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

and Applied Analysis 3 the approximate solution m n . Multiplying equality (16) by g i and summing for 1 ≤ i ≤ n, we have 1 2 󵄩󵄩󵄩󵄩mn (⋅, t) 󵄩󵄩󵄩󵄩 2 L 2 (T)


Introduction
We study the fractional Landau-Lifshitz equation where (, ) is a three-dimensional vector representing the magnetization and ,  1 ,  2 ≥ 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 R 3valued vectors.The first term  × Λ 2  is the gyromagnetic term and the second term  × ( × Λ 2 ) is called the Gilbert damping term.The fractional diffusion operator Λ 2 is nonlocal except  = 0, 1, 2, 3, . .., which means that Λ 2 () depends not only on () for  near  but also on () for all .
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][2][3][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][8][9].The fractional Landau-Lifshitz equation shares some similar difficulties with quasigeostrophic 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  2 = 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.

Abstract and Applied Analysis
In this paper, we will study local existence of weak solutions in the spatial domain (0, 2) with  2 = 0 and  ∈ [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  1 = 1.Actually, (1) can be written as in which with initial condition and the periodic boundary condition It is straightforward to check the following conclusions.
(1) The matrix () is "zero definite"; namely, (2) The matrix () is singular; that is, Hence ( 2) is quite different from usual quasilinear parabolic equations for the above reasons.
To approximate (2), we consider the following mollified equation: which can be written as 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, positivedefinition and uniform ellipticity of matrix  − () and choice of norm space  ∞ ensure that Leray-Schauder fixedpoint 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.

Cauchy Problem for the Corresponding Linear Equation
Our starting point is the linear equation (, 0) =  0 () , on T  , where T  = R  /  is the flat torus and (, ) and  0 () are N-dimensional vector-valued functions.We have the following theorem about existence of solution to (10)- (11).
Theorem 1. Suppose that × matrix (, ) defined on T  × (0, ) is measurable, bounded, and uniformly elliptic; namely, there exists a constant  such that for all -dimensional vectors, (, ) ∈  2 (0, ;  2 (T  )), and  0 () ∈   (T  ).Then there exists a unique vector-valued solution to (10)-( 11) such that Proof of Theorem 1.We apply the Galerkin method: let {  } be an orthogonal basis of  2 (T  ) consisting of all the eigenfunctions for the operator We are looking for approximate solutions   (, ) to ( 10)- (11) under the form where   are vector-valued functions, such that, for 1 ⩽  ⩽ , there holds These relations produce an ordinary differential system that can be writeen as where  = ( 1 ,  2 , . . .,   ) and  0 is the projection of  0 on ( 1 ,  2 , . . .  ).The existence of a local solution to system (18) is a classical matter.We now proceed to estimate the approximate solution   .Multiplying equality (16) by   and summing for 1 ≤  ≤ , we have Multiplying equality ( 16) by     and summing for 1 ≤  ≤ , we have Adding ( 19) to ( 20), we get Since , by Gronwall's inequality, we have sup Taking the inner product of   / and (10) and integrating over T  × (0, ), we have Hence Actually, if the matrix (, ) is retrained to a small class of good function matrix, one can get higher regularity of solution to ( 10)- (11).

Cauchy Problem for the Mollified Equation
To get existence of weak solution to (2), we consider the following approximate equation: which is called mollified equation.In this section and next section, we assume that the spatial variable  ∈ (0, 2).By Leray-Schauder fixed-point theorem, we have the following theorem.
Obviously, for all , the mapping   is continuous; and for any bounded closed set of  ∞ (  ),   is uniformly continuous with respect to 0 ≤  ≤ 1.
To apply Leray-Schauder fixed-point theorem, we make a priori estimate on all fixed points of   .
Taking the inner product of (, ) and equation we have Integrating (40) over   (0 ≤  ≤ ), we get in which  1 is a constant independent of , .

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  > 0 and  ∈ (1, +∞).If ,  ∈ S, the Schwartz class, then with  2 ,  3 ∈ (1, +∞) such that where Ẇ, is the homogeneous Sobolev space and the Ẇ,norm of  is defined by The following Lemma is due to Gagliardo and Nirenberg, see [11].
From (45), we conclude the following.
in which 1/2 <  < 1 and  is independent of .