Existence of partially regular weak solutions to Landau–Lifshitz–Maxwell equations

Abstract

In this paper, the authors establish the existence of partially regular weak solutions to the Landau-Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg-Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95-117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau-Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615-1644] from Landau-Lifshitz equations to Landau-Lifshitz-Maxwell equations.

Analysis of a coupled spin drift–diffusion Maxwell–Landau–Lifshitz system

“…with the initial and boundary conditions (2), (5)- (6), and (10)- (12). In the Maxwell equations (21)- (22), R x ε and R t ε are two families of linear regularization operators acting on functions of x and t, respectively, satisfying for all u ∈ L 2 (Ω) and v ∈ L 2 (0, T ),…”

“…with the initial and boundary conditions (2), (5)-(6), (10)- (12), satisfying the regularity properties stated in Theorem 1 and the constraint |m| = 1 in ω × (0, T ).…”

Blow-up solutions to Landau-Lifshitz-Maxwell systems

“…Recently, Ding and Lin [20] extended the result in [1] and proved that there exists a weak solution which is smooth with exception of at most finitely many singular points for Landau-Lifshitz-(quasi)Maxwell system. In dimensions three case, Ding and Guo [21,22] proved partial regularity theorems for (stationary) weak solutions to Landau-Lifshitz equation with a nonlocal term. Author in [23] studied the regularity properties of the solutions to the Maxwell-Landau-Lifshitz system in weighted Sobolev spaces.…”

Solitary Wave Solutions to the Multidimensional Landau–Lifshitz Equation

“…A variety of powerful methods have been used to study the nonlinear evolution equations, for the analytic and numerical solutions. Some of these methods, the Riccati Equation method [1], Hirota's bilinear operators [2], exponential rational function method [3], the Jacobi elliptic function expansion [4], the homogeneous balance method [5], the tanh-function expansion [6], first integral method [7,8], the subequation method [9], the expfunction method [10], the Backlund transformation, and similarity reduction [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], are used to obtain the exact solutions of NLPDE.…”