2008 # 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…

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“…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 ),…”

confidence: 99%

“…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 ),…”

confidence: 99%

“…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 ).…”

confidence: 99%

“…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.…”

confidence: 99%

“…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.…”

confidence: 99%