Existence and Hölder Regularity of the Fractional Landau-Lifshitz Equation without Gilbert Damping Term

Abstract

Existence and Hölder regularity of weak solutions to the fractional Landau-Lifshitz equation without Gilbert damping term is proved through viscosity approximation. Since the nonlinear term is nonlocal and of full order of the equation, a commutator is constructed to get the convergence of the approximating solutions.

1. Introduction

We study the fractional Landau-Lifshitz equation 𝜕𝑚𝜕𝑡=−𝜆₁𝑚×Λ^2𝛼𝑚+𝜆₂𝑚×𝑚×Λ^2𝛼𝑚, (1) where 𝑚(𝑥,𝑡) is a three-dimensional vector representing the magnetization and 𝛼,𝜆₁,𝜆₂≥0 are real numbers. Λ=(−Δ)^1/2 is the square root of the Laplacian and the so-called Zygmund operator and × denotes the cross product of ℝ³-valued vectors. The first term 𝑚×Λ^2𝛼𝑚 is the gyromagnetic term and the second term 𝑚×(𝑚×Λ^2𝛼𝑚) is called the Gilbert damping term. The fractional diffusion operator Λ^2𝛼 is nonlocal except 𝛼=0,1,2,3,…, which means that Λ^2𝛼𝑢(𝑥) depends not only on 𝑢(𝑦) for 𝑦 near 𝑥 but also on 𝑢(𝑦) for all 𝑦.

Equation (1) plays a fundamental role in the understanding of nonequilibrium magnetism, which is an interesting problem from both scientific and technological points of view. Besides their traditional applications in the magnetic recording industry, these films are also currently being explored as alternatives to semiconductors as magnetic memory devices (MRAMs), which has given greater incentive to study this subject. Since defects, impurities, and thermal noise play important roles in the dynamics of the magnetization field in nanometer thick films, they also make an ideal playground for studying some of the nanoscale physics branches [1–4].

Fractional differential equations, which appear in several branches of physics such as viscoelasticity, electrochemistry, control, porous media, and electromagnetic, now attract the interests of many mathematicians; see, for example, [5, 6]. A good case in point is the quasi-geostrophic equation with fractional dissipation, which has been extensively studied in the last decade see [7–9]. The fractional Landau-Lifshitz equation shares some similar difficulties with quasi-geostrophic equation; however, the equation studied here is much more complicated in several ways. The derivative in the nonlinear convective term is local in the quasi-geostrophic equation and the fluid velocity is divergence free, but here for 𝜆₂=0, (1) is degenerate and even worse the derivative in the nonlinear term is nonlocal and of the same order as the equation, which brings new difficulties in the convergence of the approximate solutions. Hence subtle techniques must be used to overcome the difficulties.

Let us recall some previous results of the equation. When 𝛼=1, (1) becomes the standard Landau-Lifshitz equation introduced first by Landau and Lifshitz in [10], which was widely studied in [11–16]. For general 𝛼∈(0,1], the interested reader can refer to [17] for mathematical theory. When 𝜆₂=0, (1) corresponds to Schrödinger flow which represents the conservation of angular momentum [18–21]. Numerical treatments can be found in [22, 23].

In this paper, we will study local existence of weak solutions in the spatial domain (0,2𝜋) with 𝜆₂=0 and 𝛼∈[1/2,1]. The main difficulty, as in many partial differential equations, is the convergence of the nonlinear terms. In our situation, we even face the problem of nonlocal differential operators, degeneracy, and nonlocal nonlinear term. For these reasons, the structure of (1) must be explored in detail.

Without loss of generality, we assume that 𝜆₁=1. Actually, (1) can be written as 𝜕𝑚(𝑚)Λ𝜕𝑡=𝐵^2𝛼𝑚, (2) in which 𝑚𝑚=𝑚¹₂𝑚₃,𝐵(𝑚)=0𝑚₃−𝑚₂−𝑚₃0𝑚₁𝑚₂−𝑚₁0, (3) with initial condition 𝑚(𝑥,0)=𝑚₀(𝑥) (4) and the periodic boundary condition 𝑚(𝑥,𝑡)=𝑚(𝑥+2𝜋,𝑡). (5) It is straightforward to check the following conclusions.(1)The matrix 𝐵(𝑚) is “zero definite”; namely, 𝜉^𝜏𝐵(𝑚)𝜉=0,∀𝜉,𝑚∈ℝ³. (6)(2)The matrix 𝐵(𝑚) is singular; that is, det𝐵(𝑚)=0,∀𝑚∈ℝ³. (7)Hence (2) is quite different from usual quasilinear parabolic equations for the above reasons.

To approximate (2), we consider the following mollified equation: 𝜕𝑚𝜕𝑡=−𝜀Λ^2𝛼𝑚−𝑚×Λ^2𝛼𝑚, (8) which can be written as 𝜕𝑚+((𝑚Λ𝜕𝑡𝜀𝐸−𝐵))^2𝛼𝑚=0. (9)

The rest of this paper is divided into three parts: first, we consider the corresponding linear equation and get the regularity as a preparation to deal with (9); second, positive-definition and uniform ellipticity of matrix 𝜀𝐸−𝐵(𝑚) and choice of norm space 𝐿^∞ ensure that Leray-Schauder fixed-point theorem can be applied to prove the existence of weak solution to (9), and the necessary a priori estimates in order to guarantee convergence are obtained; finally, existence and Hölder regularity of weak solution to (2) is proved by taking the limit of the solution to (9), in which a commutator is constructed to get the convergence.

2. Cauchy Problem for the Corresponding Linear Equation

Our starting point is the linear equation 𝜕𝑚()Λ𝜕𝑡+𝐴𝑥,𝑡^2𝛼𝑚=𝑓(𝑥,𝑡),in𝕋^𝑛×(),𝑚()0,𝑇𝑥,0=𝑚₀(𝑥),on𝕋^𝑛, (10)(11) where 𝕋^𝑛=ℝ^𝑛/𝑍^𝑛 is the flat torus and 𝑚(𝑥,𝑡) and 𝑚₀(𝑥) are N-dimensional vector-valued functions. We have the following theorem about existence of solution to (10)-(11).

Theorem 1. Suppose that 𝑁×𝑁 matrix 𝐴(𝑥,𝑡) defined on 𝕋^𝑛×(0,𝑇) is measurable, bounded, and uniformly elliptic; namely, there exists a constant 𝐾 such that 𝐴(𝑥,𝑡)𝜂⋅𝜂≥𝐾𝜂², (12) for all 𝑁-dimensional vectors, 𝑓(𝑥,𝑡)∈𝐿²(0,𝑇;𝐿²(𝕋^𝑛)),and𝑚₀(𝑥)∈𝐻^𝛼(𝕋^𝑛). Then there exists a unique vector-valued solution to (10)-(11) such that 𝑚(𝑥,𝑡)∈𝐿^∞0,𝑇;𝐻^𝛼𝕋^𝑛𝐿²0,𝑇;𝐻^2𝛼𝕋^𝑛,𝜕𝑚𝜕𝑡∈𝐿²0,𝑇;𝐿²𝕋^𝑛. (13)

Proof of Theorem 1. We apply the Galerkin method: let {𝜑_𝑗} be an orthogonal basis of 𝐿²(𝕋^𝑛) consisting of all the eigenfunctions for the operator Λ^2𝛼𝜑_𝑗=𝜆_𝑗𝜑_𝑗,𝜑_𝑗(0)=𝜑_𝑗(2𝜋). (14) We are looking for approximate solutions 𝑚_𝑛(𝑥,𝑡) to (10)-(11) under the form 𝑚_𝑛(𝑥,𝑡)=^𝑛_𝑗=1𝑔_𝑗(𝑡)𝜑_𝑗(𝑥), (15) where 𝑔_𝑗 are vector-valued functions, such that, for 1⩽𝑖⩽𝑛, there holds _Ω𝜕𝑚_𝑛()Λ𝜕𝑡+𝐴𝑥,𝑡^2𝛼𝑚_𝑛−𝑓(𝑥,𝑡)𝜑_𝑖𝑑𝑥=0,_Ω𝑚_𝑛(𝑥,0)−𝑚₀(𝑥)𝜑_𝑖𝑑𝑥=0. (16)(17) These relations produce an ordinary differential system that can be writeen as 𝜕𝑔𝑔(0)𝜕𝑡=𝐹,𝑔=𝑔₀, (18) where 𝑔=(𝑔₁,𝑔₂,…,𝑔_𝑛) and 𝑔₀ is the projection of 𝑚₀ on (𝜑₁,𝜑₂,…𝜑_𝑛). The existence of a local solution to system (18) is a classical matter. We now proceed to estimate the approximate solution 𝑚_𝑛. Multiplying equality (16) by 𝑔_𝑖 and summing for 1≤𝑖≤𝑛, we have 12𝑚_𝑛(⋅,𝑡)²_{𝐿2(𝕋𝑛)}=^𝑇_0𝕋𝑛𝑓⋅𝑚_𝑛−𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛⋅𝑚_𝑛+1𝑑𝑥𝑑𝑡2𝑚₀²_{𝐿2(𝕋𝑛)}. (19) Multiplying equality (16) by 𝜆_𝑖𝑔_𝑖 and summing for 1≤𝑖≤𝑛, we have 12Λ^𝛼𝑚_𝑛(⋅,𝑡)²_{𝐿2(𝕋𝑛)}+^𝑇_0𝕋𝑛𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛⋅Λ^2𝛼𝑚_𝑛=𝑑𝑥𝑑𝑡_{𝑇0𝕋𝑛}𝑓⋅Λ^2𝛼𝑚_𝑛1𝑑𝑥𝑑𝑡+2Λ^𝛼𝑚₀²_{𝐿2(𝕋𝑛)}. (20) Adding (19) to (20), we get 12𝑚_𝑛(⋅,𝑡)²_{𝐿2(𝕋𝑛)}+12Λ^𝛼𝑚_𝑛(⋅,𝑡)²_{𝐿2(𝕋𝑛)}+^𝑇_0𝕋𝑛𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛⋅Λ^2𝛼𝑚_𝑛=1𝑑𝑥𝑑𝑡2𝑚₀²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑚₀²_{𝐿2(𝕋𝑛)}+^𝑇_0𝕋𝑛𝑓⋅𝑚_𝑛+𝑓⋅Λ^2𝛼𝑚_𝑛−𝑑𝑥𝑑𝑡_{𝑇0𝕋𝑛}𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛⋅𝑚_𝑛≤1𝑑𝑥𝑑𝑡2𝑚₀²_{𝐿2(𝕋𝑛)}+Λ^𝛼𝑚₀²_{𝐿2(𝕋𝑛)}+𝐶𝑓²_{𝐿2(𝕋𝑛×(0,𝑇))}+𝐶^𝑇_0𝕋𝑛𝑚_𝑛²𝐾𝑑𝑥𝑑𝑡+2Λ^2𝛼𝑚_𝑛²_{𝐿2(𝕋𝑛×(0,𝑇))}. (21) Since ∫^𝑇₀∫_𝕋𝑛𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛⋅Λ^2𝛼𝑚_𝑛𝑑𝑥𝑑𝑡≥𝐾‖Λ^2𝛼𝑚_𝑛‖²_𝐿2(𝕋^𝑛_×(0,𝑇)), by Gronwall's inequality, we have sup_{0≤𝑡≤𝑇}𝑚_𝑛(⋅,𝑡)²_{𝐻𝛼(𝕋𝑛)}+𝑚_𝑛²_{𝐿2(𝕋𝑛×(0,𝑇))}≤𝐶𝑚₀²_{𝐻𝛼(𝕋𝑛)}+𝑓²_{𝐿2(𝕋𝑛×(0,𝑇))},Λ^2𝛼𝑚_𝑛²_{𝐿2(𝕋𝑛×(0,𝑇))}≤𝐶𝑚₀²_{𝐻𝛼(𝕋𝑛)}+𝑓²_{𝐿2(𝕋𝑛×(0,𝑇))}. (22) Taking the inner product of 𝜕𝑚_𝑛/𝜕𝑡 and (10) and integrating over 𝕋^𝑛×(0,𝑇), we have ^𝑇_0𝕋𝑛𝜕𝑚_𝑛𝜕𝑡²=𝑑𝑥𝑑𝑡_{𝑇0𝕋𝑛}𝑓−𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛⋅𝜕𝑚_𝑛𝜕𝑡𝑑𝑥𝑑𝑡≤𝐶^𝑇_0𝕋𝑛𝑓²𝑑𝑥𝑑𝑡+𝐶^𝑇_0𝕋𝑛Λ^2𝛼𝑚_𝑛²+1𝑑𝑥𝑑𝑡2^𝑇_0𝕋^𝑛𝜕𝑚_𝑛𝜕𝑡²𝑑𝑥𝑑𝑡. (23) Hence 𝜕𝑚_𝑛𝜕𝑡²_𝐿²_(𝕋^𝑛_×(0,𝑇))≤𝐶𝑚₀²_{𝐻𝛼(𝕋𝑛)}+𝑓²_{𝐿2(𝕋𝑛×(0,𝑇))}. (24)
Actually, if the matrix 𝐴(𝑥,𝑡) is retrained to a small class of good function matrix, one can get higher regularity of solution to (10)-(11).
Since the right-hand member of equality (22) and (24) is uniformly bounded, thus the solution g can be extended to all time, and we can extract from 𝑚_𝑛 a subsequence (still denoted by 𝑚_𝑛) such that 𝑚_𝑛⇀𝑚𝑠𝑤,in𝐿²0,𝑇;𝐻^2𝛼𝕋^𝑛weakly^∗,𝜕𝑚_𝑛⇀𝜕𝑡𝜕𝑚𝜕𝑡,in𝐿²0,𝑇;𝐿²𝕋^𝑛weakly. (25) Hence, we know that [24] 𝑚_𝑛⟶𝑚,stronglyin𝐿²0,𝑇;𝐿²𝕋^𝑛a.e.in𝕋^𝑛×[0,𝑇]. (26) Passing to the limit (𝑛→∞), we find a weak solution to (10)-(11). From (16), and taking the limit 𝑛→∞, we deduce that, for all 𝜑 in vectors (𝜑₁,𝜑₂,…,𝜑_𝑛), there holds _𝕋𝑛𝜕𝑚_𝑛𝜕𝑡𝜑+_𝕋𝑛𝐴(𝑥,𝑡)Λ^2𝛼𝑚_𝑛𝜑−_𝕋𝑛𝑓(𝑥,𝑡)𝜑𝑑𝑥=0. (27) By a density argument, we also obtain formula (27) for all 𝜑 in 𝐻^𝛼(𝑄_𝑇)𝑄_𝑇=𝕋^𝑛×[0,𝑇].

Theorem 2. Suppose that:
(i)  𝑁×𝑁 matrix 𝐴(𝑥,𝑡) defined on 𝕋^𝑛×(0,𝑇) is measurable, bounded, and uniformly elliptic; namely, there exists a constant 𝐾 such that 𝐴(𝑥,𝑡)𝜂⋅𝜂≥𝐾𝜂², (28) for all N-dimensional vectors 𝜂, Δ𝐴(𝑥,𝑡) is bounded, and |∇𝐴(𝑥,𝑡)|_∞≤𝐾/√5𝛿, where 𝛿 is the Sobolev embedding constant satisfying ∇𝑓_{𝐿2(𝕋𝑛×(0,𝑇))}≤𝛿Δ𝑓_{𝐿2(𝕋𝑛×(0,𝑇))}; (29)
(ii)  𝑓(𝑥,𝑡)∈𝐿²(0,𝑇;𝐻²(𝕋^𝑛)),𝑚₀(𝑥)∈𝐻^𝛼+2(𝕋^𝑛). Then there exists a unique vector-valued solution to (10)-(11) such that 𝑚(𝑥,𝑡)∈𝐿^∞0,𝑇;𝐻^𝛼+2𝕋^𝑛𝐻¹𝕋^𝑛×(0,𝑇),sup_{0≤𝑡≤𝑇}‖𝑚(⋅,𝑡)‖²_{𝐻𝛼+2(𝕋𝑛)}+𝜕𝑚𝜕𝑡²_𝐿²_(𝕋^𝑛_×(0,𝑇))+‖𝑚‖²_{𝐿20,𝑇;𝐻2𝛼+2(𝕋𝑛)}≤𝐶𝑚₀²_{𝐻𝛼+2(𝕋𝑛)}+𝑓²_{𝐿20,𝑇;𝐻2(𝕋𝑛)}, (30) in which constant 𝐶 is dependent 𝐴(𝑥,𝑡) independent of 𝑚(𝑥,𝑡).

Proof. Let the operator Λ² act on (10); we have −Δ𝑚_𝑡−Δ𝐴(𝑥,𝑡)Λ^2𝛼𝑚+𝐴(𝑥,𝑡)Λ^2𝛼+2𝑚−∇𝐴(𝑥,𝑡)⋅∇Λ^2𝛼𝑚=−Δ𝑓(𝑥,𝑡). (31) Taking the inner product of Λ^2𝛼+2𝑚 and (31) and integrating over 𝕋^𝑛×(0,𝑇), we have 12_𝕋^𝑛Λ^𝛼+2𝑚²𝑑𝑥+^𝑇_0𝕋𝑛𝐴(𝑥,𝑡)Λ^2𝛼+2𝑚⋅Λ^2𝛼+2𝑚𝑑𝑥𝑑𝑡=−^𝑇_0𝕋𝑛Δ𝑓⋅Λ^2𝛼+2+𝑚𝑑𝑥𝑑𝑡_{𝑇0𝕋𝑛}Δ𝐴(𝑥,𝑡)Λ^2𝛼𝑚⋅Λ^2𝛼+2+𝑚𝑑𝑥𝑑𝑡_{𝑇0𝕋𝑛}∇𝐴(𝑥,𝑡)∇Λ^2𝛼𝑚⋅Λ^2𝛼+2+1𝑚𝑑𝑥𝑑𝑡2_𝕋^𝑛Λ^𝛼+2𝑚₀²≤1𝑑𝑥2Λ^𝛼+2𝑚₀²_{𝐿2(𝕋𝑛)}+1𝐾Δ𝑓²_{𝐿2(𝕋𝑛×(0,𝑇))}+𝐾4Λ^2𝛼+2𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}+1𝐾Δ𝐴(𝑥,𝑡)Λ^2𝛼𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}+𝐾4Λ^2𝛼+2𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}+1𝐾∇𝐴(𝑥,𝑡)∇Λ^2𝛼𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}+𝐾4Λ^2𝛼+2𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}. (32) Note that |∇𝐴(𝑥,𝑡)|_∞<𝐾√,5𝛿∇Λ^2𝛼𝑚_{𝐿2(𝕋𝑛×(0,𝑇))}≤𝛿Λ^2𝛼+2𝑚_{𝐿2(𝕋𝑛×(0,𝑇))}(),andΔ𝐴𝑥,𝑡isbounded. (33) Reporting (33) in (32) and taking into account ^𝑇_0𝕋𝑛𝐴(𝑥,𝑡)Λ^2𝛼+2𝑚⋅Λ^2𝛼+2Λ𝑚𝑑𝑥𝑑𝑡≥𝐾^2𝛼+2𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}, (34) we have sup_{0≤𝑡≤𝑇}Λ^𝛼+2𝑚(⋅,𝑡)²_{𝐿2(𝕋𝑛)}+Λ^2𝛼+2𝑚²_{𝐿2(𝕋𝑛×(0,𝑇))}≤𝐶𝑚₀²_{𝐻𝛼+2(𝕋𝑛)}+𝑓²_{𝐿20,𝑇∶𝐻2(𝕋𝑛)}. (35)

3. Cauchy Problem for the Mollified Equation

To get existence of weak solution to (2), we consider the following approximate equation: 𝜕𝑚𝜕𝑡=−𝜀Λ^2𝛼𝑚−𝑚×Λ^2𝛼𝑚, (36) which is called mollified equation. In this section and next section, we assume that the spatial variable 𝑥∈(0,2𝜋). By Leray-Schauder fixed-point theorem, we have the following theorem.

Theorem 3. Suppose that 𝑚₀(𝑥)∈𝐻^𝛼(0,2𝜋), then there exists a unique weak solution to (36) with initial-boundary condition (4) and (5) such that 𝑚(𝑥,𝑡)∈𝐿^∞0,𝑇;𝐻^𝛼(0,2𝜋), (37) where 𝑄_𝑇=(0,2𝜋)×(0,𝑇).

Proof. First, the mapping 𝑇_𝜆∶𝐿^∞(𝑄_𝑇)→𝐿^∞(𝑄_𝑇) is defined as follows. For each 𝑢∈𝐿^∞(𝑄_𝑇), 𝑚=𝑇_𝜆(𝑢) is a solution to 𝜕𝑚𝜕𝑡=−𝜀Λ^2𝛼𝑚−𝜆𝑢×Λ^2𝛼𝑚 (38) with initial condition (4), in which 0≤𝜆≤1. By Theorem 1, we know that 𝑚=𝑇_𝜆(𝑢) is the unique solution to (38) with initial-boundary condition (4) and (5); moreover, 𝑚∈𝐿^∞(0,𝑇;𝐻¹(0,2𝜋)).
Obviously, for all 𝜆, the mapping 𝑇_𝜆 is continuous; and for any bounded closed set of 𝐿^∞(𝑄_𝑇), 𝑇_𝜆 is uniformly continuous with respect to 0≤𝜆≤1.
To apply Leray-Schauder fixed-point theorem, we make a priori estimate on all fixed points of 𝑇_𝜆.
Taking the inner product of 𝑚(𝑥,𝑡) and equation 𝜕𝑚𝜕𝑡=−𝜀Λ^2𝛼𝑚−𝜆𝑚×Λ^2𝛼𝑚, (39) we have 𝑚⋅𝜕𝑚𝜕𝑡=−𝜀𝑚⋅Λ^2𝛼𝑚−𝜆𝑚×Λ^2𝛼𝑚⋅𝑚. (40) Integrating (40) over 𝑄_𝜏(0≤𝜏≤𝑇), we get ‖𝑚(⋅,𝜏)‖²_{𝐿2()0,2𝜋}+2𝜀Λ^𝛼𝑚²_{𝐿2𝑄𝜏}≤𝑚₀²_{𝐿2()0,2𝜋}, (41) in which 0≤𝜆≤1,  0≤𝜏≤𝑇. Hence sup_{0≤𝑡≤𝑇}‖𝑚(⋅,𝑡)‖_𝐿²_(0,2𝜋)≤𝐶₁, (42) in which 𝐶₁ is a constant independent of 𝜀,𝜆.
Taking the inner product of Λ^2𝛼𝑚(𝑥,𝑡) and (39), we obtain Λ^2𝛼𝑚⋅𝜕𝑚𝜕𝑡=−𝜀Λ^2𝛼𝑚⋅Λ^2𝛼𝑚−𝜆Λ^2𝛼𝑚⋅𝑚×Λ^2𝛼𝑚. (43) Integrating (43) over (0,2𝜋) with respect to variable 𝑥 leads to 12𝑑Λ𝑑𝑡^𝛼𝑚(⋅,𝑡)²_{𝐿2()0,2𝜋}+𝜀₀^2𝜋Λ^2𝛼𝑚²𝑑𝑥≤0. (44) Obviously, sup_{0≤𝑡≤𝑇}Λ^𝛼𝑚(⋅,𝑡)_{𝐿2(0,2𝜋)}≤𝐶₂, (45) in which 𝐶₂ is a constant independent of 𝜆,𝜀. From (42), for each 𝜀>0, we have ‖𝑚‖_𝐿²_(0,𝑇;𝐻^2𝛼_(0,2𝜋))≤𝐶. (46) In view of (42), (45), and (46), Sobolev embedding theorem gives the desired result.

For small initial data 𝑚₀(𝑥), we can get higher regularity of the solution to (36).

Theorem 4. Suppose that 𝑚₀(𝑥)∈𝐻^𝛼+2(0,2𝜋),𝛼∈(1/2,1], and ‖𝑚₀(𝑥)‖²_𝐻𝛼+2≤1/𝑀𝑇, where 𝑀=𝑜(1/𝜀) is a certain constant, then there exists a unique weak solution to (36) with initial-boundary condition (4) and (5), such that 𝑚(𝑥,𝑡)∈𝐿^∞0,𝑇;𝐻^𝛼+2(0,2𝜋). (47)

Proof. Let the operator Λ² act on (39) we get −Δ𝑚_𝑡=−𝜀Λ^2𝛼+2𝑚+𝜆Δ𝑚×Λ^2𝛼𝑚−𝜆𝑚×Λ^2𝛼+2𝑚+𝜆∇𝑚×∇Λ^2𝛼𝑚. (48) Taking the inner product of Λ^2𝛼+2𝑚 and (48) and integrating over (0,2𝜋), we have 12𝑑𝑑𝑡₀^2𝜋Λ^𝛼+2𝑚²𝑑𝑥+𝜀₀^2𝜋Λ^2𝛼+2𝑚²𝑑𝑥=𝜆₀^2𝜋Δ𝑚×Λ^2𝛼𝑚⋅Λ^2𝛼+2𝑚𝑑𝑥+𝜆₀^2𝜋∇𝑚×∇Λ^2𝛼𝑚⋅Λ^2𝛼+2≤1𝑚𝑑𝑥𝜀Λ^2𝛼𝑚²_𝐿∞‖Δ𝑚‖²_𝐿2+1𝜀‖∇𝑚‖²_𝐿^∞∇Λ^2𝛼𝑚²_𝐿2+𝜀2Λ^2𝛼+2𝑚²_𝐿2≤𝑀2Λ^𝛼+2𝑚⁴_𝐿2+𝜀2Λ^2𝛼+2𝑚²_𝐿2. (49) Note that 𝑀=𝑜(1/𝜀) is a constant; hence by Lemma 5, which will be proved later, we have sup_{0≤𝑡≤𝑇}Λ^𝛼+2𝑚²_{𝐿2()0,2𝜋}≤𝐶₃, (50) in which 𝐶₃ is independent of 𝜀,𝜆. From (49), for each 𝜀>0, we have ‖𝑚‖_𝐿²_(0,𝑇;𝐻^2𝛼+2_(0,2𝜋))≤𝐶. (51)

Lemma 5. Let 𝑓(𝑡) be nonnegative continuous functions for 0≤𝑡≤𝑇. Suppose that 𝑓(0)<1/𝑘𝑇 and 𝑓(𝑡)≤𝑘^𝑡₀𝑓²(𝜏)𝑑𝜏+𝑓(0),0≤𝑡≤𝑇, (52) where 𝑘 is a constant. Then 𝑓(𝑡)≤𝑓(0)1−𝑘𝑇𝑓(0) (53) holds for 0≤𝑡≤𝑇.

Proof. Define 𝑣(𝑡)=𝑘^𝑡₀𝑓²(𝜏)𝑑𝜏+𝑓(0). (54) Then the function 𝑣(𝑡) is nondecreasing, 𝑣(0)=𝑓(0), and 𝑑𝑣(𝑡)𝑑𝑡=𝑘𝑓²(𝑡)≤𝑘𝑣²(𝑡), (55) since 𝑓(𝑡)≤𝑣(𝑡)≤𝑣(𝑇). According to (55), the function 𝑧(𝑡)=−1/𝑣(𝑡) satisfies 𝑑𝑧(𝑡)=𝑣𝑑𝑡^′(𝑡)𝑣²(𝑡)=𝑘𝑓²(𝑡)𝑣²(𝑡)≤𝑘. (56) Integrating (56) from 0 to 𝑡 yields 𝑧(𝑡)≤𝑘𝑡+𝑧(0), (57) or −1𝑣(𝑡)1≤𝑘𝑡−𝑣(0); (58) that is 𝑣(𝑡)≤𝑣(0)(0).1−𝑘𝑡𝑣 (59) Hence 𝑓(𝑡)≤𝑓(0)(0).1−𝑘𝑇𝑓 (60)

4. Convergence Process

Before we prove existence of weak solution to the fractional Landau-Lifshitz model without Gilbert term (2), we first recall two Lemmas in [25, 26], respectively.

Lemma 6. Suppose that 𝑠>0 and 𝑝∈(1,+∞). If 𝑓,𝑔∈𝒮, the Schwartz class, then Λ^𝑠𝑓𝑔−𝑓Λ^𝑠𝑔_𝐿𝑝≤𝐶∇𝑓_𝐿𝑝1𝑔_{̇𝑊2𝑠−1,𝑝}+𝑓_{̇𝑊3𝑠,𝑝}𝑔_𝐿𝑝4,Λ^𝑠𝑓𝑔_𝐿𝑝≤𝐶𝑓_𝐿𝑝1𝑔_{̇𝑊2𝑠,𝑝}+𝑓_{̇𝑊3𝑠,𝑝}𝑔_𝐿𝑝4, (61) with 𝑝₂,𝑝₃∈(1,+∞) such that 1𝑝=1𝑝₁+1𝑝₂=1𝑝₃+1𝑝₄, (62) where 𝑊̇^𝑘,𝑝 is the homogeneous Sobolev space and the 𝑊̇^𝑘,𝑝-norm of 𝑓 is defined by 𝑓_{̇𝑊𝑘,𝑝}=ℱ⁻¹𝜉^𝑠𝑓(𝜉)_𝐿𝑝. (63)

The following Lemma is due to Gagliardo and Nirenberg, see [11].

Lemma 7. Let Ω be ℝ^𝑛 or a bounded Lipschitz domain in ℝ^𝑛 with 𝜕Ω, and let 𝑢 be any function in 𝑊^𝑚,𝑟(Ω)⋂𝐿^𝑞(Ω), 1≤𝑟,𝑞≤+∞. For any integer 𝑗, 0≤𝑗<𝑚, and for any number 𝑎 in the interval 𝑗/𝑚≤𝑎≤1, set 1𝑝=𝑗𝑛1+𝑎𝑟−𝑚𝑛+()11−𝑎𝑞. (64) If 𝑚−𝑗−(𝑛/𝑟) is not a nonnegative integer, then ∇^𝑗𝑢_𝐿𝑝(Ω)≤𝐶‖𝑢‖^𝑎_𝑊^𝑚,𝑟_(Ω)‖𝑢‖_𝐿^1−𝑎𝑞_(Ω). (65) If 𝑚−𝑗−(𝑛/𝑟) is a nonnegative integer, then (65) holds for 𝑎=𝑗/𝑚. The constant 𝐶 depends only on 𝑟,𝑞,𝑚,𝑗,𝑎, and the shape of Ω.

From (45), we conclude the following.

Lemma 8. Solutions to (36) satisfy sup_{0≤𝑡≤𝑇}𝑚^𝜀(⋅,𝑡)_{𝐻𝛼(0,2𝜋)}≤𝐶, (66) in which 𝐶 is independent of 𝜀.

For the uniform bound of 𝜕𝑚^𝜀/𝜕𝑡, we have the following lemma.

Lemma 9. 𝜕𝑚^𝜀/𝜕𝑡 in (36) satisfies sup_{0≤𝑡≤𝑇}𝜕𝑚^𝜀(𝜕𝑡⋅,𝑡)_{𝐻−𝑘(0,2𝜋)}≤𝐶, (67) in which 1/2<𝑘<1 and 𝐶 is independent of 𝜀.

Proof. For all 𝜓(𝑥)∈(𝐻^𝛼(0,2𝜋))³, we have ^+∞_−∞𝜓⋅𝜕𝑚^𝜀𝜕𝑡𝑑𝑥=−𝜀^+∞_−∞Λ^𝛼𝜓(𝑥)⋅Λ^𝛼𝑚^𝜀−𝑑𝑥_+∞−∞𝑚^𝜀×Λ^2𝛼𝑚^𝜀⋅𝜓(𝑥)𝑑𝑥=−𝜀^+∞_−∞Λ^𝛼𝜓(𝑥)⋅Λ^𝛼𝑚^𝜀−𝑑𝑥_+∞−∞𝜓(𝑥)×𝑚^𝜀⋅Λ^2𝛼𝑚^𝜀𝑑𝑥=−𝜀^+∞_−∞Λ^𝛼𝜓(𝑥)⋅Λ^𝛼𝑚^𝜀−𝑑𝑥_+∞−∞Λ^𝛼𝜓(𝑥)×𝑚^𝜀⋅Λ^𝛼𝑚^𝜀≤𝜓𝑑𝑥_𝐿∞Λ^𝛼𝑚^𝜀2_𝐿2+Λ^𝛼𝑚^𝜀_𝐿2Λ^𝛼𝜓_𝐿4𝑚^𝜀_𝐿4≤𝐶𝜓_{𝐻𝑘(0,2𝜋)}, (68) in which formula (𝑎×𝑏)⋅𝑐⃗=(𝑐⃗×𝑎)⋅𝑏; Sobolev embedding theorem and Lemma 6 are used.

The following lemma is about the uniform bound of the Hölder norm of 𝑚^𝜀.

Lemma 10. Solutions to (36) satisfy 𝑚^𝜀(𝑥,𝑡)_{𝐶(2𝛼−1)/(𝛼+1),(2𝛼−1)/(2(𝛼+1))(𝑄𝑇)}≤𝐶, (69) where 𝐶 is independent of 𝜀.

Proof. Denote 𝜔^𝜀=∫^𝑥₀𝑚^𝜀(𝜉,𝑡)𝑑𝜉∀𝜓(𝑥)∈𝐻^𝑘₀(0,2𝜋_02𝜋),𝜓^′(𝑥)𝜔^𝜀_𝑡(𝑥,𝑡)𝑑𝑥=₀^2𝜋𝜓(𝑥)𝑚^𝜀_𝑡()𝜓(𝑥𝑥,𝑡𝑑𝑥≤𝐶)_{𝐻𝑘(0,2𝜋)}≤𝐶𝜓^′(𝑥)_{𝐿2(0,2𝜋)}. (70) Therefore sup_{0≤𝑡≤𝑇}𝜔^𝜀_𝑡(𝑥,𝑡)_𝐿2≤𝐶. (71) Hence 𝜔^𝜀(𝑥,𝑡)isboundedin𝐿^∞0,𝑇;𝐻^𝛼+1(𝑊0,2𝜋)_1,∞0,𝑇;𝐻^1−𝛼(0,2𝜋). (72) Therefore, by Lemma 7, 𝑚^𝜀𝑥,𝑡₂−𝑚^𝜀𝑥,𝑡₁=𝜔^𝜀_𝑥𝑥,𝑡₂−𝜔^𝜀_𝑥𝑥,𝑡₁≤𝐶𝜔^𝜀⋅,𝑡₂−𝜔^𝜀⋅,𝑡_1𝐿^{(2𝛼−1)/2(𝛼+1)}_2()0,2𝜋×𝜔^𝜀⋅,𝑡₂−𝜔^𝜀⋅,𝑡_1𝐻^3/2(𝛼+1)_{𝛼+1(0,2𝜋)}≤𝐶𝑡₂−𝑡₁^{(2𝛼−1)/2(𝛼+1)}×sup_{0≤𝑡≤𝑇}𝑤^𝜀_𝑡(⋅,𝑡)_𝐿^{(2𝛼−1)/2(𝛼+1)}_2()0,2𝜋×𝜔^𝜀⋅,𝑡₂−𝜔^𝜀⋅,𝑡_1𝐻^3/2(𝛼+1)_{𝛼+1()0,2𝜋}. (73)

Weak solution to (2) with initial-boundary condition (4) and (5) is defined as follows.

Definition 11. 3-dimensional vector-valued function 𝑚(𝑥,𝑡) is called a weak solution to (2) with initial-boundary condition (4) and (5), if 𝑚(𝑥,𝑡)∈𝐿^∞0,𝑇;𝐻^𝛼(𝐶0,2𝜋)_{(2𝛼−1)/(𝛼+1),(2𝛼−1)/2(𝛼+1)}𝑄_𝑇,_𝑄𝑇𝜕𝑚𝜕𝑡⋅𝜑+Λ^𝛼𝜑×𝑚⋅Λ^𝛼𝑚+𝑑𝑥𝑑𝑡_02𝜋𝜑(𝑥,0)⋅𝑚₀(𝑥)𝑑𝑥=0, (74) for all test functions 𝜑∈(𝐶^∞(𝑄_𝑇))³.

Theorem 12. Suppose that 𝑚₀(𝑥)∈𝐻^𝛼(0,2𝜋); then there exists a weak solution to (2) with initial-boundary condition (4) and (5) satisfying 𝑚(𝑥,𝑡)∈𝐿^∞0,𝑇;𝐻^𝛼(𝐶0,2𝜋)_{(2𝛼−1)/(𝛼+1),(2𝛼−1)/2(𝛼+1)}𝑄_𝑇,𝜕𝑚𝜕𝑡∈𝐿^∞0,𝑇;𝐻^−𝛼(0,2𝜋). (75)

Proof. For a solution to the mollified equation (36), we have _𝑄𝑇𝜕𝑚^𝜀𝜕𝑡⋅𝜑+𝜀Λ^𝛼𝜑⋅Λ^𝛼𝑚^𝜀+Λ^𝛼𝜑×𝑚^𝜀⋅Λ^𝛼𝑚^𝜀+𝑑𝑥𝑑𝑡_02𝜋𝜑(𝑥,0)𝑚₀(𝑥)𝑑𝑥=0,∀𝜑∈𝐶^∞𝑄_𝑇³. (76)
By Lemmas 8 and 9, 𝑚^𝜀(𝑥,𝑡)isboundedin𝐿^∞0,𝑇;𝐻^𝛼(𝐶0,2𝜋)_{(2𝛼−1)/(𝛼+1),(2𝛼−1)/2(𝛼+1)}𝑄_𝑇,𝜕𝑚^𝜀(𝜕𝑡𝑥,𝑡)isboundedin𝐿^∞0,𝑇;𝐻^−𝑘(0,2𝜋). (77) Hence 𝑚(𝑥,𝑡)∈𝐶^{(2𝛼−1)/(𝛼+1),(2𝛼−1)/2(𝛼+1)}(𝑄_𝑇). We select a subsequence (still denoted as {𝑚^𝜀(𝑥,𝑡)}), such that 𝑚^𝜀(𝑥,𝑡)isweak^∗convengentto𝑚(𝑥,𝑡)in𝐿^∞0,𝑇;𝐻^𝛼(),𝑚0,2𝜋_𝜀(𝑥,𝑡)isuniformlyconvergentto𝑚(𝑥,𝑡)in𝑄_𝑇,and𝜕𝑚^𝜀(𝜕𝑡𝑥,𝑡)isweak^∗convengentto𝜕𝑚𝜕𝑡in𝐿^∞0,𝑇;𝐻^−𝑘(0,2𝜋). (78) By a theorem in [24], we have that 𝑚^𝜀(𝑥,𝑡)isstronglyconvengentto𝑚(𝑥,𝑡)in𝐿²0,𝑇;𝐿²(0,2𝜋). (79) To prove that _𝑄𝑇Λ^𝛼𝜑×𝑚^𝜀⋅Λ^𝛼𝑚^𝜀𝑑𝑥𝑑𝑡⟶_𝑄𝑇Λ^𝛼𝜑×𝑚⋅Λ^𝛼𝑚𝑑𝑥𝑑𝑡, (80) we have to construct a commutator. Actually, _𝑄𝑇Λ^𝛼𝜑×𝑚^𝜀⋅Λ^𝛼𝑚^𝜀𝑑𝑥𝑑𝑡=_𝑄𝑇Λ^𝛼𝜑×𝑚^𝜀⋅Λ^𝛼𝑚^𝜀−𝑑𝑥𝑑𝑡_𝑄𝑇𝜑×Λ^𝛼𝑚^𝜀⋅Λ^𝛼𝑚^𝜀=𝑑𝑥𝑑𝑡_𝑄𝑇Λ^𝛼𝜑×𝑚^𝜀−𝜑×Λ^𝛼𝑚^𝜀⋅Λ^𝛼𝑚^𝜀𝑑𝑥𝑑𝑡. (81) Similarly, _𝑄𝑇Λ^𝛼𝜑×𝑚⋅Λ^𝛼𝑚𝑑𝑥𝑑𝑡=_𝑄𝑇Λ^𝛼𝜑×𝑚−𝜑×Λ^𝛼𝑚⋅Λ^𝛼𝑚𝑑𝑥𝑑𝑡. (82) Denote Λ^𝛼(𝜑×𝑢)−𝜑×Λ^𝛼𝑢 by ℒ_𝜑(𝑢). To get (80), it suffices to prove that _𝑄𝑇ℒ_𝜑𝑚^𝜀Λ^𝛼𝑚^𝜀−ℒ_𝜑(𝑚)Λ^𝛼𝑚𝑑𝑥𝑑𝑡=_𝑄𝑇ℒ_𝜑𝑚^𝜀−ℒ_𝜑(𝑚)⋅Λ^𝛼𝑚^𝜀+_𝑄𝑇ℒ_𝜑(𝑚)Λ^𝛼𝑚^𝜀−Λ^𝛼𝑚𝑑𝑥𝑑𝑡=𝐼₁+𝐼₂⟶0,as𝜀⟶0. (83) Obviously 𝐼₂→0, as 𝜀→0. We need to prove that 𝐼₁→0, as 𝜀→0. Actually, by Lemma 6, we can get ℒ_𝜑𝑚^𝜀−ℒ_𝜑(𝑚)_{𝐿2(0,2𝜋)}≤𝐶∇𝜑_𝐿𝑝1𝑚^𝜀−𝑚_{̇𝑊2𝛼−1,𝑝}+𝜑_{̇𝑊3𝛼,𝑝}𝑚^𝜀−𝑚_𝐿𝑝4≤𝐶∇𝜑_𝐿𝑝1𝑚^𝜀−𝑚_𝐿2+𝜑_{̇𝑊3𝛼,𝑝}𝑚^𝜀−𝑚_𝐻𝛽≤𝐶𝜑_{𝐻𝛾(0,2𝜋)}𝑚^𝜀−𝑚_{𝐻𝛽(0,2𝜋)}, (84) where 𝑝₂,𝑝₃∈(1,+∞) are such that 1𝑝₁+1𝑝₂=12,1𝑝₃+1𝑝₄=12,𝑝0<𝛽<𝛼<1,²𝛼𝑝₂𝑝−1=2,₄=21−2𝛽,2𝛾−1>1. (85) Therefore 𝛾>𝛼+(1/2)−(1/𝑝₃) automatically.
By Hölder's inequality, we can show that _𝑄𝑇Λ^𝛼𝑚^𝜀ℒ_𝜑𝑚^𝜀Λ−𝑚𝑑𝑥𝑑𝑡≤𝐶^𝛼𝑚^𝜀_{𝐿2(𝑄𝑇)}𝜑_{𝐿2(0,𝑇;𝐻𝛾(0,2𝜋))}×𝑚^𝜀−𝑚_{𝐿2(0,𝑇;𝐻𝛽(0,2𝜋))}⟶0,as𝜀⟶0. (86) As 𝜀→0 in (76), we have _𝑄𝑇𝜕𝑚𝜕𝑡𝜑+Λ^𝛼𝜑×𝑚⋅Λ^𝛼𝑚+𝑑𝑥𝑑𝑡_02𝜋𝜑(𝑥,0)⋅𝑚₀(𝑥)𝑑𝑥=0, (87) for all test functions 𝜑∈(𝐶^∞(𝑄_𝑇))³.

Acknowledgment

The paper is supported by the National Natural Science Foundation of China (no. 11201181, no. 11201311).