STUDY OF A CLASS OF LANDAU-LIFSHITZ EQUATIONS OF FERROMAGNETISM
WITHOUT EXCHANGE ENERGY
By
Wei Deng

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Applied Mathematics
2012

ABSTRACT
STUDY OF A CLASS OF LANDAU-LIFSHITZ EQUATIONS OF
FERROMAGNETISM WITHOUT EXCHANGE ENERGY
By
Wei Deng
Landau-Lifshitz equations of ferromagnetism, which are based on several competing energy contributions, are important mathematical models for the evolution of magnetization
field m of a ferromagnetic material. Many problems, such as existence, stability, regularity,
asymptotic behavior, thin-film limit and numerical computation, have been well studied for
the Landau-Lifshitz equations that include the so-called exchange energy. However, these
problems turn out to be quite challenging for equations without the exchange energy. The
main reason is that when the exchange energy is included, one automatically has the magnetization vector m ∈ L∞ ((0, ∞); H 1 (Ω)) from energy estimates, which gives some compactness
and stability that are needed for using the standard methods; however, in the cases without
the exchange energy, one only has m ∈ L∞ ((0, ∞); L∞ (Ω)), which is too rough to get the
needed compactness and stability. In this thesis, we investigate some problems for models
of reduced Landau-Lifshitz equations with no-exchange energy.
In Chapter 1, we introduce the Landau-Lifshitz theory of ferromagnetism and summarize
the main results of the thesis. The readers can check out the main results quickly in this
chapter and then go to the corresponding chapters for details of proof, more discussions and
further references.
In Chapter 2, we study the quasi-stationary limit of a simple Landau-Lifshitz-Maxwell
system with the permittivity parameter

approaching zero and, using this quasi-stationary

limit, establish the existence of global weak solutions to the reduced Landau-Lifshitz equa-

tions with initial value m0 ∈ L∞ (Ω).
In Chapter 3, we establish a local L2 -stability theorem for the global weak solutions in
finite time. The key in the proof of stability theorem is that we split the nonlocal term
Hm into two parts: one is bounded in L∞ (Ω) and the other bounded in L2 (Ω). Using this
stability theorem, we also provide another proof for the existence of global weak solutions
for a full expression of the no-exchange energy with applied field a(x) ∈ L∞ (Ω).
In Chapter 4, we prove a higher time regularity for the regular solutions, using mainly
induction method, together with several interpolation results. In this chapter, we also study
the weak ω-limit sets for the so-called soft-case and study the asymptotic behaviors for the
special case when Ω is ellipsoid and initial values m0 are constant.
In Chapter 5, we investigate a different model called the fractional Landau-Lifshitz equations and establish the existence of global weak solutions with initial value m0 ∈ H α (Ω),
where 0 < α < 1. In this new model, in contrast to the case when only the nonlocal term Hm
is included, we have some compactness in H α (Ω), which enables us to apply the Galerkin
method to establish the existence of global weak solution.

ACKNOWLEDGMENTS

I would like to gratefully and sincerely thank Dr. Baisheng Yan for his guidance, understanding, patience, and most importantly, his friendship during my graduate studies at Michigan
State University. His mentorship was paramount in providing a well rounded experience
which is consistent with my long-term career goals. He encouraged me to not only grow as
a mathematician but also as an independent thinker.
I would like to thank the Department of Mathematics at Michigan State University,
especially those members of my doctoral committee for their input, valuable discussions and
accessibility. In particular, I would like to thank Dr. Zhengfang Zhou, Dr. Moxun Tang, Dr.
Chang Yi Wang and Dr. Ignacio Uriarte-Tuero for their assistance and guidance in getting
my graduate career started on the right foot.
Finally, and most importantly, I thank my parents, Kepei Wang and Jianping Deng,
for their faith in me and allowing me to be as ambitious as I wanted. It was under their
watchful eye that I gained so much drive and an ability to tackle challenges. Also, I thank
my graduate fellow, Hongli Gao for her helpful discussions.

iv

TABLE OF CONTENTS

Chapter 1 Introduction and Main Results . . . . .
1.1 Existence . . . . . . . . . . . . . . . . . . . . . .
1.2 Stability . . . . . . . . . . . . . . . . . . . . . . .
1.3 Existence of global solutions . . . . . . . . . . . .
1.4 Time regularity and special asymptotics . . . . .
1.5 Existence of fractional Landau-Lifshitz equations

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Chapter 2 Existence for Reduced Landau-Lifshitz Equations
2.1 Quasi-stationary limit of Landau-Lifshitz-Maxwell systems . .
2.1.1 General assumptions and preliminaries . . . . . . . . .
2.1.2 Two Compensated Compactness Lemmas . . . . . . . .
2.1.3 Strong convergence of M . . . . . . . . . . . . . . . .
2.2 Existence for rough initial data . . . . . . . . . . . . . . . . .

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Chapter 3 Local L2 -Stability of Solutions in Finite Time
3.1 Stability of weak solutions . . . . . . . . . . . . . . . . . .
3.1.1 Decomposition of Hm . . . . . . . . . . . . . . . .
3.1.2 Stability Theorem . . . . . . . . . . . . . . . . . . .
3.2 Existence of global weak solutions . . . . . . . . . . . . . .
3.2.1 Properties of f for smooth applied fields . . . . . .
3.2.2 Existence of global solution for smooth data . . . .
3.2.3 Existence of global weak solution for rough data . .
3.2.4 Proof of Theorem 3.2.4 . . . . . . . . . . . . . . . .

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Chapter 4 Higher Time Regularity and Special Asymptotics
4.1 Higher time regularity . . . . . . . . . . . . . . . . . . . . . .
4.2 Weak ω-limit sets . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The energy identity . . . . . . . . . . . . . . . . . . . .
4.2.2 Weak ω-limit sets and the estimate for soft-case . . . .
4.3 Special dynamics for constant initial data on ellipsoid domains
4.3.1 Associated ODE system on R3 . . . . . . . . . . . . .
4.3.2 Lyapunov function and the special dynamics . . . . . .

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Chapter 5 Existence for Fractional Landau-Lifshitz
5.1 Notations and preliminaries . . . . . . . . . . . . .
5.2 A priori estimates . . . . . . . . . . . . . . . . . . .
5.3 Compactness . . . . . . . . . . . . . . . . . . . . .
5.4 Existence . . . . . . . . . . . . . . . . . . . . . . .

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vi

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86

Chapter 1
Introduction and Main Results
The well-known Landau-Lifshitz theory in ferromagnetism models the state of magnetization vector m of a ferromagnetic material occupying a domain Ω in R3 based on a
formulation of the total energy E(m) consisting of several competing terms:

E(m) =

1
κ
a(x) · m dx +
ϕ(m) dx −
| m|2 dx +
|Hm |2 dx.
2 Ω
2 R3
Ω
Ω

(1.1)

We refer to [4, 34, 37, 38] for more backgrounds on such a model. The first term of E(m)
is the exchange energy, penalizing the spacial change of m; this term could also be given in
terms of a positive-definite quadratic form of

m. The second term is the anisotropy energy

due to crystallographic properties of the material. The third term is the interaction energy
due to a given applied magnetic field a(x). The last term is the magnetostatic energy of the
stray field Hm induced by m through a simplified Maxwell equations:

curl Hm = 0,

div(Hm + mχΩ ) = 0 in R3 ,

(1.2)

where χΩ is the characteristic function of domain Ω. In this theory, the (rescaled) saturation
condition: |m| = 1 is usually assumed over Ω.
The dynamic Landau-Lifshitz equation governing the evolution of magnetization m =

1

m(x, t) is given by
∂t m = γm × Heff + α

γ
m × (m × Heff )
|m|

(1.3)

on Ω × [0, ∞), where γ < 0 is the electron gyromagnetic ratio, α ≥ 0 is the Landau-Lifshitz
phenomenological damping parameter, and Heff is the total effective magnetic field defined
by the functional derivative of E(m) as

Heff = −

∂E
= κ∆m − ϕ (m) + a(x) + Hm .
∂m

(1.4)

Note that the Landau-Lifshitz equation (1.3) can be written as a Landau-Lifshitz-Gilbert
equation:
∂t m = γ(1 + α2 )m × Heff +

α
m × ∂t m.
|m|

(1.5)

Most existing studies, including existence, stability, asymptotic behavior and regularity,
on (1.3) or (1.5) are for models with the exchange energy, that is, when Heff = κ∆m −
ϕ (m) + a(x) + Hm with κ > 0; however, few results have been established for the case when
the exchange energy is excluded, because of the lack of compactness and stability in such a
case.
We consider the reduced Landau-Lifshitz equation (1.3) without exchange energy and
study properties of the corresponding solutions. In following sections, we briefly introduce
some background and our results on existence, stability, asymptotic behavior and regularity
problems for reduced Landau-Lifshitz equations.

2

1.1

Existence

There are many results for existence of solutions to (1.3) or (1.5) with the exchange energy,
see [1, 3, 7, 8, 10, 17, 22, 27, 49]. In these cases, the initial data m0 usually need to be
smooth enough (e.g. in H 1 (Ω) or H 2 (Ω)) in order to use the Galerkin method and elliptic
estimates to establish the existence of solutions. Later, we will see that the most important
key in the proof of existence is that we need strong convergence for some certain sequence
of m. When we include exchange energy, the strong convergence condition is much easier to
obtain than the case without exchange energy. Specifically, with exchange energy, most of
previous research work can easily get following bound for m by applying energy estimation.

m ∈ L1 ((0, ∞); H 1 (Ω)))

(1.6)

The bound (1.6) is also very important to many other problems, such as asymptotic behavior
of solutions. Actually, most of results about Landau-Lifshitz equation with exchange energy
based on (1.6). Later we will see that in our case (without exchange energy), from energy
estimation, the best bound we can get for m is,

m ∈ L∞ ((0, ∞); L∞ (Ω)))

That is why our problem is more challenging.

Existence of weak solution for rough initial data
As introduced above, we will prove existence of solutions to the dynamic Landau-Lifshitz
equation (1.3) for a total energy E(m) without the exchange energy or simply called reduced
3

dynamic Landau-Lifshitz equation (1.3). Therefore, we have

Heff = −ϕ (m) + a(x) + Hm

This Cauchy problem can be written as a quasi-stationary system:




∂t m = F (x, m, Hm ) in Ω × (0, ∞),





3
curl Hm = 0, div(Hm + mχΩ ) = 0 in R ,







m(x, 0) = m0 (x) on Ω,

(1.7)

with a given initial datum m0 ∈ L∞ (Ω), where F : Ω × R3 × R3 → R3 is an appropriate
function (the Landau-Lifshitz interaction function) specifically given in Chapter 2.
In order to prove the existence of global solution, we wish to have tight bound like (1.6) for
solutions or approximate solutions of system (1.7) and then easily extract strong convergent
subsequence in L2 , which would handle the nonlinear term F (x, m, Hm ).
∂m

For smooth initial data m0 ∈ H 2 (Ω) with ∂ν0 |∂Ω = 0, previous research work of Carbou and Fabrie [8] has established the existence to the similar equation without exchange
energy using the singular perturbation method: include κ∆m in Heff and let κ → 0. But
unfortunately, for initial data only in L∞ (Ω), their method does not directly work. We need
to find other ways to obtain the compactness for approximate solutions to (1.7) with rough
initial data only in L∞ .
We study the system (1.7) as a quasi-stationary limit of the following Landau-LifshitzMaxwell system of electro-magnetism when the permittivity parameter

is constant and

approaches zero; the more general case of Maxwell equations with variable permittivity has

4

been studied in Jochmann [29]. However, the system (1.8) studied below is much simpler
and the method used is quite different, but more direct.



 ∂t E − curl H = 0,








∂ (H + M χ ) + curl E = 0 in R3 × (0, ∞),
 t
Ω

(1.8)



∂ M = F (x, M, H) in Ω × (0, ∞),
 t







(E, H)|
3

t=0 = (E0 , H0 ) on R , M |t=0 = m0 on Ω,
where the initial data E0 , H0 for electric and magnetic fields E, H are any vector-fields
satisfying
E0 , H0 ∈ L2 (R3 ; R3 ),
For

div E0 = div(H0 + m0 χΩ ) = 0.

(1.9)

> 0, the existence of certain weak solutions to the Cauchy problem (1.8) can be

obtained in the same way as studied in Joly, Metivier and Rauch [31] with = 1. Therefore,
for any

> 0, there exists a global weak solution M . We have following theorem for the

convergence of sequence M as → 0.
Theorem 1.1.1 (Chapter 2, section 2.1). We have M → m strongly in both C 0 ([0, T ]; L2 (Ω))
and L2 (ΩT ) for all 0 < T < ∞.
The strong convergence result in Theorem 1.1.1 is exactly what we need to handle the
nonlinear term in the Cauchy problem (1.7). We use quite different methods to get such
compactness comparing to the case with exchange energy. From details in Chapter 2, Section
2.1 for proof of Theorem 1.1.1, one can notice that we never have the same bound as (1.6).
To the best of our knowledge, in the reduced Landau-Lifshitz model, one should not be able
to get (1.6).
5

With Theorem 1.1.1, we establish the existence of global weak solution to the Cauchy
problem (1.7).
Theorem 1.1.2 (Chapter 2, section 2.2). Let m0 ∈ L∞ (Ω) and let E0 , H0 be any functions
satisfying (1.9). Then any function m determined by the convergence in Theorem 1.1.1 is a
weak solution to the Cauchy problem (1.7).

1.2

Stability

Definition 1.2.1. Given m0 ∈ L∞ (Ω), a global weak solution to the Cauchy problem (1.7)
is a function
1,∞

m ∈ Wloc ([0, ∞); L2 (Ω; R3 )) ∩ L∞ ((0, ∞); L∞ (Ω; R3 ))

(1.10)

such that m(0) = m0 in L2 (Ω) with m = m(·, t), equation

∂t m = F (x, m, Hm ),

(1.11)

holds both in L∞ ((0, ∞); L2 (Ω)) and in the sense of distribution on Ω × (0, T ), for all
0 < T < ∞.
In this section we will investigate the stability of global weak solutions to quasi-stationary
system (1.7) including function a(x); for a similar result on the Maxwell system, see [31,
Theorem 6.1] and [13, Theorem 5.1].
Theorem 1.2.1 (Chapter 3, Section 3.1). Let 0 < R, T < ∞ be given. Then there exist
constants C = C(R, T ) > 0, c = c(R, T ) > 0 and ρ = ρ(R, T ) > 0 such that, for any
weak solution mk to the system (1.7) with applied field ak and initial datum mk (0) = mk
0
6

satisfying ak L∞ + mk L∞ ≤ R for k = 1, 2, if µ = max{ m1 −m2 L2 , a1 −a2 L2 } ≤ c,
0
0
0
then one has, for all t ∈ [0, T ],

m1 (t) − m2 (t) L2 (Ω) ≤ Cµρ .

(1.12)

This stability result also implies the uniqueness of weak solution to system (1.7).

1.3

Existence of global solutions

Based on the previous stability theorem, we present a new method for the existence of global
weak solution to (1.7) with general applied fields a and initial data m0 .
First, we show the existence of global solution to (1.7) for smooth fields a and initial
data m0 ∈ H 2 (Ω; R3 ). Define f (m) = Fa (x, m, Hm ). We show f : H 2 (Ω; R3 ) → H 2 (Ω; R3 )
and is locally Lipschitz ; the proof uses a critical estimate that Hm ∈ H 2 (Ω; R3 ) for all
m ∈ H 2 (Ω; R3 ) (see, e.g., [8, 31]). By the abstract ODE theory in Banach spaces, problem
(1.7) has a local solution belongs to H 2 (Ω; R3 ) if m0 ∈ H 2 (Ω; R3 ). Then a no-blowup result
(Theorem 3.2.4) shows that the local solution is in fact global on t ∈ [0, ∞).
We remark that in the special case when ϕ = 0 and a = 0 (thus Heff = Hm ), for smooth
∂m

initial data m0 ∈ H 2 (Ω) with ∂ν0 |∂Ω = 0, Carbou and Fabrie [8] also established the global
existence through a singular perturbation method, by including κ∆m in Heff and letting
κ → 0.
Once we have obtained the global existence for smooth data a and m0 , we use approximation and the stability result Theorem 1.2.1 to establish the existence for general data.
Theorem 1.3.1 (Chapter 3, section 3.2). Let a ∈ L∞ (Ω; R3 ). Given any initial datum

7

m0 ∈ L∞ (Ω; R3 ), the problem (1.7) has an unique global weak solution.

1.4

Time regularity and special asymptotics

Higher regularity in time
In this section, we introduce the higher time-regularity result of solution to LLG equation
(1.13).



m = γm × H + γαm × (m × H ) in Ω × (0, ∞),
 t
m
m

(1.13)



m(0) = m ,

0
where Ω is a bounded smooth domain in R3 and m0 ∈ H 2 (Ω; R3 ). The existence of such
regular solutions has been proved in Section 3.2 by applying abstract ODE theory.
Similar regularity problem has been studied by Cimrak and Keer [6] for LLG with exchange energy. In their proof, they highly take advantage of energy estimation from exchange
energy and then apply induction method. Now even without exchange energy, we also get
similar time regularity. Our method is actually also inspired by Cimrak and Keer, using induction method; however, because we drop off exchange energy, we do not have any a priori
estimates as being used in [6]. One can refer to Section 4.1 for details of proof of Theorem
1.4.1.
Theorem 1.4.1 (Chapter 4, Section 4.1). For any time T > 0 and initial unit vector
m0 ∈ H 2 (Ω), the regular solution, i.e. m ∈ H 1 ([0, T ]; H 2 (Ω)), satisfies

p+1

∂t

m H 2 (Ω) ≤ C

8

where C is one constant only depending on T, p, m0 H 2 (Ω) .

Asymptotic behavior for constant initial data on ellipsoid domains
We now consider asymptotic behavior of regular solutions to Landau-Lifshitz equation (1.13).
One of difficulties is that without exchange energy

m, we do not have much control for weak

sequence with respect to time t. Thus, methods used in Carbou and Fabrie [7] can not be used
in our case. Recently, research work about asymptotic behavior of global weak solutions to
Landau-Lifshitz equation (1.13) has been done by Yan ([51],[52]). In [52], without exchange
energy, Yan studied asymptotic behaviors of equation (1.13) in the weak-star convergence of
L∞ (Ω, R3 ). The equilibrium set of equation (1.13) is

m × Hm = 0

on Ω.

Yan [51] has investigated the equilibrium set under the general framework of partial differential inclusions and the vectorial calculus of variations based on the notation quasiconvexity
in Ball [2], Dacorogna [11] and Morrey [40]. He has proved the following theorem.
Theorem 1.4.2 (Yan [51]). Let mj

lim

j→∞ Ω

m weak-star in L∞ (Ω; R3 ) as j → ∞. If

(|mj |2 + 2|mj × Hmj | − 1)+ dx = 0

then the weak-star limit m satisfies |m|2 + 2|m × Hm | ≤ 1 a.e. on Ω.
It is very obvious that all points in equilibrium set satisfies condition in Theorem 1.4.2,
thus Yan [52] described the behavior of equilibrium set in some sense.
In Section 4.2, we devote to prove a similar result as Theorem 1.4.2 in a different way. We
9

first derive an energy identity for the global weak solutions to the Landau-Lifshitz equation
(1.3).
Theorem 1.4.3. The global weak solution m to (1.7) with bounded initial data satisfies the
energy identity
t

E(m(t)) − E(m(s)) = γα
s

Ω

|m × Heff |2 dxdτ

∀ 0 ≤ s ≤ t < ∞.

(1.14)

Furthermore, if γα < 0, then mt ∈ L2 ((0, ∞); L2 (Ω; R3 )).
Therefore, the global-in-time regularity for weak solutions (even for regular solutions) is
that
m ∈ L∞ ((0, ∞); L∞ (Ω; R3 )) with mt ∈ L2 ((0, ∞); L2 (Ω; R3 )).
But this regularity is not enough to have strong convergence as t → ∞; it would be enough if
one has mt ∈ L1 ((0, ∞); L2 (Ω; R3 )) (see [32]). Therefore, it is quite challenging to study the
asymptotic behaviors for even the regular solutions. The solution orbits for general initial
data may not have strong ω-limit points; we thus study the weak ω-limit set:

˜
ω ∗ (m0 ) = {m | ∃ tj ↑ ∞ such that m(tj )

˜
m weakly in L2 (Ω; R3 )}.

(1.15)

We give an estimate of ω ∗ (m0 ) for the so-called soft-case, where there is no anisotropy energy
(ϕ = 0).
Theorem 1.4.4. Let γα < 0, ϕ = 0 and a ∈ L∞ (Ω; R3 ). Then, for any m0 ∈ L∞ (Ω; R3 )
with |m0 (x)| = 1 a.e. on Ω, it follows that

˜
˜
˜
ω ∗ (m0 ) ⊆ {m ∈ L∞ (Ω; R3 ) | |m|2 + 2|m × (a + Hm )| ≤ 1 a.e. on Ω}.
˜
10

(1.16)

This theorem generalizes some results in [52]. For more results on further special case
when a = 0, see [51, 52].
Instead of considering the asymptotic behavior of global weak solutions, in this section,
we only try to determine asymptotic behavior of solutions to Landau-Lifshitz equation (1.13)
under a special case when initial value m0 is constant over Ω and Ω set is ellipsoid. It mainly
involves two step.
The first step is that we need to write explicit form for nonlocal term Hm under our
special geometry with ellipsoid domains. After some work has been done in Chapter 4,
Section 4.2.1, we can write
Hm = −Λm on Ω,
where Λ is a positive definite matrix; the details of how we determine matrix Λ can be found
in Chapter 4, Section 4.2.1.
Then, we will use Lyapunov Theorem to determine global stable equilibrium point as
follows,
Theorem 1.4.5 (Chapter 4, Section 4.2). Let us assume that m0 is constant and Ω is
ellipsoid,
3

Ω = {x ∈

R3 |
i=1

x2
i < 1}
ai

b1 , b2 , b3 are positive numbers determined by (4.23). If bk = min{b1 , b2 , b3 }, then ±ek are
asymptotically stable critical points to Landau-Lifshitz equation (1.13), where {e1 , e2 , e3 } are
the standard basis vectors of R3 .

11

1.5

Existence of fractional Landau-Lifshitz equations

Motivation
Before we go in details about the existence result for our new model called fractional LandauLifshitz equations, let us first see motivation behind it. The micromagnetic energy is given
by,
E(m) =

κ
1
| m|2 +
ϕ(m) −
a(x) · m +
|Hm |2 .
2 Ω
2 R3
Ω
Ω

(1.17)

One popular research topic for functional energy (1.17) is to explore global minimizers under
non-convexity constraints |m| = 1. Another interesting problem is to investigate its limiting
behavior in different asymptotic regimes. Those regimes are different with relations among
following terms,

t = thickness of the film
l = length scale of the cross section

(1.18)

d = characteristic length scale of the magnetic material
There are two well studied regimes: (a) The large-body limit, in which d → 0 while t is
l
l
fixed. See [28], [18], [48] and [42]. (b) The small-aspect-ratio limit, in which t → 0 while d
l
l
is fixed. In this case, when the external filed is constant, the asymptotic variational problem
predicts a uniform magnetization; see [21].
DeSimone et al [15] considered different regimes: (a) Ω is a cylindrical domain of thickness
t with cross section Ω , such as Ω = Ω × (0, t); (b) m does not depend on the thickness
direction x3 ; (c) t

l. Under such regime, DeSimone et al [15] derived following convergent

12

magnetostatic energy through Fourier transform:

R3

|Hm |2 dx =

1

R2

| ∧− 2 (

m )|2 dx

(1.19)

1

where m = (m1 , m2 ) and ∧ denotes the square root of Laplacian (− ) 2 . It is pretty
interesting that the exchange energy is dissipative under this regime and magnetostatic
energy reduced to (1.19). This result motivated some research work associated with energy
(1.19) applied to Landau-Lifshitz equation; see [23], [24], [25], [26].

Fractional Landau-Lifshitz equation
To extend the models considered in [23], [24], [25], [26], we would like to prove the existence
of global weak solution to fractional Landau-Lifshitz equation (1.20) including magnetostatic
term with periodic boundary condition:

mt = γm × Fm + γm × (m × Fm )

in Rn ,

(1.20)

where
Fm = ∧2α m + Hm

0 < α < 1.

(1.21)

We focus on the existence of global weak solution(defined in Chapter 5, see definition
(5.0.1) ) in special domain Ω = [0, 2π]n with periodic boundary conditions.
Now, we define Hm in energy term (1.21) as,

 (ξ · m) · ξ

,

|ξ|2
Hm (ξ) =


 0, ξ = 0.
13

ξ = 0.
(1.22)

1
The operator ∧ denotes the square root of Laplacian (− ) 2 , so ∧β m can be understood in

terms of Fourier transform:
F(∧β m(x)) = |ξ|β m(ξ)

(1.23)

A priori estimates
We mainly use Galerkin method to prove existence. Therefore, we first project our problem
into finite space in L2 (Ω) and the solution can be easily established by standard ODE
method. Then we are going to prove a priori estimation in order to extract strong convergence
subsequence.
Lemma 1.5.1 (Chapter 5, Section 5.2). Let m0 ∈ H α (Ω), then for any 0 < T < ∞, the
approximate solutions mN to systems (5.8) in Chapter 5, satisfy,

max

0≤t≤T

mN 2 α (Ω) ≤ C1
H

d
where C1 only depends on initial data m0 2 α (Ω) . Moreover, for 1 ≤ r ≤ r∗ = d−α , where
H

d is dimension.
∂mN
≤ C2
r
∂t L (QT )
d
and for 1 < r ≤ r∗ = d−α

r−1

mN (t1 ) − mN (t2 ) Lr (Ω) ≤ C2 |t2 − t1 | r

where C2 only depends on initial data m0 2 α (Ω) and time T , QT = (0, T ) × Ω.
H
It is not enough to extract strong convergence subsequence; we still need following compactness lemma,
14

Lemma 1.5.2 ([39]). Let B0 , B, B1 be three Banach space such that,

B0 ⊂ B ⊂ B1

where the injections are continuous and B0 , B1 are reflexive and B0 → B is compact. Denote

W = {v|v ∈ Lp0 (0, T ; B0 ),

dv
∈ Lp1 (0, T ; B1 )}
dt

for T < ∞ and 1 < p0 , p1 < ∞. Then W equipped with the norm
dv
v Lp0 (0,T ;B ) +
p
0
dt L 1 (0,T ;B1 )
is a Banach space and the embedding W → Lp0 (0, T ; B) is compact.
Now, we can proved that there exists some m ∈ L∞ (0, T ; H α (Ω)) such that,

mN

m weakly in Lp (0, T ; H α (Ω)) for 1 < p < ∞

mN → m strongly in Lp (0, T ; H β (Ω)) for 1 < p < ∞, 0 ≤ β ≤ α
∂mN
∂t

∂m
∂t

(1.24)

weakly in Lr (QT ) for 1 < r

Existence
With compactness result (1.24), it is easy to establish existence.
Theorem 1.5.3 (Chapter 5, section 5.4). Let 0 < α < 1 and m0 ∈ H α (Ω), then for any
0 < T < ∞, then there exists at least one global weak solution to fractional Landau-Lifshitz

15

equation (1.20) such that

m ∈ L∞ (0, T ; H α (Ω))

d
for 1 < r ≤ r∗ = d−α , where d is dimension.

16

r−1

C 0, r (0, T ; Lr (Ω))

Chapter 2
Existence for Reduced
Landau-Lifshitz Equations
In this chapter, we study the existence for reduced Landau-Lifshitz equations given by

∂t m = γm × Heff + α

γ
m × (m × Heff ),
|m|

(2.1)

where
Heff = −ϕ (m) + a(x) + Hm

(2.2)

with
curl Hm = 0,

div(Hm + mχΩ ) = 0 in R3 .

(2.3)

We establish the existence of global weak solution using two methods for different initial
data. First, we study the quasi-stationary limit of a simple Landau-Lifshitz-Maxwell system
with the permittivity parameter

approaching zero and, using this quasi-stationary limit,

establish the existence of global weak solutions to the reduced Landau-Lifshitz equations
with initial value m0 ∈ L∞ (Ω). We then give a different method for the existence when the
initial datum is more regular (e.g., in H 2 (Ω)) by considering the problem as an abstract
ordinary differential equation in an appropriate Banach space.

17

2.1

Quasi-stationary limit of Landau-Lifshitz-Maxwell
systems

We study the Cauchy problem for equations (2.1)-(2.3) with a given bounded initial datum
m(x, 0) = m0 (x). This Cauchy problem can be written as a quasi-stationary system:



∂ m = F (x, m, H ) in Ω × (0, ∞),
 t

m




3
curl Hm = 0, div(Hm + mχΩ ) = 0 in R ,






m(x, 0) = m0 (x) on Ω,


(2.4)

with a given initial datum m0 ∈ L∞ (Ω), where F : Ω × R3 × R3 → R3 is an appropriate
function (the Landau-Lifshitz interaction function), which can be written in a general form:

F (x, M, H) = F(M )H + a(x, M )

(2.5)

with certain specific functions F(M ) and a(x, M ); our results are valid for more general
interaction functions F (x, M, H) of this form satisfying some conditions to be specified later
(see (2.13)-(2.14) below).
Definition 2.1.1. Given m0 ∈ L∞ (Ω), a weak solution to the Cauchy problem (2.4) is a
function
m ∈ W 1,∞ ((0, ∞); L2 (Ω)) ∩ L∞ ((0, ∞); L∞ (Ω))

(2.6)

such that m(0) = m0 in L2 (Ω) and, with Hm defined by (2.3) with m = m(·, t), equation

∂t m = F (x, m, Hm ),
18

(2.7)

holds both in L∞ ((0, ∞); L2 (Ω)) and in the sense of distribution on Ω × [0, ∞).
Remark 1. We remark that, for the Landau-Lifshitz equation (2.1) with Heff given by (2.2),
since F (x, m, Hm ) · m = 0, the length |m(x, t)| is preserved for weak solutions; therefore, if
the initial datum m0 satisfies the saturation condition |m0 (x)| = 1 then any weak solution
m(x, t) to the Cauchy problem (2.4) will also satisfy the saturation condition: |m(x, t)| = 1
for all t > 0. In general, our assumptions on interaction function F will guarantee the
L∞ -norm of m(·, t) be non-increasing for t > 0 (see Section 2.1.1).
∂m

In the special case of Heff = Hm , for the regular initial data m0 ∈ H 2 (Ω) with ∂ν0 |∂Ω =
0, certain weak solution of (2.4) has been obtained in [8] as limit of the regular solution mκ
to the Landau-Lifshitz equation (2.1) with Heff = κ∆m + Hm when κ → 0+ . However, this
singular perturbation method does not work for rough initial data m0 in L∞ (Ω).
We study the Cauchy problem (2.4) as a quasi-stationary limit of the following LandauLifshitz-Maxwell system when the permittivity parameter

approaches zero:




 ∂t E − curl H = 0,








∂ (H + M χ ) + curl E = 0 in R3 × (0, ∞),
 t
Ω

(2.8)



∂ M = F (x, M, H) in Ω × (0, ∞),
 t







(E, H)|
3

t=0 = (E0 , H0 ) on R , M |t=0 = m0 on Ω,
where the initial data E0 , H0 for electric and magnetic fields E, H are any vector-fields
satisfying
E0 , H0 ∈ L2 (R3 ; R3 ),
For

div E0 = div(H0 + m0 χΩ ) = 0.

(2.9)

> 0, the existence of certain weak solutions to the Cauchy problem (2.8) can be
19

obtained in the same way as studied in [31] for = 1. More general Landau-Lifshitz-Maxwell
systems with variable dielectric permittivity (x) and magnetic permeability µ(x) have been
studied in [29], together with the quasi-stationary limit as the variable (x) → 0. Long-time
asymptotic problems for such systems have been addressed in [7, 30, 32].
One of the main purposes is to present a more direct way to study the asymptotic behavior
of weak solution (E , H , M ) to the Cauchy problem (2.8) as

→ 0+ . We prove that, for

all 0 < T < ∞, a subsequence M converges strongly to a function m in C 0 ([0, T ]; L2 (Ω))
and L2 (Ω × (0, T )) (see Theorem 2.1.3 below) and, furthermore, the limit m = m(x, t) is
a weak solution to Cauchy problem (2.4) (see Theorem 2.2.1 below). Our direct proof is
motivated by the methods of [31] with, however, a completely new parameter asymptotics,
and is much different in nature from the studies based on semi-group theory in [29].
The rest of section is organized as follows. We will first give the assumptions on the
interaction functions F (x, M, H) appearing in the general Cauchy problem (2.4) above and
present some preliminaries, including two useful compensated compactness results. Then,
we prove the strong convergence of M as

→ 0+ using a direct approach motivated by

the work [31], which is much different from the abstract semi-group techniques used in [29].
Finally, we prove the existence of weak solution to (2.4) directly from the strong convergence
theorem.

2.1.1

General assumptions and preliminaries

We assume Ω is a bounded domain in R3 ; the boundedness of Ω will simplify many technical
assumptions otherwise needed as in [29, 31].
Let F (x, M, H) = F(M )H + a(x, M ) be defined as in (2.5), where we assume F : R3 →

20

R3×3 and a : Ω × R3 → R3 satisfy the following condition (see also [29]):

(a) F(M )T M = 0,

a(x, M ) · M ≤ 0 ∀ M ∈ R3 , x ∈ Ω;

(b) a(x, 0) ∈ L2 (Ω);
(2.10)
|a(x, M ) − a(x, M )| + |F(M ) − F(M )| ≤ C1 (R) |M − M |
(c)
∀ R > 0, x ∈ Ω, |M |, |M | ≤ R, where C1 (R) is constant.
In particular, we have

F (x, M, H) · M ≤ 0 ∀ x ∈ Ω, M, H ∈ R3

(2.11)

|F (x, M, H)| ≤ C2 (R)(|H| + 1) + |a(x, 0)| ∀ |M | ≤ R

(2.12)

for all x ∈ Ω, H ∈ R3 , R ≥ 0, where C2 (R) is a constant depending on R.
Consider the Landau-Lifshitz function defined by L(0, H) = 0 and

L(M, H) = γM × H + α

γ
M × (M × H),
|M |

M = 0,

(2.13)

where α, γ are constants. Then the function F (x, M, H) appearing in (2.4) is given by
F (x, M, H) = F(M )H + a(x, M ) with

F(M )H = L(M, H),

a(x, M ) = L(M, a(x)) − L(M, ϕ (M )).

(2.14)

It is easily seen that this particular function F satisfies all assumptions in condition (2.10);
in fact, one has F (x, M, H)·M = 0 for all x ∈ Ω and M, H ∈ R3 . This property in particular
yields the equality in the length estimate (2.18) below.

21

The orthogonal decomposition L2 (R3 ; R3 ) = L2 (R3 ; R3 ) ⊕ L2 (R3 ; R3 ) is standard, where
⊥
L2 (R3 ; R3 ), L2 (R3 ; R3 ) are the subspaces of curl-free or divergence-free functions in the
⊥
sense of distributions, respectively. This decomposition can be explicitly given in terms of
ˆ
the Fourier transform f of f ∈ L2 (R3 ; R3 ): f = f + f⊥ , where

ˆ
ˆ
f = (ξ · f )ξ/|ξ|2 ,

ˆ
ˆ
ˆ
fˆ = f − (ξ · f )ξ/|ξ|2 = −ξ × (ξ × f )/|ξ|2 .
⊥

The projection operator P (f ) = f also extends to a bounded linear operator on Lp (R3 ; R3 )
for all 1 < p < ∞, with operator norm bounded by C0 p when p ≥ 2, where C0 is independent
of p ≥ 2; see [45].

2.1.2

Two Compensated Compactness Lemmas

We give two compensated compactness results in a suitable form to be used later; both are
special case of the more general results [20, 46, 47].
∞
Let b ∈ Cc (R3 ) and [b, P ] = bP −P b : L2 (R3 ; R3 ) → L2 (R3 ; R3 ) be the commutator.

We have the following special compactness result from the well-known div-curl lemma [46].
Lemma 2.1.1. Let f k

0 weakly in L2 (R3 ; R3 ) and Lp (R3 ; R3 ) for some p > 2. Then

g k = [b, P ]f k → 0 strongly in L2 (Ω; R3 ) for all bounded sets Ω.
Proof. From
k
g k = bf k − (bf k ) = (bf k )⊥ − bf⊥ ,
k
we have div g k = − b·f⊥ , curl g k =

b×f k in the sense of distributions on R3 , and so both

{div g k } and {curl g k } are compact in H −1 (R3 ). By the div-curl lemma, |g k |2 = g k · g k → 0
22

in the sense of distributions on R3 . Since {g k } is bounded in Lp (R3 ) with p > 2, we have
g k → 0 strongly in L2 (Ω) for all bounded sets Ω.
Let U be any domain in R3 and, for 0 < T ≤ ∞, let UT = U × (0, T ). Let c ∈ R be any
constant. Denote
˜
that ∆ =

−1

cu

= ∆u − cutt , where ∆ is the Laplacian with respect to x ∈ R3 . Note

is the Laplacian operator with respect to (x, t) ∈ R3 × R and − 1 is the

usual wave operator on R3 × R. The following lemma, with c = 0, will be used later; the
result is a special case of more general analyses on micro-local defect measures or H-measures
and orthogonality of sequences [20, 47].
Lemma 2.1.2. Let uk

u and v k

v weakly in L2 (UT ) as k → ∞. Suppose { c uk } is

compact in H −2 (UT ) and {∂t v k } is bounded in L2 (UT ). Then uk v k → uv in the sense of
distributions on UT .
∞
Proof. By considering uk − u and v k − v, we can assume u = v = 0. Given any ζ ∈ Cc (UT ),

we need to show
ζ(x, t)uk (x, t)v k (x, t) dxdt = 0.

lim

k→∞

(2.15)

UT

Let A, B be two regular domains in UT such that supp ζ ⊂⊂ A ⊂⊂ B ⊂⊂ UT . We solve the
following Dirichlet problem:


˜ k
∆w = ∆wk + wk = v k in B

tt

(2.16)


 k
w = 0 on ∂B.

1
By the standard elliptic theory, this problem has a unique solution wk ∈ H0 (B) ∩ H 2 (B),

which also satisfies the estimate wk H 2 (B) ≤ C1 v k L2 (B) . Hence {wk } is bounded in
H 2 (B). By the Rellich compactness theorem, there exists a subsequence, still denoted by
23

k, such that {wk } converges strongly in H 1 (B) as k → ∞. From (2.16), it follows that
k
˜ k
∆(wt ) = vt ∈ L2 (B) and hence by the (interior) L2 -estimates,

k
k
k
wt H 2 (A) ≤ C2 ( vt L2 (B) + wt L2 (B) ).

k
This shows that {wt } is bounded in H 2 (A) and hence for a subsequence, denoted again by
k
k
k, {wt } converges strongly in H 1 (A). This implies the strong convergence of {wtt } in L2 (A)

as k → ∞. Extending wk by zero onto UT , we write

ζuk v k dxdt =
UT

=
UT

UT

k
(1 + c)ζuk wtt dxdt +

k
ζuk (∆wk + wtt ) dxdt

UT

where the first term Ik → 0 as k → ∞ since uk

ζuk c wk dxdt := Ik + IIk ,
k
0 and {wtt } converges strongly in L2 (A).

The second term can be written as

IIk =

uk ( c (ζwk ) − wk c ζ − 2 ζ ·
UT

uk c (ζwk ) dxdt −

=
UT

k
wk + 2cζt wt ) dxdt

uk (wk c ζ + 2 ζ ·
UT

k
wk − 2cζt wt ) dxdt.

The second integral on the right-hand side approaches zero as k → ∞ because uk

0 and

{wk } converges strongly in H 1 (B). Since {ζwk } is bounded in H 2 (UT ), we can estimate the
first integral as follows:

uk

c (ζw

k ) dxdt

UT

24

=

k
k
c u , ζw

≤

cu

k

H −2 (UT )

ζwk H 2 (U ) → 0
T

2
as k → ∞, where ·, · stands for the duality pairing between H −2 (UT ) and H0 (UT ). This

completes the proof of (2.15). Finally, the lemma is proved.

2.1.3

Strong convergence of M

Let 0 < < 1. Let (E , H , M ) be the weak solution to the Cauchy problem (2.8) with the
1,∞

property (E , H ) ∈ C 0 ([0, ∞); L2 (R3 )) and M ∈ Wloc ([0, ∞); L2 (Ω))∩L∞ ([0, ∞); L∞ (Ω));
loc
the existence of such a weak solution has been established in [29, 31].

Estimates and weak convergence of M and H
Standard estimates using (2.9), (2.11) and (2.12) and Gronwall’s inequality, yield that, for
all t > 0,

div E = div(H + M χΩ ) = 0 in R3 ,
|M (x, t)| ≤ |m0 (x)| a.e. x ∈ Ω,

R3

(2.17)
(2.18)

( |E (t)|2 + |H (t)|2 )dx ≤ C3 + eC3 t

R3

( |E0 |2 + |H0 |2 )dx,

(2.19)

where C3 is a constant depending only on a(·, 0) L2 and m0 L∞ .
Since {M } is bounded in L∞ (Ω × (0, ∞)) and L∞ ((0, ∞); L2 (Ω)), there exist a function
m ∈ L∞ (Ω × (0, ∞)) ∩ L∞ ((0, ∞); L2 (Ω)) and a subsequence of

→ 0, still denoted by ,

such that
M

m weakly* in L∞ (Ω × (0, ∞)) ∩ L∞ ((0, ∞); L2 (Ω)).

(2.20)

Let 0 < T < ∞ and ΩT = Ω × (0, T ) and UT = R3 × (0, T ). By energy estimate (2.19),
25

third equation of (2.8) and condition (2.12), it follows that the sequences {

√

E }, {H }

are bounded in L∞ ((0, T ); L2 (R3 )) and L2 (UT ) and that the sequence {∂t M } is bounded
˜ ˜
˜
in L∞ ((0, T ); L2 (Ω)) and L2 (ΩT ). Let E, H and S be the limit of any weakly convergent
subsequence of {

√

E }, {H } and {∂t M } along the chosen → 0 in the respective Banach

˜
˜
spaces. Clearly S = ∂t m. Note that Et → 0 and H → H also in the sense of distributions
on UT . Hence, from the first equation of system (2.8), using (2.17), we have

˜
curl H(t) = 0,

˜
div(H(t) + m(t)χΩ ) = 0

˜
in the sense of distributions on R3 for almost every t ∈ (0, T ); therefore, H(t) = −P (m(t)χΩ )
˜
for almost every t ∈ (0, ∞). This implies that any weak limit H is uniquely determined by
m. This uniqueness also shows that the whole sequence {H } (along the chosen

→ 0)

converges weakly in both L∞ ((0, T ); L2 (R3 )) and L2 (UT ) for all 0 < T < ∞; the limit is
given by
H ∞ (t) = −P (m(t)χΩ ) = Hm(t) ,

(2.21)

where Hm(t) is defined by m(t) through (1.2) above. Moreover, since M ∈ W 1,∞ ((0, T ); L2 (Ω)) ⊂
C 0 ([0, T ]; L2 (Ω)), we have

m ∈ W 1,∞ ((0, T ); L2 (Ω)) ⊂ C 0 ([0, T ]; L2 (Ω)) ∀ 0 < T < ∞;

hence, we can also assume (2.21) holds for all t > 0.

Strong convergence of M
The main result of this section is the following.

26

(2.22)

Theorem 2.1.3. We have M → m strongly in both C 0 ([0, T ]; L2 (Ω)) and L2 (ΩT ) for all
0 < T < ∞.
A closer look at the proof of [29, Theorem 1.1] shows that our theorem follows from
that proof since the conductivity σ is zero here. However the proof in [29] involves other
complicated techniques aimed for handling the variable permittivity (x) and permeability
µ(x). We present a different and direct proof of this result in our setting, which is motivated
by the methods in [31].
The rest of this section is devoted to the proof of Theorem 2.1.3.

Weighted energy estimates
Assume T > 0 and , δ → 0 are the chosen subsequence. We want to estimate some weighted
norm of M (t) − M δ (t) in L2 (Ω) for 0 ≤ t ≤ T.
In what follows, we use Cj to denote various constants depending on m0 L∞ and
perhaps T or other quantities to be specified as needed. We write

∂t (M − M δ ) = F (x, M , H ) − F (x, M δ , H δ )
= F(M )H − F(M δ )H δ + a(x, M ) − a(x, M δ )
(2.23)
= (F(M

) − F(M δ ))H ∞

+ a(x, M

) − a(x, M δ )

+ F(M )(H − H ∞ ) − F(M δ )(H δ − H ∞ ).

27

Hence
1∂
|M − M δ |2 ≤ |(F(M ) − F(M δ ))H ∞ | · |M − M δ |
2 ∂t
+ |a(x, M ) − a(x, M δ )| · |M − M δ |
+ (F(M )(H − H ∞ ) − F(M δ )(H δ − H ∞ )) · (M − M δ )

(2.24)

≤C1 |M − M δ |2 (|H ∞ | + 1)
+ (F(M )(H − H ∞ ) − F(M δ )(H δ − H ∞ )) · (M − M δ ),
where C1 = C1 (R) is the constant in (2.10-c) with R = m0 L∞ . Let a(x, t) be the function
defined by
a(x, t) = |x|2 + C1

t

(1 + |H ∞ (x, s)|) ds (x ∈ R3 , t ≥ 0).

(2.25)

0

For all t ≥ 0, since e−a(·,t) ∈ L2 (R3 ) and M (·, t) ∈ L∞ (Ω), it follows that

e−a(t) M (t)

e−a(t) m(t) weakly in L2 (Ω) ∀ t ≥ 0.

(2.26)

Furthermore, by (2.24),
1 ∂ −2a
(e
|M − M δ |2 )
2 ∂t
≤

e−2a (F(M

)(H

(2.27)

− H ∞ ) − F(M δ )(H δ

28

− H ∞ )) · (M

− M δ ).

Integrating (2.27) with respect to x ∈ Ω and on time-interval (0, t), we get
1 −a(t)
e
(M (t) − M δ (t)) 2 2
L (Ω)
2
t

≤
0

Ω
t

e−2a F(M )(H − H ∞ ) · (M − M δ ) dxds
(2.28)

−
0

Ω

e−2a F(M δ )(H δ

− H ∞ ) · (M

− M δ ) dxds

:=f ,δ (t) − f δ,δ (t),
ρ

where functions f ,δ (t) (with ρ = , δ) are defined by
t

ρ

f ,δ (t) =

0

Ω

e−2a F(M ρ )(H ρ − H ∞ ) · (M − M δ )dxds.

(2.29)

ρ

To analyze f ,δ (t), we split it into two terms:
t

ρ

f ,δ (t) =

ρ

0

Ω
t

∞
e−2a F(M ρ )(H⊥ − H⊥ ) · (M − M δ )dxds
ρ

+
0

Ω

e−2a F(M ρ )(H − H ∞ ) · (M − M δ )dxds

(2.30)

ρ

ρ

:=g ,δ (t) + h ,δ (t).
ρ
˜
˜
˜
˜
By (2.17) and (2.21), H − H ∞ = −P (M ρ − m), where M ρ = M ρ χΩ and m = mχΩ . So,
ρ

the function h ,δ (t) in (2.30) can be rewritten as
t

ρ

h ,δ (t) = −

0

Ω
t

=−
0

Ω

t

−
0

Ω

˜
˜
e−2a F(M ρ )(P (M ρ − m)) · (M − M δ )dxds
˜
˜
F(M ρ )(P (e−a (M ρ − m))) · e−a (M − M δ )dxds
(2.31)

˜
˜
F(M ρ )([e−a , P ](M ρ − m)) · e−a (M − M δ )dxds

:= −I3 − I4 ,
29

where [e−a , P ]f = e−a f − (e−a f ) denotes the commutator operator. For I3 , we have:
t

|I3 | =

0
t

≤ C3

Ω

e−a (M ρ − m) 2 2
+ e−a (M − M δ ) 2 2
ds
L (Ω)
L (Ω)

0
t

≤ 4C3

˜
˜
F(M ρ )(P (e−a (M ρ − m))) · e−a (M − M δ )dxds

0

(2.32)

e−a (M − m) 2 2
+ e−a (M − M δ ) 2 2
ds.
L (Ω)
L (Ω)

For I4 , we have the following result.
Lemma 2.1.4. For any η > 0, there exists ξ = ξ(η, T, R) > 0 such that

|I4 | < η/4

∀ 0 < , δ < ξ,

∀ t ∈ [0, T ].

(2.33)

Proof. Note that
T

|I4 | ≤ C3

0

˜
˜
[e−a(s) , P ](M ρ − m) L2 (Ω) ds.

(2.34)

We claim

˜
˜
[e−a(s) , P ](M ρ (s) − m(s)) L2 (Ω) < ∞
0≤t≤T
˜
˜
lim [e−a(s) , P ](M ρ (s) − m(s)) L2 (Ω) = 0 ∀ s ∈ (0, T ).
sup

ρ→0

(2.35)
(2.36)

Once (2.35) and (2.36) are proved, by Lebesgue dominated convergence theorem, we have
T

lim

ρ→0 0

˜
˜
[e−a(s) , P ](M ρ − m) L2 (Ω) ds = 0

and hence (2.33) follows by (2.34). To prove (2.35), note that, by the boundedness of P on

30

L4 (R3 ),
˜
˜
[e−a , P ](M ρ − m) L2 (Ω) ≤ C0 e−a L4 (Ω) M ρ − m L4 (Ω) .
To prove the point-wise convergence (2.36), fix s ∈ [0, T ] and note that

˜
˜
[e−a(s) , P ](M ρ − m) L2 (Ω) ≤ I5 + I6

˜
˜
˜
˜
:= [e−a(s) − b, P ](M ρ − m) L2 (Ω) + [b, P ](M ρ − m) L2 (Ω) ,
∞
where b ∈ Cc (Ω). Again, by the boundedness of P on L4 (R3 ), we have

I5 ≤ C0 e−a(s) − b L4 (Ω) M ρ − m L4 (Ω) ≤ C4 e−a(s) − b L4 (Ω) .

∞
Choose b ∈ Cc (Ω) so that e−a(s) − b L4 (Ω) and hence I5 are arbitrarily small. Once b

˜
is chosen, since M ρ (s)

˜
m(s) weakly in Lp (R3 ) for all p ≥ 2, by Lemma 2.1.1, we have

I6 → 0 as ρ → 0. Hence (2.36) is proved.
Combining (2.28)–(2.33) above, we have, for all t ∈ [0, T ] and 0 < , δ < ξ, with ξ =
ξ(η, T, R) determined in Lemma 2.1.4,

e−a(t) (M (t) − M δ (t)) 2 2 ≤ η + 2(g ,δ (t) − g δ,δ (t))
L
t

+C5

e−a(s) (M

0

− M δ) 2 2
L

+

e−a(s) (M

− m)

2
L2

(2.37)
ds,

ρ

where g ,δ (t) is the function defined in (2.30) by
t

ρ

g ,δ (t) =

0

ρ

Ω

∞
e−2a F(M ρ )(H⊥ − H⊥ ) · (M − M δ )dxds.

31

(2.38)

From (2.37), using Gronwall’s inequality, we have
t

t

e−a(s) (M − M δ ) 2 2 ds ≤
L

0

0

+ T eT C5 η + C5

2e(t−s)C5 (g ,δ (s) − g ,δ (s)) ds
t
0

e−a(s) (M − m) 2 2 ds .
L

Plugging this inequality into the right-hand of (2.37), we have obtained the following result.
Proposition 2.1.5. For each η > 0, there exists ξ = ξ(η, T, R) > 0 such that, for all
0 < , δ < ξ and t ∈ [0, T ],

e−a(t) (M (t) − M δ (t)) 2 2 ≤ G ,δ (t) − Gδ,δ (t)
L
t

+ C6 η +

e−a(s) (M

− m)

0

(2.39)
2 ds
L2

,

where C6 = C6 (R, T ) is a constant depending only on R, T and, for ρ = , δ,

ρ

t

ρ

G ,δ (t) = 2g ,δ (t) +

0

ρ

2e(t−s)C5 g ,δ (s) ds.

(2.40)

ρ

Estimates of g ,δ (t)
ρ

∞
We now study the function g ,δ (t) defined by (2.38). Since curl H ∞ = 0 and so H⊥ = 0, we

have thus
t

ρ

g ,δ (t) =

0

ρ

Ω

e−2a(s) F(M ρ )(H⊥ ) · (M − M δ )dxds.

(2.41)

e−2a(s) F(M )(H⊥ ) · (M − m)dxds.

(2.42)

Let
t

g (t) =
0

Ω

We have the following result.

32

Proposition 2.1.6. There exists a constant C7 = C7 (R, T ) such that

|g ,δ (t)| + |g δ,δ (t)| ≤ C7
lim g δ,δ (t) = 0

δ→0

∀ 0 < < 1, t ∈ [0, T ]

lim g ,δ (t) = g (t)

δ→0

∀ 0 < , δ < 1, t ∈ [0, T ]

(2.43)
(2.44)

∀ 0 < < 1, t ∈ [0, T ]

(2.45)

lim g (t) = 0 ∀ t ∈ [0, T ].

(2.46)

→0

Proof. It is easy to see that
T

ρ

|g ,δ (t)| ≤ C8

H ρ (s) L2 (R3 ) ds ≤ C7

0

by (2.19).

The convergence (2.45) follows easily from the weak convergence M δ

M ∞ . The proofs of

(2.44) and (2.46) are similar; so we only give the proof of (2.44). To this end, we write

g δ,δ (t) =

t
0

Ω
t

=
0

Ω
t

=
0

Ω

δ
e−2a F(M δ )(H⊥ ) · (M − M δ ) dxds
δ
e−2a (F(M δ )T (M − M δ )) · H⊥ dxds

(2.47)

δ
e−2a (F(M δ )T M ) · H⊥ dxds.

Let Ωt = Ω × (0, t). Since e−2a ∈ L2 (Ωt ) for all 0 < t ≤ T, the limit (2.44) will be proved if
we show that
δ
(F(M δ )T M ) · H⊥

0 weakly in L2 (Ωt ) as δ → 0.

(2.48)

δ
Since the sequence {(F(M δ )T M ) · H⊥ } is bounded in L2 (Ωt ), to show its weak convergence
δ
to 0, we only need to show (F(M δ )T M )·H⊥ → 0 in the sense of distributions on Ωt as δ → 0.

We prove this by using Lemma 2.1.2 above. It is easy to see function g(m, n) = F(m)T n
33

restricted to the set BR = {(m, n) | |m| ≤ R, |n| ≤ R}, where R = m0 L∞ (Ω) , is Lipschitz
continuous with Lipschitz constant ≤ C9 (R). We can thus extend this function g(m, n) to
a Lipschitz function G(m, n) on whole (m, n) ∈ R3 × R3 with Lipschitz constant ≤ C9 (R).
δ
δ
Note that (F(M δ )T M ) · H⊥ = G(M , M δ ) · H⊥ . Hence

G(M , M δ ) L2 (Ω ) ≤ C10 < ∞.
t

(2.49)

Since G is Lipschitz and ∂s M and ∂s M δ both exist as integrable functions on Ωt , we have
∂s (G(M , M δ )) exists as an integrable function on Ωt and

|∂s (G(M , M δ ))| ≤ C9 (R) (|∂s M | + |∂s M δ |)
≤ C9 (R) (|F (x, M , H )| + |F (x, M δ , H δ )|)

and hence
∂s (G(M , M δ )) L2 (Ω ) ≤ C11 < ∞.
t

(2.50)

From first two equations of the Maxwell system (2.8) for δ > 0, with s as time-variable, we
deduce
δ
δ
δ
2 ˜δ
˜δ
∆(H⊥ ) = δ∂ss (M⊥ + H⊥ ) = δ(M⊥ + H⊥ )ss

(2.51)

˜
in the sense of distributions on R3 × (0, ∞), where again M δ = M δ χΩ . Since the sequences
δ
˜δ
{H⊥ } and {M⊥ } are bounded in L2 (Ωt ), we have

δ
0 H⊥

δ
Clearly, H⊥

δ
= ∆H⊥ → 0 strongly in H −2 (Ωt )

as δ → 0.

(2.52)

∞
H⊥ = 0 weakly in L2 (Ωt ) as δ → 0, by (2.50), (2.52) and Lemma 2.1.2

34

δ
above, we have that G(M , M δ ) · H⊥ → 0 in the sense of distributions on Ωt , as δ → 0;

hence, (2.48) is proved. This completes the proof of Proposition 2.1.6.

Proof of Theorem 2.1.3
We now complete the proof of Theorem 2.1.3. Taking the limit as δ → 0 in (2.39), using
(2.26), Fatou’s lemma and (2.43)–(2.45), we have

e−a(t) (M (t) − m(t)) 2 2
L (Ω)
t

≤ G (t)+C6 η +

e−a(s) (M

− m)

0

(2.53)
2
ds
L2 (Ω)

for all 0 < < ξ and all t ∈ [0, T ], where
t

G (t) = 2g (t) +

2e(t−s)C5 g (s) ds.

0

From (2.53), another use of Gronwall’s inequality yields
T
0

T

e−a(t) (M − m) 2 2 dt ≤ C12 η +
e(T −t)C6 G (t) dt
L (Ω)
0

(2.54)

for all 0 < < ξ. Since η > 0 is arbitrary, using (2.46), (2.40), we have
T

lim

→0 0

e−a(t) (M − m) 2 2 dt = 0.
L (Ω)

Hence the sequence {e−a M } converges to e−a m strongly in L2 (ΩT ) and L2 ((0, T ); L2 (Ω)).
Since a(x, t) is finite almost everywhere and e−a is not zero almost everywhere, for any
sequence of {M } we can extract a subsequence converging point-wise almost everywhere on

35

Ω; the limit must be m from the strong convergence of {e−a M }. By Lebesgue’s dominated
convergence theorem, we have that the whole sequence {M } converges to m strongly in
L2 (ΩT ). This and (2.53) imply the convergence: M (t) − m(t) L2 → 0 for all t ∈ [0, T ].
On the other hand, the third equation of system (2.8) and (2.12) imply that:

M (t1 ) − M (t2 ) L2 (Ω) ≤ C13 |t1 − t2 | ∀ t1 , t2 ∈ [0, T ].

Thus {M } is equi-continuous in C 0 ([0, T ]; L2 (Ω)). This, combined with the point-wise convergence of {M (t)} to m(t) for all t ∈ [0, T ], implies that M → m in C 0 ([0, T ]; L2 (Ω)).
The proof is now completed.

2.2

Existence for rough initial data

We finish proving the existence of solutions to the Cauchy problem (2.4).
Theorem 2.2.1. Let m0 ∈ L∞ (Ω) and let E0 , H0 be any functions satisfying (2.9). Then
any function m determined by the convergence (2.20) is a weak solution to the Cauchy
problem (2.4).
Proof. By (2.22), m ∈ W 1,∞ ((0, ∞); L2 (Ω)). By Theorem 2.1.3, M → m in C 0 ([0, T ]; L2 (Ω));
hence m(0) = m0 because M (0) = m0 . Also, by the strong convergence of M
m and weak convergence of H
F (x, M , H )

→

H ∞ in L2 (ΩT ), we have the weak convergence of

F (x, m, H ∞ ) in L2 (ΩT ) along the chosen sequence

→ 0. From the e-

quation ∂t M = F (x, M , H ), it follows that ∂t m = F (x, m, H ∞ ) in L∞ ((0, ∞); L2 (Ω))
and in the sense of distribution on Ω × (0, ∞). Finally, by (2.21), H ∞ = −P (mχΩ ) = Hm
and hence m is a weak solution to (2.4).

36

Chapter 3
Local L2-Stability of Solutions in
Finite Time
From Chapter 2, we have proved the existence of solutions with very rough initial value
condition. In this chapter, we will continue to prove the stability result for global weak
solutions. We also consider initial value problem (2.1)-(2.3) in Chapter 2 as a quasi-stationary
system:



∂ m = F (x, m, H ) in Ω × (0, ∞),
 t

a
m





3
curl Hm = 0, div(Hm + mχΩ ) = 0 in R for all t ∈ [0, ∞),






m(x, 0) = m0 (x) on Ω,


(3.1)

where Fa (x, m, H), specifying the dependence on applied field a, is the Landau-Lifshitz
interaction function given by

Fa (x, m, H) = L(m, −ϕ (m) + a(x) + H),

with L(m, n) linear in n and defined by

L(m, n) = γm × n + γαm × (m × n),

37

m, n ∈ R3 .

(3.2)

In this chapter, we aim to prove the stability of global weak solutions(see Definition in
Section 1.2) to quasi-stationary system (3.1); we first introduce the important Lemma about
how we split nonlocal term Hm , then show the main stability result. The last section will be
one of applications of stability theorem, which proves the existence of global weak solution
to reduced Landau-Lifshitz equation with a(x) ∈ L∞ (Ω) for rough initial data.

3.1
3.1.1

Stability of weak solutions
Decomposition of Hm

The following lemma aims to handle the nonlocal term Hm . Actually, the stability result
will be trivial if we have Hm ∈ L∞ (Ω). However, such bound is valid only when m ∈ H 2 (Ω)
(see Lemma 3.2.1). In our case, we only require initial value m ∈ L∞ (Ω); therefore, Hm
can not be bounded in L∞ . Following lemma enables us to split Hm into two parts: one is
bounded in L∞ ; the other is bounded in L2 (Ω) (also see [13, Lemma 5.2] and [31, Lemma
6.2]).
The projection operator used below is the same as Helmholtz Decomposition introduced in
Chapter 2, Section 2.1 and it is pretty easy to see that H = −P (mχΩ ) with this operator.
˜
Lemma 3.1.1. Let m ∈ L∞ (Ω; R3 ) and H = −P (mχΩ ) = −m . Then, for all λ ≥ e,
H = H λ + (H − H λ ) on R3 , where H λ is a function such that

H λ L∞ ≤ C ln λ,

H − H λ L2 ≤

with constant C = C0 m L∞ for an absolute constant C0 .

38

C|Ω|1/2
,
λ

(3.3)

Proof. Define H λ by
H λ (x) =




H(x) |H(x)| ≤ C ln λ



0


otherwise

˜
˜
Since H = −m , where m = mχΩ , we have, for all p ≥ 2,

H − Hλ 2 2 =
L

˜
|m |2 dx
˜
|m |>C ln λ
p−2

˜ L
˜
≤ m 2 p |{x : |m | > C ln λ}| p
1
˜ L
˜ p−2
≤ m 2p
m Lp
(C ln λ)p−2
1
˜ p
= m Lp
.
(C ln λ)p−2
However, the boundedness of P on Lp yields that, for all p ≥ 2,

˜
˜
m Lp ≤ C0 p m Lp ≤ C0 p m L∞ (Ω) |Ω|1/p ,

where C0 is an abstract constant independent of p for all p ≥ 2; see [45]. Hence
|Ω|(C1 p)p
H − Hλ 2 2 ≤
,
L
(C ln λ)p−2

where C1 = C0 m L∞ . We choose C = 4eC1 and p = 4 ln λ ≥ 4 to obtain
|Ω|(C1 p)p
(C ln λ)2
H − Hλ 2 2 ≤
= |Ω|
L
(C ln λ)p−2
λ4

and hence
H − H λ L2 ≤

C|Ω|1/2 ln λ
C|Ω|1/2
≤
,
λ
λ2
39

using ln λ ≤ λ for λ ≥ e. This proves (3.3).

3.1.2

Stability Theorem

We now prove our main stability result, Theorem 1.2.1, which generalizes our previous
result [13, Theorem 5.1] to the case of different applied fields a(x). A similar stability result
including the different anisotropy functions ϕ(m) can also be proved; for a similar result on
Maxwell systems, see [31, Theorem 6.1].
Assume mk (k = 1, 2) is any weak solution to the problem (3.1) with given applied field
ak and initial datum mk satisfying
0

ak L∞ + mk L∞ ≤ R for k = 1, 2,
0

(3.4)

where R > 0 is a given constant. Then, Theorem 1.2.1 will be proved once we prove the
following result.
Theorem 3.1.2. Given any 0 < T < ∞, there exist constants C = C(R, T ) > 0, c =
c(R, T ) > 0 and ρ = ρ(R, T ) > 0 such that, if µ = max{ m1 − m2 L2 , a1 − a2 L2 } ≤ c,
0
0
then one has, for all t ∈ [0, T ],

m1 (t) − m2 (t) L2 (Ω) ≤ C µρ .

(3.5)

Proof. Step 1. Let δm = m1 (t) − m2 (t) and δF = Fa1 (x, m1 , H1 ) − Fa2 (x, m2 , H2 ), where
Hk = Hmk for k = 1, 2. Then ∂t (δm) = δF and hence

∂t ( δm(t) L2 ) ≤ ∂t (δm(t)) L2 = δF (t) L2 .
40

So we have
t

δm(t) L2 − δm0 L2 ≤
0

δF (s) L2 ds.

(3.6)

Step 2. The function L(m, n) defined by (3.2) above can be written as

L(m, n) = B(m) · n,

(3.7)

where B(m) is a 3×3-matrix for each m ∈ R3 ; note that each element of B(m) is a quadratic
function of m. Given any mk , nk ∈ R3 (k = 1, 2), letting δm = m1 − m2 , δn = n1 − n2 , by
virtue of L(m1 , n1 ) − L(m2 , n2 ) = [L(m1 , n1 ) − L(m2 , n1 )] + L(m2 , n1 − n2 ), one can write

L(m1 , n1 ) − L(m2 , n2 ) = A(m1 , m2 , n1 ) · δm + B(m2 ) · δn,

(3.8)

where A(m1 , m2 , n1 ) is a matrix function given by

A(m1 , m2 , n1 ) =

1

∂L
(tm1 + (1 − t)m2 , n1 ) dt.
0 ∂t

(3.9)

Step 3. By Remark 1 in Section 2.1, it follows that mk (t) L∞ ≤ R (k = 1, 2) for all
t ≥ 0. From Fak (x, mk , Hk ) = −L(mk , ϕ (mk )) + L(mk , ak (x)) + L(mk , Hk ), by (3.4) and
(3.8), we obtain the following point-wise estimate for δF :

|δF | ≤ A|δH| + B(|H1 | + 1)|δm| + D|δa|,

(3.10)

where δH = H1 − H2 = Hδm , δa = a1 (x) − a2 (x), and A = A(R), B = B(R), D = D(R) are
constants depending only on R. We apply Lemma 3.1.1 to function H1 (t) = −P (m1 (t)χΩ ).
λ
λ
λ
For any λ ≥ e, let H1 = H1 + (H − H1 ), where H1 is given in Lemma 3.1.1 with constant

41

C = C0 m1 (t) L∞ ≤ C0 R. So, by (3.10), we have the L2 (Ω)-norm estimate:

δF L2 ≤A δH L2 + B(C ln λ + 1) δm L2
1

C|Ω| 2
+B
δm L∞ + D δa L2
λ
C
≤(A + B ln λ) δm L2 +
+ D δa L2 ,
λ

(3.11)

using δH L2 ≤ Hδm L2 (R3 ) ≤ δm L2 , where constants A , B , C depend on R.
Step 4. From (3.6) and (3.11), it follows that
t

δm(t) L2 − δm0 L2 ≤

δF (s) L2 ds

0
t

≤
0

C
(A + B ln λ) δm(s) L2 +
+ D a L2
λ

ds

t
Ct
δm(s) L2 ds.
+ δa L2 Dt + (A + B ln λ)
λ
0

=

From this, a Gronwall inequality yields

δm(t) L2 ≤
≤

Ct
δm0 L2 +
+ δa L2 Dt eA t+B t ln λ
λ
Ct
δm0 L2 + DT δa L2 +
λ

e A t λB t

(3.12)

∀ 0 ≤ t ≤ T.

Step 5. We consider two cases.
Case 1. Assume both δm0 = 0 and δa = 0. Then, by (3.12),

δm(t) L2 ≤ C teA t λB t−1 .

Let t0 =

1 .
B +1

(3.13)

If 0 ≤ t ≤ t0 , then B t − 1 < 0 and hence, by (3.13) with λ → ∞, we have
42

δm(t) = 0 for all t ∈ [0, t0 ]. With mk (t0 ) as initial datum at time t0 , we obtain δm(t) = 0
on [t0 , 2t0 ]; eventually, we have δm(t) = 0 for all t ≥ 0; hence (3.5) holds. This also shows
the uniqueness of the weak solution to the system.
Case 2. Assume 0 < δm0 L2 + DT δa L2 ≤ 1/e < 1. In this case, setting λ =
( δm0 L2 + DT δa L2 )−1 ≥ e in (3.12), we obtain

δm(t) L2 ≤ (1 + C t)eA t ( δm0 L2 + DT δa L2 )1−B t .

Let t1 =

1
2(B +1)

1
and C1 = (1 + C t1 )eA t1 > 1. Then 1 − B t ≥ 2 for all 0 ≤ t ≤ t1 ; hence

1

δm(t) L2 ≤ C1 ( δm0 L2 + DT δa L2 ) 2

∀ 0 ≤ t ≤ t1 .

Adding DT δa L2 to both sides, we obtain
1

δm(t) L2 + DT δa L2 ≤ C2 ( δm0 L2 + DT δa L2 ) 2

∀ 0 ≤ t ≤ t1 ,

(3.14)

where C2 = C1 + 1 depends only on R.
Step 6. Combining Cases 1 and 2 in Step 5 above, with the constants t1 = t1 (R) and
C2 = C2 (R) > 1 determined above, we have that, if δm0 L2 + DT δa L2 ≤ 1/e, then
1

δm(t) L2 + DT δa L2 ≤ C2 ( δm0 L2 + DT δa L2 ) 2

∀ 0 ≤ t ≤ t1 .

(3.15)

Assume
1

C2 ( δm0 L2 + DT δa L2 ) 2 ≤ 1/e.

(3.16)

Then, by (3.15), δm(t1 ) L2 + DT δa L2 ≤ 1/e. With mk (t1 ) as initial datum at time t1 ,
43

we apply (3.15) again to obtain
1

δm(t1 + t) L2 + DT δa L2 ≤C2 ( δm(t1 ) L2 + DT δa L2 ) 2
1+ 1
≤C2 2 (

1

∀ 0 ≤ t ≤ t1 .

δm0 L2 + DT δa L2 ) 4

We have thus proved that, if (3.16) holds then
1

1
1+
δm(t) L2 + DT δa L2 ≤ C2 2 ( δm0 L2 + DT δa L2 ) 4

∀ 0 ≤ t ≤ 2t1 .

By induction, we obtain that, for k = 1, 2, · · · , if
1
1+ 2 +···+

C2

1
2k−1

1

( δm0 L2 + DT δa L2 ) 2k ≤ 1/e,

(3.17)

then

δm(t) L2 + DT δa L2 ≤

1
1+ 2 +···+ 1
2k (
C2

1
2k+1
)
L2

δm0 L2 + DT δa
1

2
≤ C2 ( δm0 L2 + DT δa L2 ) 2k+1

∀ 0 ≤ t ≤ 2k t1 .(3.18)

Step 7. In this step, we complete the proof of the theorem. Let k be the integer such
that 2k−1 t1 < T ≤ 2k t1 . Define

ρ = ρ(R, T ) = 1/(2k+1 ),

k

2
c = c(R, T ) = (C2 e)−2 /(1 + DT ).

Assume µ = max{ δm0 L2 , δa L2 } ≤ c. Then
k

2
δm0 L2 + DT δa L2 ≤ (1 + DT )µ ≤ (C2 e)−2 ,

44

from which it is easily seen that (3.17) holds; so, by (3.18),

2
δm(t) L2 + DT δa L2 ≤ C2 ( δm0 L2 + DT δa L2 )ρ

∀ 0 ≤ t ≤ T.

Therefore,
2
δm(t) L2 ≤ C2 (1 + DT )ρ µρ

∀ t ∈ [0, T ];

2
this proves (3.5) with constant C = C2 (1 + DT )ρ .

3.2

Existence of global weak solutions

In this section, we present a proof for the existence of global weak solution to (3.1) based on
the stability theorem proved above. To this end, we introduce a nonlinear function

f (m) = Fa (x, m, Hm ) = −L(m, ϕ (m)) + L(m, a(x)) + L(m, Hm )

(3.19)

for m ∈ L∞ (Ω; R3 ), where Hm is defined by Maxwell equation in (3.1) and L is defined by
(3.2). As before, we always assume the anisotropy function ϕ : R3 → R3 is smooth.

3.2.1

Properties of f for smooth applied fields

¯
In this subsection, we assume the applied field a belongs to C ∞ (Ω; R3 ) and show that, in
this case, f : H 2 (Ω; R3 ) → H 2 (Ω; R3 ) and is locally Lipschitz. We need some estimates.
Lemma 3.2.1. Let Ω ⊂ R3 be a bounded domain with smooth boundary. Then the following

45

estimates hold on H 2 (Ω; R3 ) :

m L∞ (Ω) + m W 1,p (Ω) ≤ C0 m H 2 (Ω)

∀ 1 ≤ p ≤ 6,

Hm H 2 (Ω) ≤ C1 m H 2 (Ω) .

(3.20)
(3.21)

Proof. We omit the proof, but only mention that (3.20) is a simple consequence of the well1

¯
known embeddings: H 2 (Ω) ⊂ W 1,6 (Ω) ⊂ C 2 (Ω) ⊂ L∞ (Ω) for bounded smooth domain
Ω ⊂ R3 , and that estimate (3.21) has been, e.g., proved in [8]. Finally, we remark that, from
(3.20) and (3.21), it follows that, with constant C2 = C0 C1 ,

Hm L∞ (Ω) ≤ C2 m H 2 (Ω)

∀ m ∈ H 2 (Ω; R3 ).

(3.22)

The main result of the subsection is the following local Lipschitz property of f on
H 2 (Ω; R3 ).
Proposition 3.2.2. Function f maps space H 2 (Ω; R3 ) into itself and is locally Lipschitz on
H 2 (Ω; R3 ).
Proof. Since f (0) = 0, the self-mapping property of f will follow from the local Lipschitz
property of f on H 2 (Ω; R3 ).
To prove the local Lipschitz property of f , given any two functions m1 , m2 ∈ H 2 (Ω; R3 )
satisfying
max{ m1 H 2 (Ω) , m2 H 2 (Ω) } ≤ R,

46

(3.23)

where R < ∞ is a constant, we need to show that

f (m1 ) − f (m2 ) H 2 (Ω) ≤ L m1 − m2 H 2 (Ω)

(3.24)

for a (local Lipschitz) constant L = L(R) < ∞ depending on R.
By (3.19), we write f (m1 ) − f (m2 ) = I1 + I2 , where

I1 = L(m1 , a − ϕ (m1 )) − L(m2 , a − ϕ (m2 ))

and I2 = L(m1 , Hm1 ) − L(m2 , Hm2 ). Let δm = m1 − m2 . Then, by (3.8),

I2 = A(m1 , m2 , Hm1 ) · δm + B(m2 ) · Hδm ,

(3.25)

where A, B are functions defined in Step 2 of the proof of Theorem 3.1.2 above. We also
write I1 as
1

I1 =

d
L(m2 + tδm, a − ϕ (m2 + tδm)) dt = C(m1 , m2 , a) · δm,
dt
0

(3.26)

where C(m1 , m2 , a) is certain smooth function of (m1 , m2 , a) ∈ R3 × R3 × R3 . Note that
C is linear in a. We aim to show

Ik H 2 (Ω) ≤ L(R) δm H 2 (Ω)

(k = 1, 2)

for some constant L(R) depending on R. By (3.23), (3.22) and Lemma 3.2.1, it follows that,

47

for k = 1, 2,

mk L∞ (Ω) + Hmk L∞ (Ω) + Hmk H 2 (Ω) +

mk L4 (Ω) ≤ C3 R.

(3.27)

We proceed with several steps.
Step 1: Estimation of I1 . Clearly, by (3.26) and (3.27),

I1 L2 (Ω) ≤ C(m1 , m2 , a) L∞ δm L2 ≤ L(R) δm L2 (Ω) .

We estimate the H 2 -norm. Denote by ∂j the first partial derivative with respect to xj and
2
by ∂ij the second partial derivative with respect to xj and xi (i, j = 1, 2, 3). Note that

∂j (I1 ) = ∂j (C(m1 , m2 , a)) · δm + C(m1 , m2 , a) · (δm)xj

and
2
2
∂ij (I1 ) = ∂ij (C(m1 , m2 , a)) · δm + ∂j (C(m1 , m2 , a)) · (δm)xi

+ ∂i (C(m1 , m2 , a)) · (δm)xj + C(m1 , m2 , a) · (δm)xi xj .
Since ∂j (C(m1 , m2 , a)) = (∂m1 C) · m1 + (∂m2 C) · m2 + (∂a C) · axj has L2 -norm controlled
x
x
j

j

by R, we have

∂j (I1 ) L2 ≤ ∂j (C(m1 , m2 , a)) L2 δm L∞ + C(m1 , m2 , a) L∞ (δm)xj L2

≤ L(R) δm H 2 (Ω) .

48

2
Similarly, ∂ij (C(m1 , m2 , a)) contains terms up to second derivatives of a and terms like
q

2
(∂mp mq C) · mk · ml and (∂mp C) · mx x , with certain choices of p, q, k, l ∈ {1, 2} and
xi
xj
i j
2
i , j ∈ {i, j}. Hence ∂ij (C(m1 , m2 , a)) L2 is bounded by the quantity

C(R)

| m1 |2 L2 + | m2 |2 L2 +

which, due to | m|2 L2 =

2 m1

L2

+

2 m2

L2

+ a H2 ,

m 2 4 ≤ C m 2 2 , is in fact bounded by another constant
H

L

2
C(R). From this, similar to the term ∂j (I1 ), the L2 -norm of the first or fourth term of ∂ij (I1 )
2
is bounded by L(R) δm H 2 (Ω) . The second and third terms of ∂ij (I1 ) can be estimated as

follows:

∂j (C(m1 , m2 , a)) · (δm)xi + ∂i (C(m1 , m2 , a)) · (δm)xj L2
≤2

(C(m1 , m2 , a)) L4 ·

≤ C(R) ( m1 W 1,4 + m2 W 1,4 +

(δm) L4
a L4 ) · δm W 1,4

≤ L(R) δm H 2 (Ω) .

This proves I1 H 2 (Ω) ≤ L(R) δm H 2 (Ω) .
Step 2: Estimation of I2 . We write I2 = I21 + I22 with

I21 = A(m1 , m2 , Hm1 ) · δm,

I22 = B(m2 ) · Hδm .

The term I21 is more like term I1 , except the constant field a is replaced by the fields Hm1 .
Since Hm1 ∈ H 2 (Ω) and Hm1 L∞ + Hm1 H 2 (Ω) ≤ C1 m1 H 2 (Ω) ≤ C1 R, estimation
resulting from Hm1 in A can be handled in a much similar way as the term a in C of I1 .

49

The term I22 is simpler but slightly different than I1 in that Hδm is in place of δm.
Nevertheless this term can also be estimated in a similar fashion as I1 , using the following
estimate on Hδm :

Hδm L∞ (Ω) +

Hδm L4 (Ω) + Hδm H 2 (Ω) ≤ C5 δm H 2 (Ω) .

We eventually obtain I2 H 2 (Ω) ≤ L(R) m H 2 (Ω) . This completes the proof.

3.2.2

Existence of global solution for smooth data

¯
We continue to assume a ∈ C ∞ (Ω; R3 ) in this subsection. Let X = H 2 (Ω; R3 ). With
function f : X → X defined above, we formulate the problem (1.7) as an abstract ODE on
X by


 dm

 dt = f (m),

(3.28)



m(0) = m .

0
A solution m to (3.28) on [0, T ] is a function m ∈ C([0, T ]; X) that satisfies
t

m(t) = m0 +

f (m(s)) ds ∀ 0 ≤ t ≤ T.
0

We say m is a solution to (3.28) on [0, T ) if m is a solution on [0, T ] for all 0 < T < T (in
this case T could be ∞).
Theorem 3.2.3. Given any m0 ∈ X, (3.28) has a unique solution m on [0, ∞). This
solution is also a global weak solution to problem (3.1).
Proof. Given m0 ∈ X, since f is locally Lipschitz on X, from the abstract theory, there

50

exists T > 0 such that (3.28) has a unique solution m on [0, T ]. Let

T∗ = sup T > 0

(3.28) has a unique solution on [0, T ] .

We claim that T∗ = ∞, which implies that (3.28) has a unique global solution m defined
on [0, ∞). Clearly, this solution is also a global weak solution to the Cauchy problem (1.7)
above.
Suppose T∗ < ∞. Then, by the elementary ODE theory, a solution m to (3.28) would
exist on [0, T∗ ) and satisfy
lim

−
t→T∗

m(t) X = ∞.

The following asserts that this finite-time blowup is impossible; this completes the proof of
Theorem 3.2.3.
Theorem 3.2.4. Given any T > 0, if m is a solution to (3.28) on [0, T ), then

sup
t∈[0,T )

< ∞.
m(t) X ≤ CT, m
0 X

(3.29)

The proof of this theorem involves lots of technical estimates and will be delayed to the
next individual section.

3.2.3

Existence of global weak solution for rough data

In this subsection, we assume both applied field a and initial datum m0 are in L∞ (Ω; R3 ).

51

¯
Let a , m0 ∈ C ∞ (Ω; R3 ) be such that

a L∞ + m0 L∞ ≤ R ∀

> 0,

lim ( a − a L2 + m0 − m0 L2 ) = 0,

→0+

a → a, m0 → m0

pointwise in Ω.

(3.30)
(3.31)
(3.32)

Consider the Cauchy problem (3.1) with applied field a and initial datum m0 . Then, by
Theorem 3.2.3, for each > 0, (3.1) has a global weak solution m . Since m · f (m ) = 0, it
follows that ∂t (|m (x, t)|2 ) = 0 and hence |m (x, t)| = |m0 (x)| for a.e. x ∈ Ω and all t > 0.
This implies
m (t) L∞ = m0 L∞ ≤ R.

(3.33)

For each n ∈ {1, 2, 3, · · · }, our stability result (Theorems 1.2.1 and 3.1.2) implies that
sequence {m } is Cauchy in Banach space C([0, n]; L2 (Ω; R3 )) as → 0+ . Therefore, m →
m in C([0, n]; L2 (Ω; R3 )) as

→ 0+ for some m ∈ C([0, n]; L2 (Ω; R3 )). (Presumably, m =

mn depends on n.) Hence, by (3.31),

m(0) = m0 .

(3.34)

We also have Hm → Hm in C([0, n]; L2 (Ω; R3 )). It follows that m → m and Hm → Hm
also in L2 (Ω × (0, n)) as → 0+ . Using a subsequence, we can assume

m (x, t) → m(x, t),

Hm (x, t) → Hm (x, t) pointwise in Ω × (0, n).

Therefore, Fa (x, m , Hm ) → Fa (x, m, Hm ) pointwise in Ω × (0, n). This shows ∂t m =

52

Fa (x, m, Hm ) in the sense of distribution on Ω × (0, n).
Note also that Fa (x, m , Hm ) ∈ L2 (Ω; R3 ) uniformly on

and t ∈ (0, n); this implies

that ∂t m = Fa (x, m, Hm ) holds in L∞ ((0, n); L2 (Ω)) and that m ∈ W 1,∞ ([0, n); L2 (Ω; R3 )).
Combining with (3.34), we have proved that m = mn is a weak solution to (3.1) on Ω×(0, n).
By the uniqueness of weak solutions, we have mn+1 = mn on Ω × (0, n); therefore, the
sequence {mn }∞ defines a unique function m by setting m(x, t) = mn (x, t) with n = [t] + 1.
1
It is easy to see that m is a global weak solution to (3.1).
Finally, we have proved the following theorem.
Theorem 3.2.5. Let a ∈ L∞ (Ω; R3 ). Given any initial datum m0 ∈ L∞ (Ω; R3 ), the problem (3.1) has a unique global weak solution.

3.2.4

Proof of Theorem 3.2.4

In this separate section, we give the proof of Theorem 3.2.4. This involves the special form
of function L(m, n) and several estimates.
¯
In what follows, assume a ∈ C ∞ (Ω; R3 ), 0 < T < ∞ and m is a solution to (3.28) on
[0, T ) with initial datum m0 ∈ H 2 (Ω; R3 ). Assume

m0 L∞ (Ω) = R > 0.

Then, similar to (3.33) above, we have

m(t) L∞ = m0 L∞ = R,

1

m(t) L2 = m0 L2 ≤ R|Ω| 2

53

∀ 0 ≤ t < T.

(3.35)

We would like to show

sup
t∈[0,T )

m(t) H 2 (Ω) ≤ CT, m
< ∞.
0 H2

(3.36)

Let
y(t) = 1 + m(t) 2 2
= 1 + m(t) 2 2 +
H (Ω)
L

m(t) 2 2 +
L

2 m(t) 2 .
L2

The goal is to show
y (t) ≤ Cy(t)(1 + ln y(t)) ∀ 0 < t < T,

(3.37)

where C = C(R) is a constant depending on R. Once (3.37) is proved, one easily obtains
that
ln(y(t)) ≤ (ln(y(0)) + 1) eCT < ∞ ∀ t ∈ [0, T ),
from which (3.36) follows.
The rest of the section is devoted to proving (3.37).

Energy estimates
It is convenient to use the special structure of function L to write function f (m) as follows:

f (m) = B(m) · a − B(m) · ϕ (m) + B(m) · Hm ,

where B(m) is a 3 × 3 matrix defined in (3.7) above, whose elements are quadratic functions
of m; hence B (m) = D is a constant tensor. However, this special structure of B is not
used; in fact, the following arguments are valid for arbitrary smooth functions B.

54

Differentiating equation in (3.28) with respect to xi yields
dmxi
= B (m) · mxi · a + B(m) · axi
dt
− B (m) · mxi · ϕ (m) − B(m) · ϕ (m) · mxi

(3.38)

+ B (m) · mxi · Hm + B(m) · (Hm )xi .
Again, differentiating equation (3.38) with respect to xj yields
dmxi xj
dt

= D · mxj · mxi · a + B · mxi xj · a + B · mxi · axj
+ B · mxj · axi + B · axi xj
− D · mxj · mxi · ϕ − B · mxi xj · ϕ + B · mxi · ϕ · mxj

(3.39)

− B · mxj · ϕ · mxi − B · ϕ · mxj · mxi − B · ϕ · mxi xj
+ D · mxj · mxi · Hm + B · mxi xj · Hm + B · mxi · (Hm )xj
+ B · mxj · (Hm )xi + B · (Hm )xi xj .
Dot-product of (3.38) with mxi and of (3.39) with mxi xj yields the following identities:
1d
2 dt
1d
2 dt

mxi 2 2
L

=

mxi xj 2 2
L

=

Ω

Ω

dmxi
dx,
dt
dmxi xj
dx.
mxi xj ·
dt

mxi ·

(3.40)
(3.41)

The energy estimates involve estimating the right-hand sides of (3.40) and (3.41) with terms
dmxi dmxi xj
dt ,
dt

given by the right-hand sides of (3.38) and (3.39).

55

More subtle inequalities
To handle the terms involved in the integrals on the right-hand sides of (3.40) and (3.41),
more subtle inequalities are needed.
Lemma 3.2.6. Let Ω ⊂ R3 be a bounded domain with smooth boundary. Then
1

1

2
n L4 ≤ C6 n L∞ n 2 2 ,
H

Hn L∞ (Ω) ≤ C n ∞
L

∀ n ∈ H 2 (Ω; R3 ),

(1 + ln+ (

(3.42)

n H 2 )),

where ln+ t = max{ln t, 0} for t > 0 and C n ∞ < ∞ depends on n L∞ (Ω) .
L
Proof. The first inequality of (3.42) is a consequence of the well-known Gagliardo-Nirenberg
inequality:
jf

Lq (Rn )

≤ C f 1−θ n )
Lr (R

lf θ
Lp (Rn ) ,

where θ = j/l ∈ (0, 1) and 1/q = θ/p + (1 − θ)/r, 1 ≤ p, r ≤ ∞. Here j = 1, l = 2, p = 2, q =
4, r = ∞ and θ = 1/2. While the second inequality of (3.42) is a Judovic-type inequality
proved, e.g., in [31, Lemma 7.2].
The following result is an immediate consequence of this lemma and (3.35).
Proposition 3.2.7. For the solution m(t), with y(t) defined above, it follows that

m(t) 4 4
≤ C7 y(t),
L (Ω)
Hm (t) L∞ (Ω) ≤ C8 (1 + ln y(t)),
where C7 , C8 are constants depending on R = m0 L∞ .

56

∀ 0 ≤ t < T,

(3.43)

Energy estimates (continued) and proof of (3.37)
First of all, the integral on right-hand side of (3.40) is bounded by

| m|2 + | m| + | m|2 |Hm | + | m|| (Hm )| dx.

C(R)
Ω

The third term is bounded by C(R) Hm L∞ (Ω)

m 2 2 and hence, by (3.43b), is bounded
L

by C(R)y(t)(1 + ln y(t)), while all the other terms are bounded by C(R) m 2 2 and hence
H
by C(R)y(t). Therefore,

d
dt

mxi 2 2 ≤ C(R)y(t)(1 + ln y(t)),
L

∀ 0 < t < T.

(3.44)

Similarly, the integrand of the right-hand side of (3.41) is bounded by a constant C(R)
times
| m|2 | 2 m| + | 2 m|2 + | m|| 2 m| + | 2 m| + | 2 m|| 2 (Hm )|
+| m|2 |Hm || 2 m| + |Hm || 2 m|2 + | m|| 2 m|| (Hm )|.
Integrals of terms in the first group can all be bounded by Cy(t). Integrals of the first two
terms in the second group can be bounded by constant times

Hm L∞ (Ω) (

m 44 +
L

2 m 2 ),
L2

which, by (3.43a-b), is bounded by Cy(t)(1 + ln y(t)). Finally, the integral of the last term
in the second group can be estimated as follows:

| m|| 2 m|| (Hm )|dx ≤ | m| · | (Hm )| L2 (Ω) | 2 m| L2 (Ω)
Ω
57

≤

(Hm ) L4 (Ω) | 2 m| L2 (Ω) ,

m L4 (Ω)

which, by using Lemma 3.2.6, is bounded by
1

1

1

2
≤C m 22
Hm L∞ (Ω) Hm 2 2
m H 2 (Ω)
H (Ω)
H (Ω)
1

1

1

1

2
2
≤C m 22
Hm L∞ (Ω) m 2 2
m H 2 (Ω) = C m 2 2
Hm L∞ (Ω)
H (Ω)
H (Ω)
H (Ω)
1

≤ C y(t) · (1 + ln y(t)) 2 ≤ C y(t)(1 + ln y(t)).
Therefore, by (3.41), we have obtained that

d
dt

mxi xj 2 2 ≤ C(R)y(t)(1 + ln y(t)),
L

∀ 0 < t < T.

(3.45)

Summing up i, j = 1, 2, 3 in (3.44) and (3.45) and using (3.33), we obtain (3.37).
Remark 2. By the local Lipschitz property of f (m), from (3.36), one easily obtains

sup
t∈[0,T )

mt H 2 (Ω) ≤ CT, m
< ∞.
0 H2

In next chapter, we prove higher time-regularity for solutions.

58

(3.46)

Chapter 4
Higher Time Regularity and Special
Asymptotics

4.1

Higher time regularity

The higher time regularity has been studied for Landau-Lifshitz equation with exchange
energy by Cimrak and Keer [6]. We study a higher time-regularity of weak solutions for
simple Landau-Lifshitz equation



m = γm × H + γαm × (m × H ) in Ω × (0, ∞),
 t
m
m

(4.1)



m(0) = m ,

0
where Ω is a bounded smooth domain in R3 and m0 ∈ H 2 (Ω; R3 ).
Theorem 4.1.1. For any time T > 0, the solution m to (4.1) satisfies, for p = 0, 1, 2, · · · ,

sup
t∈[0,T ]

p+1

∂t

m H 2 (Ω) ≤ C < ∞,

(4.2)

where C is constant depending on T, p and m0 H 2 (Ω) .
Proof. We use induction on p. The case for p = 0 is already mentioned in Remark 2 in
Section 3.2.4. Let us assume (4.2) holds for all powers up to p − 1. We consider the case for
59

i
p. Note that ∂t (Hm ) = H∂ i m and hence, by (3.22),
t

i
H∂ i m H 2 (Ω) ≤ C ∂t m H 2 (Ω) .
t

Therefore, by the induction assumption, it follows that, for all t ∈ [0, T ],

i
i
∂t (Hm ) H 2 (Ω) ≤ C ∂t m H 2 (Ω) ≤ CT,p, m
< ∞ ∀ 0 ≤ i ≤ p.
0 H2

(4.3)

Taking pth -derivatives with respect to t to equation (4.1) yields

p+1

∂t

j

i+j=p
p+1

We need to prove ∂t

j

i
∂t m × ∂t Hm + γα

m=γ

i
k
∂t m × (∂t m × ∂t Hm ).

(4.4)

i+j+k=p

m H 2 (Ω) ≤ CT,p, m
< ∞.
0 H2

Estimation of ∂tp+1 m

L2 (Ω)

Since, ∀ 0 ≤ i ≤ p,

i
i
∂t (Hm ) L∞ (Ω) ≤ C ∂t (Hm ) H 2 (Ω) ≤ CT,p, m
< ∞,
0 H2

the L2 -norm of each term on the right-hand side of (4.4) can be bounded by the L∞ -norms
of its factors, which are in turn bounded by constant CT,p, m
. Hence we have
0 H2
p+1

∂t

m L2 (Ω) ≤ CT,p, m
.
0 H2

60

(4.5)

Estimation of ∂tp+1 m

L2 (Ω)

Taking ∂l = ∂xl on equation (4.1) yields

∂l mt = γ∂l (m × Hm ) + γα∂l (m × (m × Hm ))
= γmxl × Hm + γm × (Hm )xl + γαm × (mxl × Hm )

(4.6)

+ γαm × (m × (Hm )xl ) + γαmxl × (m × Hm )
Taking pth derivative with respect to t on Eq. (4.6) yields

p+1

∂t

j

j

i
∂t m × ∂t (Hm )xl

i
∂t mxl × ∂t Hm + γ

mxl =γ

i+j=p

i+j=p

j

j

i
k
∂t m × (∂t mxl × ∂t Hm )

i
k
∂t m × (∂t m × ∂t (Hm )xl ) + γα

γα

i+j+k=p

i+j+k=p
j

i
k
∂t mxl × (∂t m × ∂t Hm )

+ γα
i+j+k=p
p+1

In order to estimate ∂t

mxl L2 (Ω) , it is sufficient to estimate the following L2 -norms.

j

i
∂t mxl × ∂t Hm L2 (Ω) .
i+j=p
j

i
∂t m × ∂t (Hm )xl L2 (Ω) .
i+j=p
j

k
i
∂t m × (∂t m × ∂t (Hm )xl ) L2 (Ω) .
i+j+k=p
j

i
k
∂t m × (∂t mxl × ∂t Hm ) L2 (Ω) .
i+j+k=p
j

i
k
∂t mxl × (∂t m × ∂t Hm ) L2 (Ω) .
i+j+k=p

61

All these norms can be estimated in the same way: For each of the individual cross-product
integrands, use the L2 -norm of a sole factor with xl -derivative and use the L∞ -norms for
the other factor or factors. All these norms can be bounded by constant CT,p, m
< ∞.
0 H2
Finally, summing up l = 1, 2, 3, we have proved

p+1

m L2 (Ω) ≤ CT,p, m
< ∞.
0 H2

∂t

Estimation of ∂tp+1 m

(4.7)

L2 (Ω)

Differentiating (4.6) with respect to xl and summing up over l = 1, 2, 3 yields that

mt = γ m × Hm + γm ×

mxl × (Hm )xl

Hm + γ
l

+ γα[ m × (m × Hm ) + m × ( m × Hm ) + m × (m ×

(4.8)

Hm )]

[mxl × (mxl × Hm ) + mxl × (m × (Hm )xl ) + m × (mxl × (Hm )xl )].

+ γα
l

p+1

Differentiating equation (4.8) p times with respect to t will yield a formula for ∂t
p+1

To estimate ∂t

m.

m L2 (Ω) , we do not need to estimate every single term because lots of

them are similar; it is sufficient to estimate the following 4 L2 -norms:

i
∂t m × ∂t Hm L2 (Ω) .

j

(4.9)

j

(4.10)

i+j=p
i
∂t mxl × ∂t (Hm )xl L2 (Ω) .
i+j=p
j

(4.11)

j

(4.12)

i
k
∂t m × (∂t m × ∂t Hm ) L2 (Ω) .
i+j+k=p
i
k
∂t mxl × (∂t mxl × ∂t Hm ) L2 (Ω) .
i+j+k=p

62

For (4.9), we use

j

j

i
i
∂t m × ∂t Hm L2 (Ω) ≤ ∂t Hm L∞ ∂t m L2 (Ω) .

For (4.10), we use

j

j

i
i
∂t mxl × ∂t (Hm )xl L2 (Ω) ≤ ∂t m L4 (Ω) ∂t

Hm L4 (Ω) .

For (4.11), we use

j

j

i
k
k
i
∂t m × (∂t m × ∂t Hm ) L2 (Ω) ≤ ∂t m L∞ ∂t Hm L∞ ∂t m L2 (Ω) .

For (4.12), we use

j

j

i
k
k
i
∂t mxl × (∂t mxl × ∂t Hm ) L2 ≤ ∂t Hm L∞ ∂t m L4 ∂t

m L4 .

Finally, from these estimates, we obtain

p+1

∂t

m L2 (Ω) ≤ CT,p, m
< ∞.
0 H2

(4.13)

Combining (4.5), (4.7) and (4.13), we have shown that

p+1

∂t

m H 2 (Ω) ≤ CT,p, m
< ∞.
0 H2

This completes the induction process and hence the proof.
Remark 3. Theorem 4.1.1 is also valid for the general equation (3.28) with smooth functions
63

ϕ(m) and a(x); the proof should be similar.

4.2

Weak ω-limit sets

We first study the energy decay for global weak solutions to the Landau-Lifshitz equation
(4.1). We write the initial value problem as



m = L(m, H ) in Ω × (0, ∞),
 t
eff

(4.14)



m(0) = m ,

0
in terms of the Landau-Lifshitz interaction function L defined by (3.2), where the effective
magnetic field Heff is given by (2.2).

4.2.1

The energy identity

Let E(m) be defined by (1.1). Assume a, m0 ∈ L∞ (Ω; R3 ).
Theorem 4.2.1. The global weak solution m to (4.14) satisfies the energy identity
t

E(m(t)) − E(m(s)) = γα
s

Ω

|m × Heff |2 dxdτ

∀ 0 ≤ s ≤ t < ∞.

(4.15)

Furthermore, if γα < 0, then mt ∈ L2 ((0, ∞); L2 (Ω; R3 )).
Proof. By the definition of Heff , (4.15) follows from the identity L(m, n) · n = −αγ|m × n|2
for all m, n ∈ R3 and the property

d
(E(m(t))) = −
Heff · mt dx a.e. (0, ∞).
dt
Ω
64

∞
If γα < 0, by (4.15), it follows that 0 Ω |m × Heff |2 dxdt ≤ E(m0 )/|αγ| < ∞. Also from

the equation (4.14),

|mt |2 =|L(m, Heff )|2
=γ 2 |m × Heff |2 + (γα)2 |m|2 |m × Heff |2
≤C |m × Heff |2 ,
where constant C depends on m0 L∞ . Hence mt ∈ L2 ((0, ∞); L2 (Ω; R3 )).
Remark 4. The stability result and all the regularity estimates previously proved are for
finite time; the only global space for the solutions (even for the regular solutions) is

m ∈ L∞ ((0, ∞); L∞ (Ω; R3 )) with mt ∈ L2 ((0, ∞); L2 (Ω; R3 )).

But this space is not enough to have strong convergence as t → ∞; it would be enough if
one has mt ∈ L1 ((0, ∞); L2 (Ω; R3 )) (see [32]). Therefore, it is quite challenging to study
the asymptotic behaviors of even regular solutions. The solution orbits for general initial
data may not have strong ω-limit points. See [52] for some recent results on the estimation
of weak ω-limit set.

4.2.2

Weak ω-limit sets and the estimate for soft-case

Given m0 ∈ L∞ (Ω; R3 ), let m be the global weak solution to the initial value problem (4.14)
and define the weak ω-limit set for m to be

˜
ω ∗ (m0 ) = {m | ∃ tj ↑ ∞ such that m(tj )

65

˜
m weakly in L2 (Ω; R3 )}.

(4.16)

We give an estimate of ω ∗ (m0 ) for the so-called soft-case, where there is no anisotropy
energy (ϕ = 0). For more results on further special case when a = 0, see [51, 52].
Theorem 4.2.2. Let γα < 0, ϕ = 0 and a ∈ L∞ (Ω; R3 ). Then, for any m0 ∈ L∞ (Ω; R3 )
with |m0 (x)| = 1 a.e. on Ω, it follows that

˜
˜
˜
ω ∗ (m0 ) ⊆ {m ∈ L∞ (Ω; R3 ) | |m|2 + 2|m × (a + Hm )| ≤ 1 a.e. on Ω}.
˜

(4.17)

Proof. Let m be the global weak solution to (4.14) with the given initial datum m0 . Then
˜
m weakly in L2 (Ω; R3 ) for a sequence

|m(t)| = 1 a.e. on Ω for all t ≥ 0. Assume m(tj )
tj ↑ ∞. In the following, we show that

˜
˜
|m|2 + 2|m × (a + Hm )| ≤ 1 a.e. on Ω.
˜

(4.18)

Let e(t) = E(m(t)). Then, by (4.15), e(t) is non-increasing and bounded and hence e(t) has
limit as t → ∞; this again by (4.15) implies

e(tj + 1) − e(tj ) = γα

tj +1
tj

m(t) × (a + Hm(t) ) 2 2 dt → 0.
L

Hence there exists some sj ∈ [tj , tj + 1] such that

m(sj ) × (a + Hm(s ) ) L2 (Ω) → 0.
j

(4.19)

By Theorem 4.2.1, mt ∈ L2 ((0, ∞); L2 (Ω; R3 )); hence

sj

m(sj ) − m(tj ) L2 ≤
tj

mt (t) L2 dt ≤
66

1
(sj − tj ) 2

sj
tj

mt 2 2 dt
L

1
2

→ 0,

which yields m(sj )

˜
m weakly in L2 (Ω; R3 ). Therefore, by (4.19), (4.18) follows from the

following lemma with mj = m(sj ). This completes the proof.
Lemma 4.2.3. Let mj

(a)

˜
m weakly in L2 (Ω; R3 ) and satisfy the following conditions:

|mj | = 1 a.e. Ω;

mj × (a + Hmj ) L2 (Ω) → 0.

(b)

˜
Then m satisfies the condition (4.18) above.
Proof. This result can be proved by a similar method as used for [51, Theorem 1.1]. However,
we present a different but direct proof based on the div-curl lemma [46]. For any m ∈
L∞ (Ω; R3 ), let Gm = mχΩ + Hm . Then div Gm = 0 on R3 . Denote

Gj = a + Gmj , Hj = a + Hmj ;

Then Gj

˜
G, Hj

˜
˜
G = a + Gm , H = a + Hm .
˜
˜

˜
H weakly in L2 (Ω; R3 ) and, by the div-curl lemma [46],

Ω

Gj · Hj φ dx →

Ω

∞
˜ ˜
G · Hφ dx ∀ φ ∈ C0 (Ω).

(4.20)

Since mj = Gj − Hj on Ω, it follows that

|mj |2 + 2|mj × (a + Hmj )| =|Gj − Hj |2 + 2|Gj × Hj |
=|Gj

|2

+ |Hj

|2

(4.21)

+ 2|Gj × Hj | − 2Gj · Hj .

Note that function f (m, n) = |m|2 + |n|2 + 2|m × n| is convex on (m, n) ∈ R3 × R3 . Hence,

67

∞
for all φ ∈ C0 (Ω) with φ ≥ 0, one has

lim inf
j→∞

Ω

(|Gj |2 + |Hj |2 + 2|Gj × Hj |)φ ≥

˜
˜
˜
˜
(|G|2 + |H|2 + 2|G × H|)φ.

(4.22)

Ω

By assumptions (a), (b), from (4.20)–(4.22), it follows that

˜
˜
˜
˜
˜ ˜
(|G|2 + |H|2 + 2|G × H| − 2G · H)φ dx ≤
Ω

φ dx
Ω

∞
for all φ ∈ C0 (Ω) with φ ≥ 0. This implies

˜
˜
˜
˜
˜ ˜
|G|2 + |H|2 + 2|G × H| − 2G · H ≤ 1 a.e. Ω,

which, exactly, is equivalent to (4.18). This completes the proof.

4.3

Special dynamics for constant initial data on ellipsoid domains

We now study a special case of (4.14) where applied field a(x) = a is constant, domain Ω
is an ellipsoid, and initial datum m0 is a constant unit vector. Therefore, in (4.14), the
effective magnetic field Heff is now given by

Heff = −ϕ (m) + a + Hm

68

as above, but with constant vector a. In what follows, we assume ellipsoid domain Ω is given
by
3

Ω = {x ∈

R3

x2 /ai < 1},
i

|
i=1

where ai > 0 are constants.

4.3.1

Associated ODE system on R3

Let bi (i = 1, 2, 3) be defined as below:
√
a1 a2 a3 dt
1 ∞
bi =
.
2 0 (ai + t) (a1 + t)(a2 + t)(a3 + t)

(4.23)

Note that, if Ω is a ball in R3 , all bi ’s are equal to 1/3.
Theorem 4.3.1 (Pedragal and Yan [44]). For each k = 1, 2, 3, the Dirichlet-Neumann
problem
ω = 0 in Ωc

(4.24)

with boundary condition

ω|∂Ω = xk ,

∂ω
1
|∂Ω = (1 − )νk
∂ν
bk

ν is the unit normal on ∂Ω pointing outward of Ω, has an unique solution ω satisfies ω ∈
1
Hloc (Ωc ) and | ω| ∈ L2 (Ωc ).

With Theorem 4.3.1, one can prove following lemma; the proof is in [44].
Lemma 4.3.2. Let ω = (ω1 , ω2 , ω3 ) be the functions determined in Theorem 4.3.1. For each

69

k = 1, 2, 3, let
uk (x) =




b x
 k k

x ∈ Ω,



b ω (x)
 k k
1
Then uk ∈ Hloc (Rn ), F k =

(4.25)
x ∈ Ωc

uk ∈ L2 (Rn ; Rn ), and

curl F k = 0,

div(−F k + ek χΩ ) = 0,

in R3 ,

(4.26)

where e1 , e2 , e3 are the standard basis vectors of R3 .
From this lemma, the magneto-static stray field Hm induced by any constant field m
has constant value on Ω given by

Hm |Ω = −Λm (∀ m ∈ R3 ),

where Λ is a diagonal matrix of positive numbers. In fact, Λ = diag (b1 , b2 , b3 ).
Let m0 be a constant unit vector. Then, problem (4.14) reduces to the following ODE
system on m ∈ R3 :



˙
m = Φ(m),

t > 0,
(4.27)



m(0) = m ,

0
where function Φ : R3 → R3 is defined by

Φ(m) = L(m, −ϕ (m) + a − Λm),

∀ m ∈ R3 .

(4.28)

Since m · Φ(m) = 0, system (4.27) also preserves the length of m(t). Thus we have
|m(t)| = 1 for all t ≥ 0. Moreover, L(m, n) = 0 if and only if m × n = 0; hence, the
70

equilibrium points of (4.27), that is, the solutions of Φ(m) = 0 on unit sphere |m| = 1, are
characterized by vectors m ∈ R3 for which there is a real number λ ∈ R such that

−ϕ (m) + a − Λm = λm,

|m| = 1.

(4.29)

This condition is equivalent to m being a critical point of the function

1
P (m) = Λm · m − a · m + ϕ(m)
2

(4.30)

on unit sphere |m| = 1.
In most cases, there will be at least two distinct equilibrium points for system (4.27); for
example, all maximum or minimum points of P on |m| = 1 (always exist) are such points.

4.3.2

Lyapunov function and the special dynamics

The dynamics of system (4.27) can be studied by the classical ODE theory. For example,
we have the following result.
Theorem 4.3.3. Function P is a Lyapunov function for system (4.27). The ω-limit set of
(4.27) is contained in the set of all critical points of P on unit sphere.
Proof. Since Heff = −ϕ (m) + a − Λm = − P (m), ∀ m ∈ R3 , it follows that for solution
m = m(t) of (4.27)

d
P (m(t)) =
dt

P (m) · mt = −Heff · L(m, Heff ) = γα|m × Heff |2 ≤ 0.

Hence P is a Lyapunov function for system (4.27).
71

¯
To show the second part of the theorem, assume m(tj ) → m for a sequence tj ↑ ∞. Let
p(t) = P (m(t)). Then p(t) is smooth and has limit as t → ∞. Hence p(tj + 1) − p(tj ) =
p (sj ) → 0 for some sj ∈ (tj , tj + 1). Since γα < 0, this implies

|m(sj ) × Heff (sj )| = |m(sj ) ×

P (m(sj ))| → 0.

(4.31)

As above, since mt ∈ L2 (0, ∞), one has

|m(sj ) − m(tj )| ≤

sj
tj

|mt |dt ≤

1
(sj − tj ) 2

¯
¯
This implies m(sj ) → m; hence, by (4.31), |m ×

sj
tj

|mt |2 dt

1
2

→ 0.

¯
¯
P (m)| = 0, which proves m is a critical

point of P on unit sphere. This completes the proof.
Finally, we prove a special result for a = 0 and ϕ = 0.
Proposition 4.3.4. Let b1 , b2 , b3 be positive numbers determined by (4.23). If bk = min{b1 , b2 , b3 },
then ±ek are asymptotically stable equilibrium points to the system (4.27), where e1 , e2 , e3
are the standard basis vectors of R3 .
Proof. Without loss of generality, let us assume b1 = min{b1 , b2 , b3 }. It is trivial to see that
P has a strict relative minimum at (±1, 0, 0). According to the Lyapunov stability theorem,
(±1, 0, 0) are the asymptotically stable equilibrium points.

72

Chapter 5
Existence for Fractional
Landau-Lifshitz Equations
In this chapter, we prove the existence of global weak solution to fractional LandauLifshitz equation including magneto-static term with periodic boundary condition:

mt = γm × Fm + γm × (m × Fm )

in Rn ,

(5.1)

where
Fm = ∧2α m + Hm

0 < α < 1.

(5.2)

Let Ω = [0, 2π]n denote the periodic box in all directions and Z denote the dual lattice
associated to Ω in Rn . The two terms in energy form (5.2), for x ∈ Ω, are defined as:

 (ξ · m) · ξ

,

|ξ|2
Hm (ξ) =


 0, ξ = 0.

ξ = 0.
(5.3)

1
The operator ∧ denotes the square root of Laplacian (− ) 2 , so ∧β m can be understood in

terms of Fourier transform:
F(∧β m(x)) = |ξ|β m(ξ)

73

(5.4)

Definition 5.0.1. Let m0 ∈ H α , |m0 | = 1, m is a global weak solution to equation (5.1) if

m ∈ L∞ ((0, T ); H α (Ω))

∂m
·ϕ=γ
(m × Fm ) · ϕ + γ
m × (m × Fm ) · ϕ
QT
QT
QT ∂t
for all ϕ ∈ C ∞ (QT ). Here QT = (0, T ) × Ω.
Remark 5. We remark that, as being the same as weak solution in Chapter 2, the length
|m(x, t)| is preserved for weak solutions; therefore, if the initial datum m0 satisfies the
saturation condition |m0 (x)| = 1 then any weak solution m(x, t) to the fractional equation
(5.1) will also satisfy the saturation condition: |m(x, t)| = 1 for all t > 0.

5.1

Notations and preliminaries

Let Lp denote the space of all the p-th integrable function f in Ω with associated norm given
by:
1

f (x) Lp = (

|f (x)|p ) p
Ω

For any r ∈ R, we define the homogeneous Sobolev space H r for all tempered distribution
function f (x) such that f H r is finite, where

· H r is defined through Fourier transform:

f H r = ∧r f (x) L2 = (
ξ∈Z

74

1

|ξ|2r |f (ξ)|2 ) 2

For 1 ≤ p ≤ ∞ and r ∈ R, the space H r,p (Ω) consists of all f (x) such that f = ∧−r g, where
g ∈ Lp (Ω). The corresponding norm is given by:

f H r,p = F−1 (|ξ|r f ) Lp

The following commutator estimate comes from a general result of Coifman and Meyer [9];
see also [35, 36].
Lemma 5.1.1. Suppose that s > 0 and p ∈ (1, ∞). Then

∧s (f g) − f ∧s g Lp ≤ C

f Lp1 g s−1,p2 + f H s,p3 g Lp4 ,
H

where p1 , p2 , p3 , p4 ∈ (1, ∞) satisfy
1
1
1
1
1
=
+
=
+
p
p1 p2
p3 p4
and f, g are such that the right hand side terms make sense.
For fractional derivative, we also have Hardy-Littlewood-Sobolev theorem; the proof can
be found in Stein [45].
Lemma 5.1.2. Suppose that q > 1, p ∈ [q, ∞] and

1 s
1
+ =
p d
q
Assume that ∧s f ∈ Lq , then f ∈ Lp and there is a constant C > 0 such that

f Lp ≤ C ∧s f Lq
75

As a corollary, if f = ∧−s g for g ∈ Lq , then

g H −s,p = ∧−s g Lp ≤ C g Lq

Later we are going to apply the Galerkin method; therefore, we need to consider following
eigenvalue problem,

 2α

 ∧ u = µu

(5.5)



 with periodic boundary conditions.
Because ∧−2α is a compact self-adjoint operator in L2 (Ω), there exists a completed orthogonal family of L2 (Ω), say {ωj }j∈N , being eigenvectors of ∧−2α ,

∧−2α ωj = µj ωj

where sequence µj is decreasing and tends to 0. By setting νj = µ−1 , we obtain,
j

∧2α ωj = νj ωj

where 0 < ν1 ≤ ν2 ≤ ... The family {ωj } satisfies

(ωj , ωk ) = δjk ,

Kronecker symbol

76

(5.6)

5.2

A priori estimates

We let Vn =

n
k=1 ωk

∈ L2 (Ω), where ωk is eigenvectors in (5.6). We look for solutions to

approximate fractional Landau-Lifshitz equation (5.1) in the form of
N

mN =

ϕk,N (t)ωk (x)

(5.7)

k=1

In order to show such {ϕk,N (t)} exist, we take inner product of equation (5.1) with each
ωk (x), k = 1, 2, ..., N ,
∂mN
mN × (mN × FmN ) · ωk (x)
· ωk (x) = γ
mN × FmN · ωk (x) + γ
Ω
Ω
Ω ∂t

(5.8)

where
FmN = ∧2α mN + PN (HmN )
We define PN as the orthogonal projection,

˙
PN : H r (Ω) → VN = {ω1 , ω2 , ..., ωN }

The local existence of solutions {ϕk,N (t)} is justified by standard Picard’s theorem. In order
to extend the local weak solution to [0, T ] and take N → ∞, we need to establish following
a priori estimate,
Lemma 5.2.1. Let m0 ∈ H α (Ω), then for any 0 < T < ∞, the approximate solutions mN
to systems (5.8) satisfy,
max

0≤t≤T

mN 2 α (Ω) ≤ C1
H

77

d
where C1 only depends on initial data m0 2 α (Ω) . Moreover, for 1 ≤ r ≤ r∗ = d−α , where
H

d is dimension,
∂mN
≤ C2
r
∂t L (QT )
d
and for 1 < r ≤ r∗ = d−α

r−1

mN (t1 ) − mN (t2 ) Lr (Ω) ≤ C2 |t2 − t1 | r

where C2 only depends on initial data m0 2 α (Ω) and time T .
H
Proof. Multiply systems (5.8) with ϕk,N (t), k = 1, 2, .., N and sum up together, then we
have,
d
|m |2 dx = 0
dt Ω N
So,
mN L2 (Ω) ≤ m0 L2 (Ω)
Multiply systems (5.8) with ϕk,N (t), k = 1, 2, .., N , where ϕk,N (t) satisfies,
N

FmN =

ϕk,N (t)ωk (x)
k=1

and sum up together,

∂mN
∂mN
· ∧2α mN +
· PN (HmN ) +
|mN × FmN |2 = 0
∂t
∂t
Ω
Ω
Ω

78

(5.9)

By using periodic boundary condition,

∂mN
1∂
| ∧α mN |2 +
· PN (HmN ) +
|mN × FmN |2 = 0
2 ∂t Ω
∂t
Ω
Ω
For mapping operator PN ,
∂mN
∂mN
· PN (HmN ) =
· HmN
Ω ∂t
Ω ∂t
Also we have,
Ω

mN · HmN =

mN · HmN =

Ω

|HmN |2

So,
1∂
1∂
|mN × FmN |2 = 0
| ∧α mN |2 +
|HmN |2 +
2 ∂t Ω
2 ∂t Ω
Ω

(5.10)

Integrating (5.10) with respect to time from [0, T ],

max

0≤t≤T

∧α mN 2 2
+ max HmN 2 2
+ mN × FmN 2 2
≤ C1 (5.11)
L (Ω) 0≤t≤T
L (Ω)
L (0,T ;L2 (Ω))

where C1 only depends on initial data

∧α m0 2 2 . By Sobolev embedding, for any
L (Ω)

2d
1 ≤ p ≤ p∗ = d−2α , we have,

max

0≤t≤T

mN (t) Lp (Ω) ≤ k1 C1

d
By Holder inequality, for 1 ≤ r ≤ r∗ = d−α ,

(

2r

1

2−r

|mN × FmN |r dx) r ≤ FmN L2 (Ω) ( |mN | 2−r dx) 2r ≤ k2 C1
Ω
Ω

79

2r
d
since 2−r ≤ p∗ when r < r∗ . Moreover, we have for 1 ≤ r ≤ r∗ = d−α ,

(

∂mN r
|mN × FmN |r +
|mN × (mN × FmN )|r
| )≤
∂t
QT
QT
QT
|

T

≤ k2 C1 +

0

2r

2−r

mN × FmN L2 (Ω) ( |mN | 2−r dx) 2r dt
Ω

(5.12)

≤ k3 C2

where C2 only depends on initial data

and time T . With the same r > 1 as
∧α m0 2 2
L (Ω)

defined above, we can further prove the approximation solution mN is continuous in time,
t2

∂mN
dt Lr (Ω)
∂t
t1
t2 ∂m
N
≤
dt
r
∂t L (Ω)
t1

mN (t1 ) − mN (t2 ) Lr (Ω) ≤

≤

r−1
|t2 − t1 | r (

(5.13)

∂mN r 1
|
| )r
∂t
QT

r−1

≤ C2 |t2 − t1 | r

Based on Lemma 5.2.1, we have following conclusions for approximate solution mN ,
Lemma 5.2.2. For arbitrage N and time T , under the condition of Lemma 5.2.1, the initial
value problem for approximation systems (5.8) has at least one time continuous and global
weak solution.

5.3

Compactness

In order to take N → ∞ with the approximation solutions, we still need strong convergence
based on Lemma 5.2.1. The compactness lemma below can be found in [39],
80

Lemma 5.3.1. Let B0 , B, B1 be three Banach space such that,

B0 ⊂ B ⊂ B1

where the injections are continuous and B0 , B1 are reflexive and B0 → B is compact. Denote

W = {v|v ∈ Lp0 (0, T ; B0 ),

dv
∈ Lp1 (0, T ; B1 )}
dt

for T < ∞ and 1 < p0 , p1 < ∞. Then W equipped with the norm
dv
v Lp0 (0,T ;B ) +
p
0
dt L 1 (0,T ;B1 )
is a Banach space and the embedding W → Lp0 (0, T ; B) is compact.
From Lemma 5.2.1, we know mN is uniformly bounded in

L∞ (0, T ; H α (Ω))

W 1,r (0, T ; Lr (Ω))

With compactness lemma 5.3.1, there exists some m ∈ L∞ (0, T ; H α (Ω)) such that,

mN

m weakly in Lp (0, T ; H α (Ω)) for 1 < p < ∞

mN → m strongly in Lp (0, T ; H β (Ω)) for 1 < p < ∞, 0 ≤ β ≤ α
∂mN
∂t

∂m
∂t

weakly in Lr (QT ) for 1 < r

81

(5.14)

5.4

Existence

We are going to prove the global existence of weak solution to fractional LLG (5.1) by
determining weak convergence limit of approximate solution mN when N → ∞. From
compactness result(5.14), it is obvious that we have,

∂m
∂mN
·ϕ→
· ϕ , ∀ϕ ∈ C ∞ (QT )
QT ∂t
QT ∂t
For the nonlocal nonlinear term, we first observe that,

mN × FmN 2 2
≤ C1
L (0,T ;L2 (Ω))

which has been proved in (5.11). So there exists η ∈ L2 (0, T ; L2 (Ω)), such that,

mN × FmN

η

weakly in L2 (0, T ; L2 (Ω))

We would like to show that η = m × Fm in L2 (0, T ; L2 (Ω)). Let ϕ ∈ H s (Ω) for s > α + d/2,

QT

mN × FmN · ϕ =

QT

mN × PN (HmN ) · ϕ +

QT

mN × ∧2α mN · ϕ

= A1 + A2
From compactness result(5.14),

A1 =

QT

mN × PN (HmN ) · ϕ →

82

m × Hm · ϕ
QT

(5.15)

For term A2 ,

A2 =

QT

mN × ∧2α mN · ϕ
(5.16)
∧α m

=−
QT

α
N · ∧ (mN

× ϕ)

In order to show that η = m × Fm in L2 (0, T ; L2 (Ω)), it is enough to prove

QT

∧α mN · ∧α (mN × ϕ)

∧α m · ∧α (m × ϕ)

weakly in L2 (0, T ; L2 (Ω))

QT

We set commutator Φϕ (m) = [∧α , ϕ]m = ∧α (m × ϕ) − ∧α m × ϕ, then it is equivalent to
prove
QT

∧α mN · Φϕ (mN − m) +

QT

∧α (mN − m) · Φϕ (m) → 0

From the compactness result (5.14), it is obvious,

QT

∧α (mN − m) · Φϕ (m) → 0

From Lemma 5.1.1 and Lemma 5.1.2,

Φϕ (mN − m) L2 (Ω) ≤ C(

ϕ Lp1 mN − m α−1,p2 + ϕ H α,p3 mN − m Lp4 )
H

≤ C(

ϕ Lp1 mN − m L2 (Ω) + ϕ H α,p3 mN − m H β (Ω) )

≤ C ϕ H s (Ω) mN − m H β (Ω)
(5.17)
where p2 , p3 ∈ (1, ∞)
1
1
1
1
1
=
+
=
+
2
p1 p2
p3 p 4
83

1−α
1
1
+
=
d
p2
2
β
1
1
+
=
d p4
2

0<β<α

d
−1
2

s>
Therefore,

|
QT

∧α mN · Φϕ (mN − m)| ≤ C ϕ H s (Ω) ∧α mN L2 (Ω ) mN − m L2 (0,T ;H β (Ω)) → 0
T

So, ∀ϕ ∈ C ∞ (QT )
QT

(mN × FmN ) · ϕ →

(m × Fm ) · ϕ
QT

Furthermore, since mN → m strongly in L2 (QT ),

QT

(mN × FmN ) × mN · ϕ →

(m × Fm ) × m · ϕ
QT

The theorem is summarized as below
Theorem 5.4.1. Let 0 < α < 1 and m0 ∈ H α (Ω), then for any 0 < T < ∞, then there
exists at least one global weak solution to fractional Landau-Lifshitz equation (5.1) such that

m ∈ L∞ (0, T ; H α (Ω))

d
for 1 < r ≤ r∗ = d−α .

84

r−1

C 0, r (0, T ; Lr (Ω))

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