GLOBAL WEAK SOLUTIONS TO LANDAU-LIFSHTIZ SYSTEMS WITH SPIN-POLARIZED TRANSPORT

. In this paper, we consider the Landau-Lifshitz-Gilbert systems with spin-polarized transport from a bounded domain in R 3 into S 2 and show the existence of global weak solutions to the Cauchy problems of such Landau- Lifshtiz systems. In particular, we show that the Cauchy problem to Landau-Lifshitz equation without damping but with diﬀusion process of the spin ac- cumulation admits a global weak solution. The Landau-Lifshtiz system with spin-polarized transport into a compact Lie algebra is also discussed and some similar results are proved. The key ingredients of this article consist of the choices of test functions and approximate equations.


1.
Introduction. In physics, the Landau-Lifshtiz equation, which is deduced in 1935 by Landau and Lifshitz [23], is a fundamental equation describing the evolution of ferromagnetic spin chain and was proposed on the phenomenological ground in studying the dispersive theory of magnetization of ferromagnets. The equation describes the Hamiltonian dynamics corresponding to the Landau-Lifshitz energy, which is defined as follows.
Let Ω be a bounded domain in the Euclidean space R 3 and u, denoting magnetization vector, be a mapping from Ω into a unit sphere S 2 ⊂ R 3 . The energy of map u is defined by where the ∇ R 3 and ∆ R 3 denotes the gradient operator and Laplace operator on R 3 respectively, and dx is the volume element. The well-known Landau-Lifshtiz equation is u t = −u × ∆ R 3 u.

ZONGLIN JIA AND YOUDE WANG
Here × is vector cross product in R 3 . The Landau-Lifshitz equation with dissipation, which can be written as was proposed by Gilbert in 1955 [16]. Here α is the damping parameter, which is characteristic of the material, and α is usually called the Gilbert damping coefficient. Hence the Landau-Lifshitz equation with damping term is also called the Landau-Lifshitz-Gilbert (LLG) equation in the literature. Generally, in physics the Landau-Lifshitz functional is defined by In the above functional, the first and second terms are the anisotropy and exchange energies, respectively. Φ(u) is a real function on S 2 . If one only considers uniaxial materials with easy axis parallel to the OX-axis, for which Φ(u) = u 2 2 + u 2 3 . The last term is the self-induced energy, and h d = −∇w is the demagnetizing field. The magnetostatic potential, w, solves the differential equation ∆w = div(uχ Ω ) in R 3 in the sense of distributions. The solution to this equation is where N (x) = − 1 4π|x| is the Newtonian potential in R 3 . In the absence of spin currents, the relaxation process of the magnetization distribution is described by the following with |u| = 1 and Neumann boundary condition: ∂u ∂ν = 0 on ∂Ω, where ν represents the outward unit normal on ∂Ω. The local field h of E(u) is just People also care much about another new physical model for the spin magnetization system which takes into account the diffusion process of the spin accumulation through the multilayer and has been presented by Zhang et al. [37,28]. Later, in [13] García-Cervera and Wang considered an extension to three dimensions of the model derived in [37], and studied this model for spin-polarized transport. For convenience, we call this model as Landau-Lifshitz equation with spin accumulation which is given by with initial-boundary conditions: Here, Ω 0 is a bounded domain and Ω ⊂ Ω 0 ⊂ R 3 , s = (s 1 , s 2 , s 3 ) ∈ R 3 is the spin accumulation, u = (u 1 , u 2 , u 3 ) ∈ S 2 is the precession of the magnetization, D 0 (x) is would like to consider the following system which takes value in a Lie algebra g with initial-boundary conditions: s(·, 0) = s 0 : Ω 0 −→ g, ∂s ∂ν 1 ∂Ω0 = 0, u(·, 0) = u 0 : Ω −→ S g (1), ∂u ∂ν 2 ∂Ω = 0.
Here, Ω and Ω 0 (Ω ⊂ Ω 0 ) are two bounded domains in Euclidean space R n with boundaries ∂Ω 0 and ∂Ω respectively, ν 1 and ν 2 are the corresponding outward unit normal vectors ∂Ω 0 and ∂Ω. It is worthy to point out that dim(Ω) = dim(g) is needed. Otherwise, the definition of h d does not make sense.
On the other hand, we can also make the following extension of the Landau-Lifshitz equation with spin accumulation. Let (T, h) be an n-dimensional closed Riemannian manifold with metric h = (h ij ). We propose to consider the following Cauchy problem: with initial conditions: s(·, 0) = s 0 : T −→ g, where (u, s) : T × R + → S g (1) × g is an unknown mapping, the coefficient matrix A(x, u) is symmetric and measurable dim(g) × dim(g) matrix defined on T × S g (1), and J e maps T × R + onto the tangent bundle T T of T which belongs to L 2 (T × R + , T T).
In recent years, there has been lots of interesting studies for the Landau-Lifshitz equation, concerning its existence, uniqueness and regularities of various kinds of solutions. Before moving on to the next step, we list only a few of the literature that are closely related to our work in the present paper.
First, let us recall some results of Landau-Lifshitz equation with spin accumulation. Garcia-Cervera and Wang [13] applied Galerkin approximation to construct a global weak solution of (1), provided α > 0, D 0 and J e meet some suitable conditions. Following the seminal work of Struwe for the heat flow of harmonic maps, Pu and Wang [27] gave a unique global weak solution to the simplified system (1) from R 2 into S 2 . They proved the uniqueness of solution to this problem under the help of Littlewood-Paley theory and the techniques of Besov spaces.
Besides, one also discussed the so-called spin-vector drift-diffusion equations which can be derived from the spinor Boltzmann equation by assuming a moderate spin-orbit coupling [18] and the scattering rates are supposed to be scalar quantities. The existence of global weak solutions to a coupled spin drift-diffusion and Maxwell-Landau-Lifshitz system is proved in [36]. Assuming that the scattering rates are positive definite Hermitian matrices, a more general matrix drift-diffusion model was derived in [35]. The global existence of weak solutions to this model was shown in [21].
The equations are considered in a two-dimensional magnetic layer structure and are supplemented with Dirichlet-Neumann boundary conditions. The spin driftdiffusion model for the charge density and spin density vector is the diffusion limit of a spinorial Boltzmann equation for a vanishing spin polarization constant. The Maxwell-Landau-Lifshitz system consists of the time-dependent Maxwell equations for the electric and magnetic fields and of the Landau-Lifshitz-Gilbert equation for the local magnetization, involving the interaction between magnetization and spin density vector. The existence proof is based on a regularization procedure, L 2 -type estimates, and Moser-type iterations which yield the boundedness of the charge and spin densities. Furthermore, the free energy is shown to be nonincreasing in time if the magnetization-spin interaction constant in the Landau-Lifshitz equation is sufficiently small.
Next, we also retrospect the work relative to the following LL equation In case α = 0 and [· , ·] is cross product in R 3 , for the above system Wang has established the existence of global solution to LL equation without Gilbert damping term defined on a closed Riemannian manifold in [33]. Later, in [12] the authors proved the existence of global weak solutions to (5) on a closed Riemannian manifold T or T = R n for the case α = 0 and [· , ·] is a compact Lie algebra. For the case Ω is a bounded domain in R 3 , Carbou and Fabrie studied a model of ferromagnetic material governed by a nonlinear Landau-Lifschitz equation coupled with Maxwell equations and proved the existence of weak solutions in [6] (also see [7]). Later, Tilioua [29] employed the penalized method to show the existence of the weak solution to the following Cauchy problem: with parameters α > 0, γ > 0, β > 0, where P denotes the orthogonal projector onto the closure of the space of gradients of smooth functions in L 2 -topology.
In this paper, we follow the idea in [33] to approach the existence problems of systems (1) and (4). One of the crucial ingredients for the presented analysis here is the choices of auxiliary approximation equations and test functions. It is the aim of the paper at hand to present a proof of the existence of global weak solution. Now we are in the position to present our main conclusions.
Then, for any α ≥ 0, the system (1) admits a global weak solution (s, u) with initial value map (s 0 , u 0 ). Theorem 1.2. Let T be an n-dimensional closed manifolds and g be a m dimensional compact Lie algebra. Suppose that the following conditions are met: (1) A(x, u) is a symmetric and measurable dim(g) × dim(g) function matrix defined on T × S g (1) and is Lipshitz continuous with respect to u, moreover, there exist two positive constants θ 1 and θ 2 such that for any (x, u) ∈ T × S g (1) and any Y ∈ g there holds (2) the initial value maps s 0 ∈ L 2 (T, g) and u 0 ∈ H 1 (T, g) with |u 0 | = 1 a.e. on T; (3) J e ∈ L 2 (T × R + , T T), D 0 : T → R + is a measurable function and there exist positive constants such that 0 < c ≤ D 0 (x) ≤ C, a.e. x ∈ T.
As a direct corollary, we have Corollary 1.3. Let T be a n-dimensional closed manifolds. Suppose that the following conditions hold true (1) A(x, u) is a symmetric and measurable 3 × 3 function matrix defined on T × S 2 and is Lipshitz continuous with respect to u, moreover, there exist two positive constants θ 1 and θ 2 such that for any (x, u) ∈ T × S 2 and any Y ∈ R 3 there holds (2) s 0 ∈ L 2 (T, R 3 ) and u 0 ∈ H 1 (T, R 3 ) with |u 0 | = 1 a.e. on T; (3) J e ∈ L 2 (T × R + , T T), D 0 : T → R + is a measurable function and there exist positive constants such that 0 < c ≤ D 0 (x) ≤ C, a.e. x ∈ T.
Then, for any α ≥ 0, the LL equation with accumulation (4) admits a global weak solution (s, u) with initial value map (s 0 , u 0 ). where and (h ij ) is the inverse of (h ij ). On the other hand, for a smooth function f defined on T, we denote We define the Sobolev space of the functions with compact Lie algebra value Moreover, we define Similarly, Let Ω ⊂ R n be a smooth domain. The operator is defined by where N (|x − y|) is the classical Newton potential. We will use the following properties of the above operator.
The fields h d (u N ) can be defined equivalently by in the sense of distributions. Multiplying the equation by any v ∈ H 1 (R n ) integrating by parts, we obtain Takeing v = w N in the above identity we get In fact, the following lemma was shown in [6] and [13] although they only need to consider the case dim(M ) = n = 3 therein. For more details we refer to page 196 in [22].
Moreover, if u N belongs to W 1,p (Ω) and p ∈ (0, +∞), the restriction of h d (u N ) to Ω belongs to W 1,p (Ω) and there exists a constant C such that Next, we give two definitions on weak solutions.

Definition 2.2.
In the case α > 0, the function pair (s, u) with Here dσ is the volume element of ∂Ω 0 and A(u) is given by (2). In the case α = 0, we say that the function pair (s, u), which satisfies s ∈ , is a weak solution to system (1) with initial values s 0 and u 0 if there hold true Definition 2.3. In the case α > 0, the function pair (s, u) of Lie algebra value, which satisfies is called a weak solutions to system (4) with initial values ( Definition 2.4. In the case α = 0, we say that the function pair (s, u) of Lie algebra 3. Initial-boundary problem on a bounded domains in R 3 . In this section, we discuss the following with initial-boundary conditions: Here, .
Ω ⊂ Ω 0 are two bounded domains in Euclidean space R 3 with boundaries ∂Ω and ∂Ω 0 respectively, ν denotes the corresponding outward unit normal on ∂Ω 0 and ∂Ω. We employ the following auxiliary approximation system with initial-boundary conditions: It should be pointed out that the above Φ(u) has been extended to the closed ball B(1) ⊂ R 3 . In fact, we can extend Φ(z) bỹ and ζ(1) = 1. It is easy to see thatΦ is C 2 -smooth on B(1). For simplicity, we still denoteΦ by Φ.
3.1. Galerkin approximation: A priori estimates. For convenience, we set be the eigenvalues of the operator −∆ on the domain and ω i j is the normalized eigenfunction corresponding to λ i . That is to say, According to Galerkin approximation, we define Here {β N i (t), ρ N i (t)} are unknown functions and assumed to satisfy the following ODE: where A(u) is given by (2). It is easy to check that (8) admits a local solution existing in the interval [0, τ ) for some τ > 0. So we get with initial conditions

ZONGLIN JIA AND YOUDE WANG
Multiplying both sides of the first equation of (9) by ρ N i and summing i from 1 to N yield 1 2 From (2) it follows that Combining the assumption 0 < c ≤ D 0 (x) ≤ C, a.e. x ∈ Ω 0 , we get 1 2 wherec is the Sobolev constant of the imbedding H 1 (Ω 0 ) → L 2 (∂Ω 0 ). For the details of the boundary trace imbedding theorem we refer to Theorem 5.36 in page 164 of [1]. We integrate the above inequality on [0, t] and then pick η = (1−θ)c 2(1+c) to derive the following Similarly, multiplying both sides of the second equation of (9) by β N i and summing i from 1 to N and integrating by parts, we can derive 1 2 where vol(Ω 1 ) is the volume of Ω 1 . Since then, from the above estimates on u N L 2 and s N L 2 we know that ρ N i (t) and β N i (t) can be extended to [0, T ] for any T > 0 and i. That is to say, {s N } and {u N } can be extended to [0, T ].
Multiplying both sides of the second equation of (9) by −λ i β N i and summing i from 1 to N , we get This implies the following 1 2 Multiplying both sides of the second equation of (9) by dβ N i dt and summing i from 1 to N and integrating by parts give Multiplying the two sides of (15) by α and then adding both sides of (14) to the two sides of (15), we obtain: where we have used Lemma 2.1 and the constant C(Φ) depends upon the values of Φ, ∇ u Φ and ∇ 2 u Φ, which are restricted in the unit closed ball B(1). Therefore, we infer from the above Integrating the above inequality on [0, t] gives From the inequality, (11), (12) and (13), we conclude that Furthermore, by the property of weak limits and Aubin-Lions Lemma, there exist two functions v ε ∈ W 2,1 and a subsequence of {u N , s N } which is also denoted by {u N , s N } such that ), here we have used Lemma 2.1; It is easy to check that, for any φ ∈ C ∞ (Ω 0 × [0, T ], R 3 ), s satisfies the following equations  (Ω 1 , R 3 )). Similarly, we also have for some constantC 13 . So, there hold true Fixing r ∈ Z + and taking any N ≥ r, we multiply two sides of the second equation of (9) by η i (t), which belongs to C ∞ ([0, T ], R 3 ), and integrate it on [0, T ]. Then, we sum the obtained identities corresponding to i from 1 to r to derive Letting N tends to ∞, for any r we get It is easy to see the functions Φ r (x, t) defined as above are dense in L 2 ([0, T ], L 2 (Ω 1 , R 3 )). Hence, we conclude that, in the sense of distribution, there holds with v ε (0, ·) = u 0 . Next, we would like to show that ∂v ε ∂ν ∂Ω1 = 0. Indeed, for any φ ∈ C ∞ (Ω 1 ), we have The arbitrariness of φ implies ∂v ε ∂ν ∂Ω1 = 0.

Lemma 3.2.
If v ε and s satisfy (19) and (20) respectively, then there exist two positive constants M 1 (T ) and M 2 (T ), which are independent of ε and α as ε and α are small enough, such that Proof. Multiplying the both sides of (19) by v ε and then integrating (19) on Note that we have gotten By the property of weak limits and (11), we obtain Multiplying the both sides of (19) by v ε t and then integrating (19) on Multiplying the two sides of (19) by ∆v ε and then integrating (19) on Ω 1 × [0, t], we are leaded to Hence, from (23) we have Substituting this identity into (24) we have where we have used the fact that h d (v ε ) = −∇w ε . By Lemma 2.1 we know that Hence, it follows is non-decreasing function with respect to t. The Gronwall inequality tells us that there holds true Furthermore, we have So, the desired estimates follows immediately. This completes the proof of the lemma.
Letting ε in (21) tends to 0, we have: On the other hand, for any t we have |u| ≤ 1 for a.e. x ∈ Ω 1 , which is implied by the fact: for any t ∈ [0, T ] there holds true |v ε | ≤ 1 for a.e. x ∈ Ω 1 . Hence, we deduce from the above that Since there holds true that for any Moreover, as ε → 0 we deduce from (20) Therefore, (s, u) is the weak solution to system (1) by the definition.
Case 2. α = 0. When α > 0, we know that u α is uniformly bounded in L ∞ loc (R + , H 1 (Ω 1 , S 2 )), where (u α , s) is the weak solution in Case 1. Therefore, there exists a subsequence of u α , which is still denoted by u α , converging to u weakly * in L ∞ loc (R + , H 1 (Ω 1 , S 2 )) and converging to u a.e. Ω 1 × R + . Besides, we also have u α → u a.e. ∂Ω 1 × R + and h d (u α ) → h d (u) a.e. R 3 × R + . On the other hand, for (27) Noting that Moreover, since As α → 0 in the above identity we deduce that (26) still holds true. Therefore, (s, u) is the weak solution to system (1) by the definition. This completes the proof.
4. Initial problems on a closed manifold. For the sake of simplification we set that D 0 (x) ≡ 1 and one will see that there is no loss of generality. First of all, we need to extend the domain of A to T × g. For arbitrary u ∈ g and u = 0, we definẽ if u = 0, we defineÃ(x, u) = Id, where Id is the identity matrix on g.
It is easy to check thatÃ is also symmetric and measurable on T × g and there exist two positive constants θ andθ such that for any (x, u) ∈ T × B g (1) and Y ∈ g, where B g (1) is the closure of the unit ball B g (1) in the Lie algebra g. Suppose that the eigenvalues ofÃ(x, u) are From (28) it follows that SinceÃ(x, u) is a symmetric matrix, we have for any u ∈ B g (1) and Y 1 , Y 2 ∈ g.
For the sake of convenience, we still set We employ the following Cauchy problem of auxiliary equation: to approach the the Cauchy problem (4), where ε is a positive constant. From now on, without confusions we always denoteÃ by A. Let λ i (i = 1, 2, ...) be the eigenvalues of the operator −∆ on the domain H 2 (T). ω i is the normalized eigenfunction corresponding to λ i . That is to say, According to Galerkin approximation, we define Here {β N i (t), ρ N i (t)} are unknown functions taking values in g and assumed to satisfy the following ODE: It is easy to check that (30) admits a local solution existing in the interval [0, τ ) for some τ > 0. So we get Multiplying both sides of the first equation of (31) by ρ N i and summing i from 1 to N yield It implies the following Similarly, multiplying both sides of the second equation of (31) by β N i and summing i from 1 to N and integrating by parts, we can derive where vol(T) is the volume of T. Since then, from the above estimates on u N L 2 and s N L 2 we know that ρ N i (t) and β N i (t) can be extended to [0, T ] for any T > 0 and i. That is to say, {s N } and {u N } can be extended to [0, T ].
Multiplying both sides of the second equation of (31) by −λ i β N i and summing i from 1 to N , we get