent electromagnetic fields

“…Since H 2 (R 3 ) is closed under pointwise multiplication v × m ∈ H 2 (R 3 ; R 3 ) is a valid test function and using the identity m × ∂ t m, v = ∂ t m, v × m we can pass to the Landau-Lifshitz formulation where Dm ⊗ Dm, D 2 (v • m) stands for i,j ∂ i m • ∂ j m, ∂ 2 ij (v • m) . Details can be found in [5]. We now assume there exist two distinct solutions to the LLG equation ( 7), m 1 and m 2 .…”