Regular solutions for Landau-Lifschitz equation in a bounded domain

A blowup criterion for nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in 2‐D

“…Subsequently, the combination of ( 8), (10), and (20) allowing to apply Gronwall's inequality to (22) arrives at (14). The proof of Lemma 5 is completed.…”

“…Ding and Wang 12 proved the same result when the mapping is from a Riemannian manifold into a complete Kähler manifold. Sulem et al 13 and Carbou and Fabrie 14,15 together proved, in 2-D and 3-D, respectively, the local existence and uniqueness of regular solutions and the global existence both under the assumption of small initial data.…”

Existence and uniqueness of weak solutions to the Landau–Lifshitz equation for large ferromagnetic bodies

“…where p ∈ [1,6] for the three-dimensional case; see, for example, Carbou and Fabrie 13 and Joly et al 37 and Chapter 5 in Ladyzhenskaya 39 for more details. By (3.12) and (3.17), we have…”

“…Indeed, () is not only a good approximate model to describe the magnetization evolution of large ferromagnetic bodies (see, e.g., previous studies 30–32 ) but also can be regarded as the singular limit of the classical Landau–Lifshitz equation (see Carbou and Fabrie 33 ) or the quasi‐stationary limit of the Landau–Lifshitz–Maxwell equation (see, e.g., Jochmann 34 and Yan 35 ). In particular, De Simone 32 discusses the problem of predicting the macroscopic response of a permanent magnet to an applied magnetic field, the treatment depends on some abstract concepts such as $$$ H $$$‐measure, Young measure, and weak limits.…”

“…In the past few decades, the well-posedness of the Landau-Lifshitz equation with exchange energy as the main energy (e.g., 1.3) has been studied extensively. The partial regularity of weak solutions has been studied in, for instance, Chen 10 and Guo and Hong 11 ; the existence of weak solutions and strong solutions has been established in, for instance, previous studies [12][13][14][15][16][17][18] ; the explicit solutions has been constructed in, for instance, previous studies [19][20][21][22][23] ; and the stability and controllability have been given in, for instance, previous studies. 9,24,25 For more details, we refer the reader to Guo and Ding 26 and the references therein.…”