On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness

Magnetization switching in small ferromagnetic ellipsoidal samples

“…Indeed, a lot more results are known in the more geometrical case of the harmonic maps equation, like explicit blow-up solutions in finite time [7,9], global regular solutions if the energy is small [17] or if the energy is non-increasing in time [11], or if the initial condition takes values in a open half-sphere [13], etc. Non-uniqueness results are known for the heat flow of harmonic maps [8] or for Landau-Lifschitz equations but only when the effective magnetic field consists of the exchange term [1]. Such results are still not known for Landau-Lifschitz equations in full generality and seem very challenging.…”

“…, the first equality of (3.5) holds for every t ∈ (0, T n ) and the proof may be finished exactly as in [1]. When H ext ∈ L 2 loc (R + , R 3 ), the first equality of (3.5) may only hold a.e., this changes a few details at the end of the proof.…”

“…For m = (m (1) , m (2) , m (3) ) t ∈ V T , we define Moreover, any function m ∈ V T solution of (2.7) satisfies at time 0 …”

“…When α 1 < α 2 = α 3 , for every m = (m (1) , m (2) , m (3) ) t ∈ H 1 ((0, T ), R 3 ), the functionm := (m (1) , −m (3) , m (2) ) satisfies J T (m) = J T (m). Thus, the optimal path is not unique, even up to the symmetry defined by (2.10).…”

“…It is well known (see [1] for examples) that, for λ > 0 fixed, there may not be uniqueness for the weak solutions of (3.15), (3.16). However, all these weak solutions converge in C 0 ([0…”