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

Authors: Lijun Wang; Jingna Li; Li Xia
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

We study the fractional Landau-Lifshitz equation(1)∂m∂t=-λ1m×Λ2αm+λ2m×(m×Λ2αm),
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×Λ2αm is the gyromagnetic term and the second term m×(m×Λ2αm) is called the Gilbert damping term. The fractional diffusion operator Λ2α is nonlocal except α=0,1,2,3,…, which means that Λ2αu(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. Whenα=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 thatλ1=1. Actually, (1) can be written as
(2)∂m∂t=B(m)Λ2αm,
in which
(3)m=(m1m2m3),B(m)=(0m3-m2-m30m1m2-m10),
with initial condition
(4)m(x,0)=m0(x)
and the periodic boundary condition
(5)m(x,t)=m(x+2π,t).
It is straightforward to check the following conclusions.(1)
The matrixB(m) is “zero definite”; namely,
(6)ξτB(m)ξ=0,∀ξ,m∈ℝ3.(2)
The matrixB(m) is singular; that is,
(7)detB(m)=0,∀m∈ℝ3.Hence (2) is quite different from usual quasilinear parabolic equations for the above reasons.To approximate (2), we consider the following mollified equation:
(8)∂m∂t=-εΛ2αm-m×Λ2αm,
which can be written as
(9)∂m∂t+(εE-B(m))Λ2αm=0.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(10)∂m∂t+A(x,t)Λ2αm=f(x,t),in𝕋n×(0,T),(11)m(x,0)=m0(x),on𝕋n,
where 𝕋n=ℝn/Zn 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 thatN×N matrix A(x,t) defined on 𝕋n×(0,T) is measurable, bounded, and uniformly elliptic; namely, there exists a constant K such that
(12)A(x,t)η·η≥K|η|2,
for all N-dimensional vectors, f(x,t)∈L2(0,T;L2(𝕋n)),andm0(x)∈Hα(𝕋n). Then there exists a unique vector-valued solution to (10)-(11) such that
(13)m(x,t)∈L∞(0,T;Hα(𝕋n))⋂‍L2(0,T;H2α(𝕋n)),∂m∂t∈L2(0,T;L2(𝕋n)).Proof of Theorem1.
We apply the Galerkin method: let{φj} be an orthogonal basis of L2(𝕋n) consisting of all the eigenfunctions for the operator
(14)Λ2αφj=λjφj,φj(0)=φj(2π).
We are looking for approximate solutions mn(x,t) to (10)-(11) under the form
(15)mn(x,t)=∑j=1n‍gj(t)φj(x),
where gj are vector-valued functions, such that, for 1⩽i⩽n, there holds
(16)∫Ω‍[∂mn∂t+A(x,t)Λ2αmn-f(x,t)]φidx=0,(17)∫Ω‍[mn(x,0)-m0(x)]φidx=0.
These relations produce an ordinary differential system that can be writeen as
(18)∂g∂t=F(g),g(0)=g0,
where g=(g1,g2,…,gn) and g0 is the projection of m0 on (φ1,φ2,…φn). The existence of a local solution to system (18) is a classical matter. We now proceed to estimate the approximate solution mn. Multiplying equality (16) by gi and summing for 1≤i≤n, we have
(19)12∥mn(·,t)∥L2(𝕋n)2=∫0T‍∫𝕋n‍(f·mn-A(x,t)Λ2αmn·mn)dxdt+12∥m0∥L2(𝕋n)2.
Multiplying equality (16) by λigi and summing for 1≤i≤n, we have
(20)12∥Λαmn(·,t)∥L2(𝕋n)2+∫0T‍∫𝕋n‍A(x,t)Λ2αmn·Λ2αmndxdt=∫0T‍∫𝕋n‍f·Λ2αmndxdt+12∥Λαm0∥L2(𝕋n)2.
Adding (19) to (20), we get
(21)12∥mn(·,t)∥L2(𝕋n)2+12∥Λαmn(·,t)∥L2(𝕋n)2+∫0T‍∫𝕋n‍A(x,t)Λ2αmn·Λ2αmndxdt=12(∥m0∥L2(𝕋n)2+∥Λαm0∥L2(𝕋n)2)+∫0T‍∫𝕋n‍(f·mn+f·Λ2αmn)dxdt-∫0T‍∫𝕋n‍A(x,t)Λ2αmn·mndxdt≤12(∥m0∥L2(𝕋n)2+∥Λαm0∥L2(𝕋n)2)+C∥f∥L2(𝕋n×(0,T))2+C∫0T‍∫𝕋n‍|mn|2dxdt+K2∥Λ2αmn∥L2(𝕋n×(0,T))2.
Since ∫0T‍∫𝕋n‍A(x,t)Λ2αmn·Λ2αmndxdt≥K∥Λ2αmn∥L2(𝕋n×(0,T))2, by Gronwall's inequality, we have
(22)sup0≤t≤T∥mn(·,t)∥Hα(𝕋n)2+∥mn∥L2(𝕋n×(0,T))2≤C(∥m0∥Hα(𝕋n)2+∥f∥L2(𝕋n×(0,T))2),∥Λ2αmn∥L2(𝕋n×(0,T))2≤C(∥m0∥Hα(𝕋n)2+∥f∥L2(𝕋n×(0,T))2).
Taking the inner product of ∂mn/∂t and (10) and integrating over 𝕋n×(0,T), we have
(23)∫0T‍∫𝕋n‍|∂mn∂t|2dxdt=∫0T‍∫𝕋n‍(f-A(x,t)Λ2αmn)·∂mn∂tdxdt≤C∫0T‍∫𝕋n‍|f|2dxdt+C∫0T‍∫𝕋n‍|Λ2αmn|2dxdt+12∫0T‍∫𝕋n‍|∂mn∂t|2dxdt.
Hence
(24)∥∂mn∂t∥L2(𝕋n×(0,T))2≤C(∥m0∥Hα(𝕋n)2+∥f∥L2(𝕋n×(0,T))2).
Actually, if the matrixA(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 mn a subsequence (still denoted by mn) such that
(25)mn⇀msw,inL2(0,T;H2α(𝕋n))weakly*,∂mn∂t⇀∂m∂t,inL2(0,T;L2(𝕋n))weakly.
Hence, we know that [24]
(26)mn⟶m,stronglyinL2(0,T;L2(𝕋n))hhhhhhhhhhhhhhhhhha.e.in𝕋n×[0,T].
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,…,φn), there holds
(27)∫𝕋n‍∂mn∂tφ+∫𝕋n‍A(x,t)Λ2αmnφ-∫𝕋n‍f(x,t)φdx=0.
By a density argument, we also obtain formula (27) for all φ in Hα(QT)QT=𝕋n×[0,T].Theorem 2.
Suppose that:
(i)N×N matrix A(x,t) defined on 𝕋n×(0,T) is measurable, bounded, and uniformly elliptic; namely, there exists a constant K such that
(28)A(x,t)η·η≥K|η|2,
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
(29)∥∇f∥L2(𝕋n×(0,T))≤δ∥Δf∥L2(𝕋n×(0,T));
(ii)f(x,t)∈L2(0,T;H2(𝕋n)),m0(x)∈Hα+2(𝕋n). Then there exists a unique vector-valued solution to (10)-(11) such that
(30)m(x,t)∈L∞(0,T;Hα+2(𝕋n))⋂‍H1(𝕋n×(0,T)),sup0≤t≤T∥m(·,t)∥Hα+2(𝕋n)2+∥∂m∂t∥L2(𝕋n×(0,T))2+∥m∥L2(0,T;H2α+2(𝕋n))2≤C(∥m0∥Hα+2(𝕋n)2+∥f∥L2(0,T;H2(𝕋n))2),
in which constant C is dependent A(x,t) independent of m(x,t).Proof.
Let the operatorΛ2 act on (10); we have
(31)-Δmt-ΔA(x,t)Λ2αm+A(x,t)Λ2α+2m-∇A(x,t)·∇Λ2αm=-Δf(x,t).
Taking the inner product of Λ2α+2m and (31) and integrating over 𝕋n×(0,T), we have
(32)12∫𝕋n‍|Λα+2m|2dx+∫0T‍∫𝕋n‍A(x,t)Λ2α+2m·Λ2α+2mdxdt=-∫0T‍∫𝕋n‍Δf·Λ2α+2mdxdt+∫0T‍∫𝕋n‍ΔA(x,t)Λ2αm·Λ2α+2mdxdt+∫0T‍∫𝕋n‍∇A(x,t)∇Λ2αm·Λ2α+2mdxdt+12∫𝕋n‍|Λα+2m0|2dx≤12∥Λα+2m0∥L2(𝕋n)2+1K∥Δf∥L2(𝕋n×(0,T))2+K4∥Λ2α+2m∥L2(𝕋n×(0,T))2+1K∥ΔA(x,t)Λ2αm∥L2(𝕋n×(0,T))2+K4∥Λ2α+2m∥L2(𝕋n×(0,T))2+1K∥∇A(x,t)∇Λ2αm∥L2(𝕋n×(0,T))2+K4∥Λ2α+2m∥L2(𝕋n×(0,T))2.
Note that
(33)|∇A(x,t)|∞<K5δ,∥∇Λ2αm∥L2(𝕋n×(0,T))≤δ∥Λ2α+2m∥L2(𝕋n×(0,T)),andΔA(x,t)isbounded.
Reporting (33) in (32) and taking into account
(34)∫0T‍∫𝕋n‍A(x,t)Λ2α+2m·Λ2α+2mdxdt≥K∥Λ2α+2m∥L2(𝕋n×(0,T))2,
we have
(35)sup0≤t≤T∥Λα+2m(·,t)∥L2(𝕋n)2+∥Λ2α+2m∥L2(𝕋n×(0,T))2≤C(∥m0∥Hα+2(𝕋n)2+∥f∥L2(0,T:H2(𝕋n))2).


## 3. Cauchy Problem for the Mollified Equation

To get existence of weak solution to (2), we consider the following approximate equation:
(36)∂m∂t=-εΛ2αm-m×Λ2αm,
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 thatm0(x)∈Hα(0,2π), then there exists a unique weak solution to (36) with initial-boundary condition (4) and (5) such that
(37)m(x,t)∈L∞(0,T;Hα(0,2π)),
where QT=(0,2π)×(0,T).Proof.
First, the mappingTλ:L∞(QT)→L∞(QT) is defined as follows. For each u∈L∞(QT), m=Tλ(u) is a solution to
(38)∂m∂t=-εΛ2αm-λu×Λ2αm
with initial condition (4), in which 0≤λ≤1. By Theorem 1, we know that m=Tλ(u) is the unique solution to (38) with initial-boundary condition (4) and (5); moreover, m∈L∞(0,T;H1(0,2π)).
Obviously, for allλ, the mapping Tλ is continuous; and for any bounded closed set of L∞(QT), Tλ 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 ofTλ.
Taking the inner product ofm(x,t) and equation
(39)∂m∂t=-εΛ2αm-λm×Λ2αm,
we have
(40)m·∂m∂t=-εm·Λ2αm-λ(m×Λ2αm)·m.
Integrating (40) over Qτ(0≤τ≤T), we get
(41)∥m(·,τ)∥L2(0,2π)2+2ε∥Λαm∥L2(Qτ)2≤∥m0∥L2(0,2π)2,
in which 0≤λ≤1,  0≤τ≤T. Hence
(42)sup0≤t≤T∥m(·,t)∥L2(0,2π)≤C1,
in which C1 is a constant independent of ε,λ.
Taking the inner product ofΛ2αm(x,t) and (39), we obtain
(43)Λ2αm·∂m∂t=-εΛ2αm·Λ2αm-λΛ2αm·(m×Λ2αm).
Integrating (43) over (0,2π) with respect to variable x leads to
(44)12ddt∥Λαm(·,t)∥L2(0,2π)2+ε∫02π‍|Λ2αm|2dx≤0.
Obviously,
(45)sup0≤t≤T∥Λαm(·,t)∥L2(0,2π)≤C2,
in which C2 is a constant independent of λ,ε. From (42), for each ε>0, we have
(46)∥m∥L2(0,T;H2α(0,2π))≤C.
In view of (42), (45), and (46), Sobolev embedding theorem gives the desired result.For small initial datam0(x), we can get higher regularity of the solution to (36).Theorem 4.
Suppose thatm0(x)∈Hα+2(0,2π),α∈(1/2,1], and ∥m0(x)∥Hα+22≤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
(47)m(x,t)∈L∞(0,T;Hα+2(0,2π)).Proof.
Let the operatorΛ2 act on (39) we get
(48)-Δmt=-εΛ2α+2m+λΔm×Λ2αm-λm×Λ2α+2m+λ∇m×∇Λ2αm.
Taking the inner product of Λ2α+2m and (48) and integrating over (0,2π), we have
(49)12ddt∫02π‍|Λα+2m|2dx+ε∫02π|Λ2α+2m|2dx=λ∫02π‍(Δm×Λ2αm)·Λ2α+2mdx+λ∫02π‍(∇m×∇Λ2αm)·Λ2α+2mdx≤1ε∥Λ2αm∥L∞2∥Δm∥L22+1ε∥∇m∥L∞2∥∇Λ2αm∥L22+ε2∥Λ2α+2m∥L22≤M2∥Λα+2m∥L24+ε2∥Λ2α+2m∥L22.
Note that M=o(1/ε) is a constant; hence by Lemma 5, which will be proved later, we have
(50)sup0≤t≤T∥Λα+2m∥L2(0,2π)2≤C3,
in which C3 is independent of ε,λ. From (49), for each ε>0, we have
(51)∥m∥L2(0,T;H2α+2(0,2π))≤C.Lemma 5.
Letf(t) be nonnegative continuous functions for 0≤t≤T. Suppose that f(0)<1/kT and
(52)f(t)≤k∫0t‍f2(τ)dτ+f(0),0≤t≤T,
where k is a constant. Then
(53)f(t)≤f(0)1-kTf(0)
holds for 0≤t≤T.Proof.
Define(54)v(t)=k∫0t‍f2(τ)dτ+f(0).
Then the function v(t) is nondecreasing, v(0)=f(0), and
(55)dv(t)dt=kf2(t)≤kv2(t),
since f(t)≤v(t)≤v(T). According to (55), the function z(t)=-1/v(t) satisfies
(56)dz(t)dt=v′(t)v2(t)=kf2(t)v2(t)≤k.
Integrating (56) from 0 to t yields
(57)z(t)≤kt+z(0),
or
(58)-1v(t)≤kt-1v(0);
that is
(59)v(t)≤v(0)1-ktv(0).
Hence
(60)f(t)≤f(0)1-kTf(0).


## 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 thats>0 and p∈(1,+∞). If f,g∈𝒮, the Schwartz class, then
(61)∥Λs(fg)-fΛsg∥Lp≤C(∥∇f∥Lp1∥g∥W˙s-1,p2+∥f∥W˙s,p3∥g∥Lp4),∥Λs(fg)∥Lp≤C(∥f∥Lp1∥g∥W˙s,p2+∥f∥W˙s,p3∥g∥Lp4),
with p2,p3∈(1,+∞) such that
(62)1p=1p1+1p2=1p3+1p4,
where W˙k,p is the homogeneous Sobolev space and the W˙k,p-norm of f is defined by
(63)∥f∥W˙k,p=∥ℱ-1(|ξ|sf^(ξ))∥Lp.The following Lemma is due to Gagliardo and Nirenberg, see [11].Lemma 7.
LetΩ be ℝn or a bounded Lipschitz domain in ℝn with ∂Ω, and let u be any function in Wm,r(Ω)⋂‍Lq(Ω), 1≤r,q≤+∞. For any integer j, 0≤j<m, and for any number a in the interval j/m≤a≤1, set
(64)1p=jn+a(1r-mn)+(1-a)1q.
If m-j-(n/r) is not a nonnegative integer, then
(65)∥∇ju∥Lp(Ω)≤C∥u∥Wm,r(Ω)a∥u∥Lq(Ω)1-a.
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
(66)sup0≤t≤T∥mε(·,t)∥Hα(0,2π)≤C,
in which C is independent of ε.For the uniform bound of∂mε/∂t, we have the following lemma.Lemma 9.
∂
m
ε
/
∂
t in (36) satisfies
(67)sup0≤t≤T∥∂mε∂t(·,t)∥H-k(0,2π)≤C,
in which 1/2<k<1 and C is independent of ε.Proof.
For allψ(x)∈(Hα(0,2π))3, we have
(68)∫-∞+∞‍ψ·∂mε∂tdx=-ε∫-∞+∞‍Λαψ(x)·Λαmεdx-∫-∞+∞‍(mε×Λ2αmε)·ψ(x)dx=-ε∫-∞+∞‍Λαψ(x)·Λαmεdx-∫-∞+∞‍(ψ(x)×mε)·Λ2αmεdx=-ε∫-∞+∞‍Λαψ(x)·Λαmεdx-∫-∞+∞‍Λα(ψ(x)×mε)·Λαmεdx≤∥ψ∥L∞∥Λαmε∥L22+∥Λαmε∥L2∥Λαψ∥L4∥mε∥L4≤C∥ψ∥Hk(0,2π),
in which formula (a→×b→)·c→=(c→×a→)·b→; Sobolev embedding theorem and Lemma 6 are used.The following lemma is about the uniform bound of the Hölder norm ofmε.Lemma 10.
Solutions to (36) satisfy
(69)∥mε(x,t)∥C(2α-1)/(α+1),(2α-1)/(2(α+1))(QT)≤C,
where C is independent of ε.Proof.
Denoteωε=∫0x‍mε(ξ,t)dξ(70)∀ψ(x)∈H0k(0,2π),|∫02π‍ψ′(x)ωtε(x,t)dx|=|∫02π‍ψ(x)mtε(x,t)dx|≤C∥ψ(x)∥Hk(0,2π)≤C∥ψ′(x)∥L2(0,2π).
Therefore
(71)sup0≤t≤T∥ωtε(x,t)∥L2≤C.
Hence
(72){ωε(x,t)}isboundedinL∞(0,T;Hα+1(0,2π))⋂‍W1,∞(0,T;H1-α(0,2π)).
Therefore, by Lemma 7,
(73)|mε(x,t2)-mε(x,t1)|=|ωxε(x,t2)-ωxε(x,t1)|≤C∥ωε(·,t2)-ωε(·,t1)∥L2(0,2π)(2α-1)/2(α+1)×∥ωε(·,t2)-ωε(·,t1)∥Hα+1(0,2π)3/2(α+1)≤C|t2-t1|(2α-1)/2(α+1)×sup0≤t≤T∥wtε(·,t)∥L2(0,2π)(2α-1)/2(α+1)hhhhhhhhh×∥ωε(·,t2)-ωε(·,t1)∥Hα+1(0,2π)3/2(α+1).Weak solution to (2) with initial-boundary condition (4) and (5) is defined as follows.Definition 11.
3-dimensional vector-valued functionm(x,t) is called a weak solution to (2) with initial-boundary condition (4) and (5), if
(74)m(x,t)∈L∞(0,T;Hα(0,2π))⋂‍C(2α-1)/(α+1),(2α-1)/2(α+1)(QT),∫‍∫QT‍[∂m∂t·φ+Λα(φ×m)·Λαm]dxdt+∫02π‍φ(x,0)·m0(x)dx=0,
for all test functions φ∈(C∞(QT))3.Theorem 12.
Suppose thatm0(x)∈Hα(0,2π); then there exists a weak solution to (2) with initial-boundary condition (4) and (5) satisfying
(75)m(x,t)∈L∞(0,T;Hα(0,2π))⋂‍C(2α-1)/(α+1),(2α-1)/2(α+1)(QT),∂m∂t∈L∞(0,T;H-α(0,2π)).Proof.
For a solution to the mollified equation (36), we have
(76)∫‍∫QT‍[∂mε∂t·φ+εΛαφ·Λαmε+Λα(φ×mε)·Λαmε]dxdt+∫02π‍φ(x,0)m0(x)dx=0,∀φ∈(C∞(QT))3.
By Lemmas8 and 9,
(77){mε(x,t)}isboundedinL∞(0,T;Hα(0,2π))⋂‍C(2α-1)/(α+1),(2α-1)/2(α+1)(QT),{∂mε∂t(x,t)}isboundedinL∞(0,T;H-k(0,2π)).
Hence m(x,t)∈C(2α-1)/(α+1),(2α-1)/2(α+1)(QT). We select a subsequence (still denoted as {mε(x,t)}), such that
(78){mε(x,t)}isweak*convengenttom(x,t)inL∞(0,T;Hα(0,2π)),{mε(x,t)}isuniformlyconvergenttom(x,t)inQT,and{∂mε∂t(x,t)}isweak*convengentto∂m∂tinL∞(0,T;H-k(0,2π)).
By a theorem in [24], we have that
(79){mε(x,t)}isstronglyconvengenttom(x,t)hhhhhhhhhhhhhhhhhhinL2(0,T;L2(0,2π)).
To prove that
(80)∫‍∫QT‍Λα(φ×mε)·Λαmεdxdt⟶∫‍∫QT‍Λα(φ×m)·Λαmdxdt,
we have to construct a commutator. Actually,
(81)∫‍∫QT‍Λα(φ×mε)·Λαmεdxdt=∫‍∫QT‍Λα(φ×mε)·Λαmεdxdt-∫‍∫QT‍(φ×Λαmε)·Λαmεdxdt=∫‍∫QT‍[Λα(φ×mε)-φ×Λαmε]·Λαmεdxdt.
Similarly,
(82)∫‍∫QT‍Λα(φ×m)·Λαmdxdt=∫‍∫QT‍[Λα(φ×m)-φ×Λαm]·Λαmdxdt.
Denote Λα(φ×u)-φ×Λαu by ℒφ(u). To get (80), it suffices to prove that
(83)∫‍∫QT‍ℒφ(mε)Λαmε-ℒφ(m)Λαmdxdt=∫‍∫QT‍[ℒφ(mε)-ℒφ(m)]·Λαmε+∫‍∫QT‍ℒφ(m)[Λαmε-Λαm]dxdt=I1+I2⟶0,asε⟶0.
Obviously I2→0, as ε→0. We need to prove that I1→0, as ε→0. Actually, by Lemma 6, we can get
(84)∥ℒφ(mε)-ℒφ(m)∥L2(0,2π)≤C(∥∇φ∥Lp1∥mε-m∥W˙α-1,p2+∥φ∥W˙α,p3∥mε-m∥Lp4)≤C(∥∇φ∥Lp1∥mε-m∥L2+∥φ∥W˙α,p3∥mε-m∥Hβ)≤C∥φ∥Hγ(0,2π)∥mε-m∥Hβ(0,2π),
where p2,p3∈(1,+∞) are such that
(85)1p1+1p2=12,1p3+1p4=12,0<β<α<1,p2αp2-1=2,p4=21-2β,2(γ-1)>1.
Therefore γ>α+(1/2)-(1/p3) automatically.
By Hölder's inequality, we can show that(86)|∫‍∫QT‍Λαmεℒφ(mε-m)dxdt|≤C∥Λαmε∥L2(QT)∥φ∥L2(0,T;Hγ(0,2π))×∥mε-m∥L2(0,T;Hβ(0,2π))⟶0,asε⟶0.
As ε→0 in (76), we have
(87)∫‍∫QT‍[∂m∂tφ+Λα(φ×m)·Λαm]dxdt+∫02π‍φ(x,0)·m0(x)dx=0,
for all test functions φ∈(C∞(QT))3.



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*Source: 757824-2013-12-09.xml*