# Global Existence of Weak Solutions to a Fractional Model in Magnetoelastic Interactions

Authors: Idriss Ellahiani; EL-Hassan Essoufi; Mouhcine Tilioua
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

The paper deals with global existence of weak solutions to a one-dimensional mathematical model describing magnetoelastic interactions. The model is described by a fractional Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We prove global existence by using Faedo-Galerkin/penalty method. Some commutator estimates are used to prove the convergence of nonlinear terms.


---

## Body

## 1. Introduction

The nonlinear parabolic hyperbolic coupled system describing magnetoelastic dynamics inQ
=
(
0
,
T
)
×
Ω (T
>
0 and Ω is a bounded open set of R
d, d
⩾
1) is given by (see [1])(1)
γ
-
1
m
t
=
-
m
×
H
e
f
f
+
m
t
.
(2)
ρ
u
t
t
-
div
⁡
S
u
+
1
2
L
m
=
0
.Equation (1), well known in the literature, is the Landau-Lifshitz-Gilbert (LLG) equation. The unknown m, the magnetization vector, is a map from Ω to S
2 (the unit sphere of R
3) and m
t is its derivative with respect to time. The symbol × denotes the vector cross product in R
3. Moreover we denote by m
i, i
=
1,2
,
3, the components of m. The constant γ represents the damping parameter. H
e
f
f represents the effective field which is given by(3)
H
e
f
f
=
a
Δ
m
+
l
m
,
u
,where a is a positive constant and the components of the vector l
(
m
,
u
) are given by (4)
l
i
=
λ
i
j
k
l
m
j
ϵ
k
l
u
.Hereϵ
i
j
(
u
)
=
1
/
2
(
∂
i
u
j
+
∂
j
u
i
) stand for the components of the linearized strain tensor ϵ, λ
i
j
k
l
=
λ
1
δ
i
j
k
l
+
λ
2
δ
i
j
δ
k
l
+
λ
3
(
δ
i
k
δ
j
l
+
δ
i
l
δ
j
k
) with δ
i
j
k
l
=
1 if i
=
j
=
k
=
l and δ
i
j
k
l
=
0 otherwise.Equation (2) describes the evolution of the displacement u, ρ is a positive constant, and the tensors S
(
u
), L
(
m
) are given by(5)
S
k
l
=
σ
i
j
k
l
ϵ
i
j
u
,
L
k
l
=
λ
i
j
k
l
m
i
m
j
.σ
=
(
σ
i
j
k
l
) is the elasticity tensor satisfying the following symmetry property: (6)
σ
i
j
k
l
=
σ
k
l
i
j
=
σ
j
i
k
l
.Many studies have been done on the fractional Landau-Lifshitz equation; we quote here, for example, [2], where the existence of weak solutions under periodical boundary condition has been proven for equation of a reduced model for thin-film micromagnetics. In [3], the main purpose is to consider the well-posedness of the fractional Landau-Lifshitz equation without Gilbert damping. The global existence of weak solutions is proved by vanishing viscosity method. Note that the existence and asymptotic behaviors of global weak solutions to the one-dimensional periodical fractional Landau-Lifshitz equation modeling the soft micromagnetic materials are studied in [4]. For the magnetoelasticity coupling, in [1], the authors study the three-dimensional case and establish the existence of weak solutions taking into account three terms of the total free energy. Existence and uniqueness of solutions have been proven in [5] for a simplified model and in [6] a one-dimensional penalty problem is discussed and the gradient flow of the associated type Ginzburg-Landau functional is studied. More precisely the authors prove the existence and uniqueness of a classical solution which tends asymptotically for subsequences to a stationary point of the energy functional. Our aim here is to study the coupled system of magnetoelastic interactions with fractional LLG equation.The rest of the paper is divided as follows. In the next section we present the model equation we will be interested in. Section3 recalls some useful lemmas. Finally in Section 4 we prove a global existence result of weak solutions to the considered model.


## 2. The Model and Main Result

We assume thatΩ is a subset of R and the displacement is only in one direction. Specifically, we consider a simple variable space x and assume that Ω
=
(
0,2
π
). We take the following system:(7)
γ
-
1
m
t
=
-
m
×
H
e
f
f
+
m
t
ρ
u
t
t
-
div
⁡
S
u
+
1
2
L
m
=
0
,with associated initial and boundary conditions(8)
u
·
,
0
=
u
0
,
u
t
·
,
0
=
u
1
,
m
·
,
0
=
m
0
,
m
0
=
1
in
Ω
,
(9)
u
=
0
,
m
x
,
t
=
m
x
+
2
π
,
t
o
n
Σ
≔
∂
Ω
×
0
,
T
.The effective field is given by(10)
H
e
f
f
=
a
Λ
2
α
m
+
l
m
,
u
,where Λ
=
(
-
Δ
)
1
/
2 denotes the square root of the Laplacian which can be defined via Fourier transformation [7]. In this paper we are interested in the case α
∈
(
1
/
2
,
1
). For the vector u, we assume that u
=
(
0,0
,
ω
) and we keep the three components of the vector m
=
(
m
1
,
m
2
,
m
3
).It is a common practice (see [5]) to replace the first equation of system (7) by the quasilinear parabolic equation (Ginzburg-Landau type equation):(11)
m
t
ε
+
γ
-
1
m
t
ε
×
m
ε
=
-
H
e
f
f
ε
-
m
ε
2
-
1
ε
m
ε
.Hereε is a positive parameter and m
ε
:
Ω
→
R
3. ε-penalization in (11) replaces the magnitude constraint m
=
1.Throughout, we make use of the following notation. ForΩ, an open bounded domain of R
3, we denote by L
p
(
Ω
)
=
(
L
p
(
Ω
)
)
3 and H
1
(
Ω
)
=
(
H
1
(
Ω
)
)
3 the classical Hilbert spaces equipped with the usual norm denoted by ·
L
p
(
Ω
) and ·
H
1
(
Ω
) (in general, the product functional spaces (
X
)
3 are all simplified to X). For all s
>
0, W
s
,
p denotes the usual Sobolev space consisting of all f such that(12)
f
W
s
,
p
≔
F
-
1
1
+
·
2
s
/
2
F
f
·
L
p
<
∞
,where F denotes the Fourier transform and F
-
1 its inverse. Let W
˙
s
,
p denote the corresponding homogeneous Sobolev space. When p
=
2, W
s
,
p corresponds to the usual Sobolev space H
s.Now we give a definition of the solution in the weak sense of problem (7)-(8)-(9).Definition 1.
Letm
0
∈
H
α
(
Ω
), m
0
=
1 a.e., ω
0
∈
H
0
1
(
Ω
), and ω
1
∈
L
2
(
Ω
). One says that the pair (
m
,
ω
) is a weak solution of problem (7)-(8)-(9) if(i)
for allT
>
0, m
∈
L
∞
(
0
,
T
;
H
α
(
Ω
)
), m
t
∈
L
2
(
0
,
T
;
L
2
(
Ω
)
), m
=
1 a.e., ω
∈
L
2
(
0
,
T
;
H
0
1
(
Ω
)
), and ω
t
∈
L
2
(
0
,
T
;
L
2
(
Ω
)
);
(ii)
for allφ
∈
L
2
(
0
,
T
;
H
α
(
Ω
)
) and ψ
∈
H
0
1
(
Q
) one has(13)
γ
-
1
∫
Q
m
t
·
φ
d
x
d
t
+
∫
Q
m
×
m
t
·
φ
d
x
d
t
=
a
∫
Q
Λ
α
m
·
Λ
α
m
×
φ
d
x
d
t
+
∫
Q
l
m
,
ω
×
m
·
φ
d
x
d
t
-
ρ
∫
Q
ω
t
ψ
t
d
x
d
t
+
∫
Q
ω
x
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
m
3
ψ
x
d
x
d
t
=
0
;
(iii)
m
(
0
,
x
)
=
m
0
(
x
) and ω
(
0
,
x
)
=
ω
0
(
x
) in the trace sense;
(iv)
for allT
>
0 we have(14)
a
2
∫
Ω
Λ
α
m
T
2
d
x
+
∫
Q
m
t
2
d
x
d
t
+
ρ
2
∫
Ω
ω
t
T
2
d
x
+
1
4
∫
Ω
ω
x
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
0
2
d
x
+
ρ
2
∫
Ω
ω
1
2
d
x
+
1
4
∫
Ω
ω
x
0
2
d
x
+
C
Ω
,
λ
,where C
(
Ω
,
λ
) is a positive constant which depends only on Ω and λ.The main result of this paper is the following.Theorem 2.
Letα
∈
(
1
/
2
,
1
), m
0
∈
H
α
(
Ω
) such that m
0
=
1 a.e., ω
0
∈
H
0
1
(
Ω
), and ω
1
∈
L
2
(
Ω
). Then there exists at least a weak solution for problem (7)-(8)-(9) in the sense of Definition 1.The proof of Theorem2 will be given in Section 4.


## 3. Some Technical Lemmas

In this section we present some lemmas which will be used in the rest of the paper. We start by recalling the following lemma due to Simon (see [8]).Lemma 3.
AssumeA, B, and C are three Banach spaces and satisfy A
⊂
B
⊂
C with compact embedding A
↪
B. Let Θ be bounded in L
∞
(
0
,
T
;
A
) and Θ
t
≔
{
f
t
;
f
∈
Θ
} be bounded in L
p
(
0
,
T
;
C
)
,
p
>
1. Then Θ is relatively compact in C
(
[
0
,
T
]
;
B
).There is another lemma whose proof can be found in [[9], page 12].Lemma 4.
LetΘ be a bounded open set of R
x
d
×
R
t, h
n and h in L
q
(
Θ
)
,
1
<
q
<
∞ such that h
n
L
q
(
Θ
)
≤
C, h
n
→
h a.e. in Θ; then h
n
⇀
h weakly in L
q
(
Θ
).The following lemma will ensure a compact embedding for the spaceW
s
,
p.Lemma 5.
LetΘ be a bounded open set of R
d, which is uniform Lipschitz. Let s
∈
[
0,1
[, p
>
1
,
d
≥
1. If s
p
<
d then the injection of W
s
,
p
(
Θ
) in L
k
(
Θ
) is compact, for any k
<
d
p
/
d
-
s
p.The proof can be found in [[10], Theorem 4.54., p 216]. We give now a lemma that will play a very important role in the convergence of approximate solutions (see [11–13] for a proof).Lemma 6 (commutator estimates).
Suppose thats
>
0 and p
∈
(
1
,
+
∞
). If f
,
g
∈
S (the Schwartz class) then(15)
Λ
s
f
g
-
f
Λ
s
g
L
p
≤
C
∇
f
L
p
1
g
W
˙
s
-
1
,
p
2
+
f
W
˙
s
,
p
3
g
L
p
4
,
(16)
Λ
s
f
g
L
p
≤
C
f
L
p
1
g
W
˙
s
,
p
2
+
f
W
˙
s
,
p
3
g
L
p
4with p
2, p
3
∈
(
1
,
+
∞
) such that 1
/
p
=
1
/
p
1
+
1
/
p
2
=
1
/
p
3
+
1
/
p
4.Here is another lemma which can be viewed as a result of the Hardy-Littlewood-Sobolev theorem of fractional integration; see [7] for a detailed proof.Lemma 7.
Suppose thatp
>
q
>
1 and 1
/
p
+
s
=
1
/
q. Assume that f
∈
L
q; then Λ
-
s
f
∈
L
p and there is a constant C
>
0 such that(17)
f
W
˙
-
s
,
p
≔
Λ
-
s
f
L
p
≤
C
f
L
q
.We finish this section with the following result (the proof can be found in [2]).Lemma 8.
Iff and g belong to H
p
e
r
2
α
(
Ω
)
≔
{
f
∈
L
2
(
Ω
)
/
Λ
2
α
f
∈
L
2
(
Ω
)
}, then(18)
∫
Ω
Λ
2
α
f
·
g
d
x
=
∫
Ω
Λ
α
f
·
Λ
α
g
d
x
.


## 4. Proof of Theorem2

Our goal is to show global existence of weak solutions for the fractional problem (7)-(8)-(9).


### 4.1. The Penalty Problem

Letε
>
0 be a fixed parameter. We construct approximated solutions m
ε converging, as ε
→
0, to a solution m of the problem. System (7) is reduced to the following problem:(19)
γ
-
1
m
t
ε
×
m
ε
+
m
t
ε
+
a
Λ
2
α
m
ε
+
l
m
ε
,
ω
ε
+
m
ε
2
-
1
ε
m
ε
=
0
ρ
ω
t
t
ε
-
ω
x
x
ε
-
λ
m
1
ε
m
3
ε
x
=
0in Q
=
Ω
×
(
0
,
T
), where the vector l
(
m
,
ω
) is given by l
(
m
,
ω
)
=
(
λ
m
3
ω
x
,
0
,
λ
m
1
ω
x
), λ
1
=
λ
2
=
0, λ
3
=
λ, and σ
1313
=
1.System (19) is supplemented with initial and boundary conditions (20)
ω
ε
·
,
0
=
ω
0
,
ω
t
ε
·
,
0
=
ω
1
,
m
ε
·
,
0
=
m
0
,
m
0
=
1
a.e
.
i
n
Ω
,
ω
ε
=
0
,
m
ε
x
,
t
=
m
ε
x
+
2
π
,
t
o
n
Σ
.We apply Faedo-Galerkin method: let{
f
i
}
i
∈
N be an orthonormal basis of L
2
(
Ω
) consisting of all the eigenfunctions for the operator Λ
2
α (the existence of such a basis can be proved as in [14], Ch. II),(21)
Λ
2
α
f
i
=
α
i
f
i
,
i
=
1,2
,
…
,
f
i
0
=
f
i
2
π
,and let {
g
i
}
i
∈
N be an orthonormal basis of L
2
(
Ω
) consisting of all the eigenfunctions for the operator -
Δ:(22)
-
Δ
g
i
=
β
i
g
i
,
i
=
1,2
,
…
,
g
i
=
0
on
∂
Ω
.We then consider the following problem in Q
=
Ω
×
(
0
,
T
):(23)
γ
-
1
m
t
ε
,
N
×
m
ε
,
N
+
m
t
ε
,
N
+
a
Λ
2
α
m
ε
,
N
+
l
m
ε
,
N
,
ω
ε
,
N
+
m
ε
,
N
2
-
1
ε
m
ε
,
N
=
0
ρ
ω
t
t
ε
,
N
-
ω
x
x
ε
,
N
-
λ
m
1
ε
,
N
m
3
ε
,
N
x
=
0with initial and boundary conditions,(24)
ω
ε
,
N
·
,
0
=
ω
N
·
,
0
,
ω
t
ε
,
N
·
,
0
=
ω
t
N
·
,
0
,
m
ε
,
N
·
,
0
=
m
N
·
,
0
,
in
Ω
,
ω
ε
,
N
=
0
,
m
ε
,
N
x
,
t
=
m
ε
,
N
x
+
2
π
,
t
on
Σ
=
∂
Ω
×
0
,
T
,
∫
Ω
ω
N
x
,
0
g
i
x
d
x
=
∫
Ω
ω
0
x
g
i
x
d
x
,
∫
Ω
ω
t
N
x
,
0
g
i
x
d
x
=
∫
Ω
ω
1
x
g
i
x
d
x
,
∫
Ω
m
N
x
,
0
f
i
x
d
x
=
∫
Ω
m
0
x
f
i
x
d
x
.We are looking for approximate solutions(
m
ε
,
N
,
ω
ε
,
N
) to (23) under the form(25)
m
ε
,
N
=
∑
i
=
1
N
a
i
t
f
i
x
,
ω
ε
,
N
=
∑
i
=
1
N
b
i
t
g
i
x
.If we multiply each scalar equation of the first equation of (23) by f
i and the second by g
i and integrate in Ω we get to a system of ordinary differential equations in the unknown (
a
i
(
t
)
,
b
i
(
t
)
), i
=
1,2
,
…
,
N
. We observe that we can write the first equation in the form(26)
-
a
Λ
2
α
m
ε
,
N
-
l
m
ε
,
N
,
ω
ε
,
N
-
m
ε
,
N
2
-
1
ε
m
ε
,
N
=
A
m
ε
,
N
m
t
ε
,
Nwith(27)
A
=
1
γ
-
1
m
3
ε
,
N
-
γ
-
1
m
2
ε
,
N
-
γ
-
1
m
3
ε
,
N
1
γ
-
1
m
1
ε
,
N
γ
-
1
m
2
ε
,
N
-
γ
-
1
m
1
ε
,
N
1
.It is clear that the matrixA is invertible which implies the system of first-order ordinary differential equations is Lipschitz locally; then there exists a local solution to the problem that we can extend on [
0
,
T
] using a priori estimates. For this, we multiply the first equation of (23) by m
t
ε
,
N and the second by ω
t
ε
,
N; integrating in Ω, we obtain (by using Lemma 8)(28)
∫
Ω
m
t
ε
,
N
2
d
x
+
a
2
d
d
t
∫
Ω
Λ
α
m
ε
,
N
2
d
x
+
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
d
x
+
1
4
ε
d
d
t
∫
Ω
m
ε
,
N
2
-
1
2
d
x
=
0
ρ
2
d
d
t
∫
Ω
ω
t
ε
,
N
2
d
x
+
1
2
d
d
t
∫
Ω
ω
x
ε
,
N
2
d
x
-
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
=
0
.We find after summing(29)
∫
Ω
m
t
ε
,
N
2
d
x
+
a
2
d
d
t
∫
Ω
Λ
α
m
ε
,
N
2
d
x
+
1
4
ε
d
d
t
∫
Ω
m
ε
,
N
2
-
1
2
d
x
+
ρ
2
d
d
t
∫
Ω
ω
t
ε
,
N
2
d
x
+
1
2
d
d
t
∫
Ω
ω
x
ε
,
N
2
d
x
+
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
-
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
=
0
.On the other hand(30)
d
d
t
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
ω
x
ε
,
N
d
x
=
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
d
x
+
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
ω
x
,
t
ε
,
N
d
x
=
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
d
x
-
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
=
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
-
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
.Now integrating in time,(31)
∫
Q
m
t
ε
,
N
2
d
x
+
a
2
∫
Ω
Λ
α
m
ε
,
N
T
2
d
x
+
1
4
ε
∫
Ω
m
ε
,
N
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
,
N
T
2
d
x
+
1
2
∫
Ω
ω
x
ε
,
N
T
2
d
x
+
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
ω
x
ε
,
N
T
d
x
=
a
2
∫
Ω
Λ
α
m
N
0
2
d
x
+
1
4
ε
∫
Ω
m
N
0
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
N
0
2
d
x
+
1
2
∫
Ω
ω
x
N
0
2
d
x
+
λ
∫
Ω
m
1
N
m
3
N
ω
x
N
0
d
x
.Omitting superscripts, we obtain for allb
>
0,(32)
∫
Ω
m
1
m
3
ω
x
d
x
≤
∫
Ω
1
2
b
m
1
m
3
2
+
b
2
ω
x
2
d
x
≤
1
2
b
∫
Ω
m
4
d
x
+
b
2
∫
Ω
ω
x
2
d
x
=
1
2
b
∫
Ω
m
2
-
1
+
1
2
d
x
+
b
2
∫
Ω
ω
x
2
d
x
≤
1
b
∫
Ω
m
2
-
1
2
d
x
+
2
π
b
+
b
2
∫
Ω
ω
x
2
d
x
=
1
b
∫
Ω
m
2
-
1
2
d
x
+
2
π
b
+
b
2
ω
x
L
2
Ω
2
.Hence, taking into account (31),(33)
∫
Q
m
t
ε
,
N
2
d
x
+
a
2
∫
Ω
Λ
α
m
ε
,
N
T
2
d
x
+
1
4
ε
-
λ
b
∫
Ω
m
ε
,
N
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
,
N
T
2
d
x
-
2
λ
π
b
+
1
-
λ
b
2
∫
Ω
ω
x
ε
,
N
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
N
0
2
d
x
+
1
4
ε
+
λ
b
∫
Ω
m
N
0
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
N
0
2
d
x
+
2
λ
π
b
+
1
+
λ
b
2
∫
Ω
ω
x
N
0
2
d
xand hence for ε small enough (ε
<
1
/
16
λ
2) and b
=
1
/
2
λ one has(34)
a
2
∫
Ω
Λ
α
m
ε
,
N
T
2
d
x
+
∫
Q
m
t
ε
,
N
2
d
x
d
t
+
1
8
ε
∫
Ω
m
ε
,
N
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
,
N
T
2
d
x
+
1
4
∫
Ω
ω
x
ε
,
N
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
N
0
2
d
x
+
3
8
ε
∫
Ω
m
N
0
2
-
1
2
d
x
+
8
π
λ
2
+
ρ
2
∫
Ω
ω
t
N
0
2
d
x
+
3
4
∫
Ω
ω
x
N
0
2
d
x
.The right-hand side is uniformly bounded. Indeed, forα
∈
(
1
/
2
,
1
), H
α
(
Ω
)
↪
L
4
(
Ω
); furthermore (for constants C
1, C
2, and C independent of N)(35)
∫
Ω
m
N
0
2
-
1
2
d
x
=
∫
Ω
m
N
0
4
d
x
-
2
∫
Ω
m
N
0
2
d
x
+
2
π
≤
m
N
0
L
4
Ω
4
+
2
π
≤
C
1
m
0
N
H
α
Ω
4
+
C
2
,
∫
Ω
ω
x
N
0
2
d
x
=
∫
Ω
ω
x
N
0
-
ω
x
0
+
ω
x
0
2
d
x
≤
2
∫
Ω
ω
x
N
0
-
ω
x
0
2
d
x
+
2
∫
Ω
ω
x
0
2
d
x
≤
2
ω
N
0
-
ω
0
H
0
1
Ω
2
+
2
ω
0
H
0
1
Ω
2
≤
C
,thanks to the strong convergence ω
N
(
·
,
0
)
→
ω
0 in H
0
1
(
Ω
). For the other term (ω
t
N
(
0
)), the estimate can be carried out in an analogous way using the strong convergence ω
t
N
(
·
,
0
)
→
ω
1 in L
2
(
Ω
). Moreover, noting that (for a constant C independent of ε and N)(36)
∫
Ω
m
ε
,
N
2
d
x
=
∫
Ω
m
ε
,
N
2
-
1
+
1
d
x
≤
1
2
∫
Ω
m
ε
,
N
2
-
1
2
d
x
+
C
,therefore, for fixedε
>
0, we have(37)
m
ε
,
N
N
is  bounded  in
L
∞
0
,
T
;
H
α
Ω
,
m
t
ε
,
N
N
is  bounded  in
L
2
0
,
T
;
L
2
Ω
,
m
ε
,
N
2
-
1
N
is  bounded  in
L
∞
0
,
T
;
L
2
Ω
,
ω
ε
,
N
N
is  bounded  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
,
N
N
is  bounded  in
L
2
0
,
T
;
L
2
Ω
.Note that (37) is due to the Poincaré lemma. Now, from classical compactness results there exist two subsequences which we still denote by (
m
ε
,
N
) and (
ω
ε
,
N
) such that for fixed ε
>
0 and for any 1
<
p
<
∞
(38)
m
ε
,
N
⇀
m
ε
weakly  in
L
p
0
,
T
;
H
α
Ω
,
m
ε
,
N
⟶
m
ε
strongly  in
C
0
,
T
,
H
β
Ω
,
a.e.  for
0
≤
β
<
α
m
t
ε
,
N
⇀
m
t
ε
w
e
a
k
l
y
i
n
L
2
0
,
T
;
L
2
Ω
,
m
ε
,
N
2
-
1
⇀
ζ
weakly  in
L
p
0
,
T
;
L
2
Ω
,
ω
ε
,
N
⇀
ω
ε
weakly  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
,
N
⇀
ω
t
ε
weakly  in
L
2
0
,
T
;
L
2
Ω
,
ω
ε
,
N
⟶
ω
ε
strongly  in
L
2
Q
.Convergence (38) is due to Lemma 3 and thanks to Lemma 4 it can be shown that ζ
=
|
m
ε
|
2
-
1. Moreover from the Sobolev embedding (Lemma 5) H
α
(
Q
)
↪
L
4
(
Q
), the further compactness result follows:(39)
m
i
ε
,
N
m
j
ε
,
N
⟶
m
i
ε
m
j
ε
strongly  in
L
2
Q
.The above estimates allow us to pass to the limit asN goes to infinity and to get the desired result. Indeed consider the variational formulation of (23):(40)
γ
-
1
∫
Q
m
t
ε
,
N
×
m
ε
,
N
·
ϕ
d
x
d
t
+
∫
Q
m
t
ε
,
N
·
ϕ
d
x
d
t
+
a
∫
Q
Λ
α
m
ε
,
N
·
Λ
α
ϕ
d
x
d
t
+
∫
Q
l
m
ε
,
N
,
ω
ε
,
N
·
ϕ
d
x
d
t
+
∫
Q
m
ε
,
N
2
-
1
ε
m
ε
,
N
·
ϕ
d
x
d
t
=
0
-
ρ
∫
Q
ω
t
ε
,
N
ψ
t
d
x
d
t
+
∫
Q
ω
x
ε
,
N
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
ε
,
N
m
3
ε
,
N
ψ
x
d
x
d
t
=
0
,for any ϕ
∈
L
2
(
0
,
T
;
H
α
(
Ω
)
) and ψ
∈
H
0
1
(
Q
). Taking N
→
∞ in (40), we find(41)
γ
-
1
∫
Q
m
t
ε
×
m
ε
·
ϕ
d
x
d
t
+
∫
Q
m
t
ε
·
ϕ
d
x
d
t
+
a
∫
Q
Λ
α
m
ε
·
Λ
α
ϕ
d
x
d
t
+
∫
Q
l
m
ε
,
ω
ε
·
ϕ
d
x
d
t
+
∫
Q
m
ε
2
-
1
ε
m
ε
·
ϕ
d
x
d
t
=
0
,
-
ρ
∫
Q
ω
t
ε
ψ
t
d
x
d
t
+
∫
Q
ω
x
ε
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
ε
m
3
ε
ψ
x
d
x
d
t
=
0
,for any ϕ
∈
L
2
(
0
,
T
;
H
α
(
Ω
)
) and ψ
∈
H
0
1
(
Q
). We proved the following result.Proposition 9.
Givenm
0
∈
H
α
(
Ω
) such that |
m
0
|
=
1 a.e., ω
0
∈
H
0
1
(
Ω
), and ω
1
∈
L
2
(
Ω
), then there exists a solution m
ε to problem (19) in the sense of distributions. Moreover we have the following energy estimate:(42)
a
2
∫
Ω
Λ
α
m
ε
T
2
d
x
+
∫
Q
m
t
ε
T
2
d
x
d
t
+
1
8
ε
∫
Ω
m
ε
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
T
2
d
x
+
1
4
∫
Ω
ω
x
ε
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
0
2
d
x
+
8
π
λ
2
+
ρ
2
∫
Ω
ω
1
2
d
x
+
1
4
∫
Ω
ω
x
0
2
d
x
.


### 4.2. Convergence of Approximate Solutions

Our aim here is to pass to the limit asε
→
0. For this, we will use estimate (42), from which we have the following:(43)
m
ε
ε
is  bounded  in
L
∞
0
,
T
;
H
α
Ω
,
m
t
ε
ε
is  bounded  in
L
2
0
,
T
;
L
2
Ω
,
m
ε
2
-
1
ε
is  bounded  in
L
∞
0
,
T
;
L
2
Ω
,
ω
ε
ε
is  bounded  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
ε
is  bounded  in
L
2
0
,
T
;
L
2
Ω
.Then there exist two subsequences which we still denote by(
m
ε
) and (
ω
ε
) such that for any 1
<
p
<
∞
(44)
m
ε
⇀
m
weakly  in
L
p
0
,
T
;
H
α
Ω
,
m
ε
⟶
m
strongly  in
C
0
,
T
,
H
β
Ω
,
a.e.  for
0
≤
β
<
α
m
t
ε
⇀
m
t
weakly  in
L
2
0
,
T
;
L
2
Ω
,
m
ε
2
-
1
⟶
0
strongly  in
L
p
0
,
T
;
L
2
Ω
,
a.e
.
ω
ε
⇀
ω
weakly  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
⇀
ω
t
weakly  in
L
2
0
,
T
;
L
2
Ω
,
ω
ε
⟶
ω
strongly  in
L
2
Q
.It can be shown from convergence (44) that m
=
1 a.e.Now in order to pass to the limitε
→
0 in (41), let φ
∈
C
∞
(
[
0
,
T
]
×
Ω
¯
), and let ϕ
=
m
ε
×
φ. We first recall the identity (
a
×
b
)
·
(
c
×
d
)
=
(
a
·
c
)
(
b
·
d
)
-
(
a
·
d
)
(
b
·
c
) for all a, b, c, and d in R
3.Asϕ is in L
2
(
0
,
T
;
H
α
(
Ω
)
), the following holds: (45)
γ
-
1
∫
Q
m
t
ε
·
m
ε
m
ε
·
φ
d
x
d
t
-
γ
-
1
∫
Q
m
t
ε
·
φ
m
ε
2
d
x
d
t
+
∫
Q
m
t
ε
×
m
ε
·
φ
d
x
d
t
+
a
∫
Q
Λ
α
m
ε
·
Λ
α
m
ε
×
φ
d
x
d
t
+
∫
Q
l
m
ε
,
ω
ε
×
m
ε
·
φ
d
x
d
t
=
0
-
ρ
∫
Q
ω
t
ε
ψ
t
d
x
d
t
+
∫
Q
ω
x
ε
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
ε
m
3
ε
ψ
x
d
x
d
t
=
0
.Note that for this choice we haveΛ
α
(
m
ε
×
φ
)
∈
L
2
(
Q
), indeed applying the multiplicative estimates (16) in Lemma 6 to m
ε and φ (for s
=
α, p
=
2, p
1
=
∞, p
2
=
2, p
3
=
2, and p
4
=
∞); we find for different constants C independent of ε:(46)
Λ
α
m
ε
×
φ
L
2
Q
≤
C
m
ε
L
∞
Q
φ
H
˙
α
Q
+
m
ε
H
˙
α
Q
φ
L
∞
Q
≤
C
;since 2
α
>
1 (1 here is the dimension) then H
α
(
Ω
)
↪
L
∞
(
Ω
) and consequently (
m
ε
)
ε is bounded in L
∞
(
Q
).Takingε
→
0 and following the idea introduced in [4] we have(47)
∫
Q
m
t
ε
·
m
ε
m
ε
·
φ
d
x
d
t
=
∫
Q
m
t
ε
·
m
ε
m
ε
·
φ
d
x
d
t
-
∫
Q
m
t
·
m
m
·
φ
d
x
d
t
=
∫
Q
m
t
ε
·
m
ε
m
ε
-
m
·
φ
d
x
d
t
+
∫
Q
m
ε
·
m
t
ε
-
m
t
m
·
φ
d
x
d
t
+
∫
Q
m
t
·
m
ε
-
m
m
·
φ
d
x
d
t
⟶
0
∫
Q
m
ε
2
m
t
ε
·
φ
d
x
d
t
=
∫
Q
m
ε
2
-
1
m
t
ε
·
φ
d
x
d
t
+
∫
Q
m
t
ε
·
φ
d
x
d
t
⟶
∫
Q
m
t
·
φ
d
x
d
t
.Now for the fourth term of the first equation, we introduce the commutator (see [4]):(48)
Γ
φ
m
≔
Λ
α
m
×
φ
-
φ
×
Λ
α
mLetI
ε
=
∫
Q
Λ
α
m
ε
·
Λ
α
(
m
ε
×
φ
)
d
x
d
t and I
=
∫
Q
Λ
α
m
·
Λ
α
(
m
×
φ
)
d
x
d
t. We will show that I
ε
→
I.First, note thatΓ
φ
(
m
)
∈
L
2
(
Q
). Indeed, applying (15) for p
1
=
1
/
1
-
α
,
p
2
=
2
/
2
α
-
1, p
3 and p
4 in (
2
,
+
∞
), we find (for different constants C)(49)
Γ
φ
m
L
2
Ω
≤
C
∇
φ
L
p
1
Ω
m
W
˙
α
-
1
,
p
2
Ω
+
φ
W
˙
α
,
p
3
Ω
m
L
p
4
Ω
≤
C
∇
φ
L
p
1
Ω
m
L
2
Ω
+
φ
W
˙
α
,
p
3
Ω
m
L
2
Ω
≤
C
m
L
2
Ω
, thanks to Lemma 7. Once again(50)
Γ
φ
m
ε
-
m
L
2
Ω
≤
C
∇
φ
L
p
1
Ω
m
ε
-
m
W
˙
α
-
1
,
p
2
Ω
+
φ
W
˙
α
,
p
3
Ω
m
ε
-
m
L
p
4
Ω
Γ
φ
m
ε
-
m
L
2
Q
≤
C
∇
φ
L
∞
0
,
T
;
L
p
1
Ω
m
ε
-
m
L
2
Q
+
φ
L
∞
0
,
T
;
W
˙
α
,
p
3
Ω
m
ε
-
m
L
2
0
,
T
;
H
β
Ω
,where the right-hand side of the last inequality tends to 0.Finally, we have(51)
I
ε
-
I
=
∫
Q
Λ
α
m
ε
·
Γ
φ
m
ε
d
x
d
t
-
∫
Q
Λ
α
m
·
Γ
φ
m
d
x
d
t
=
∫
Q
Λ
α
m
ε
·
Γ
φ
m
ε
-
m
d
x
d
t
+
∫
Q
Λ
α
m
ε
-
m
·
Γ
φ
m
d
x
d
t
≤
∫
Q
Λ
α
m
ε
·
Γ
φ
m
ε
-
m
d
x
d
t
+
∫
Q
Λ
α
m
ε
-
m
·
Γ
φ
m
d
x
d
t
≤
C
Γ
φ
m
ε
-
m
L
2
Q
+
∫
Q
Λ
α
m
ε
-
m
·
Γ
φ
m
d
x
d
t
⟶
0
.Therefore(52)
γ
-
1
∫
Q
m
t
·
φ
d
x
d
t
+
∫
Q
m
×
m
t
·
φ
d
x
d
t
=
a
∫
Q
Λ
α
m
·
Λ
α
m
×
φ
d
x
d
t
+
∫
Q
l
m
,
ω
×
m
·
φ
d
x
d
t
,
-
ρ
∫
Q
ω
t
ψ
t
d
x
d
t
+
∫
Q
ω
x
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
m
3
ψ
x
d
x
d
t
=
0
.This being true for everyφ
∈
C
∞
(
[
0
,
T
]
×
Ω
¯
) and by a standard density argument, it is true for any φ in L
2
(
0
,
T
;
H
α
(
Ω
)
). Note that, from (42), one can easily get (14). Hence (
m
,
ω
) is a solution of problem (7)-(8)-(9) in the sense of Definition 1. The proof of Theorem 2 is complete.


## 4.1. The Penalty Problem

Letε
>
0 be a fixed parameter. We construct approximated solutions m
ε converging, as ε
→
0, to a solution m of the problem. System (7) is reduced to the following problem:(19)
γ
-
1
m
t
ε
×
m
ε
+
m
t
ε
+
a
Λ
2
α
m
ε
+
l
m
ε
,
ω
ε
+
m
ε
2
-
1
ε
m
ε
=
0
ρ
ω
t
t
ε
-
ω
x
x
ε
-
λ
m
1
ε
m
3
ε
x
=
0in Q
=
Ω
×
(
0
,
T
), where the vector l
(
m
,
ω
) is given by l
(
m
,
ω
)
=
(
λ
m
3
ω
x
,
0
,
λ
m
1
ω
x
), λ
1
=
λ
2
=
0, λ
3
=
λ, and σ
1313
=
1.System (19) is supplemented with initial and boundary conditions (20)
ω
ε
·
,
0
=
ω
0
,
ω
t
ε
·
,
0
=
ω
1
,
m
ε
·
,
0
=
m
0
,
m
0
=
1
a.e
.
i
n
Ω
,
ω
ε
=
0
,
m
ε
x
,
t
=
m
ε
x
+
2
π
,
t
o
n
Σ
.We apply Faedo-Galerkin method: let{
f
i
}
i
∈
N be an orthonormal basis of L
2
(
Ω
) consisting of all the eigenfunctions for the operator Λ
2
α (the existence of such a basis can be proved as in [14], Ch. II),(21)
Λ
2
α
f
i
=
α
i
f
i
,
i
=
1,2
,
…
,
f
i
0
=
f
i
2
π
,and let {
g
i
}
i
∈
N be an orthonormal basis of L
2
(
Ω
) consisting of all the eigenfunctions for the operator -
Δ:(22)
-
Δ
g
i
=
β
i
g
i
,
i
=
1,2
,
…
,
g
i
=
0
on
∂
Ω
.We then consider the following problem in Q
=
Ω
×
(
0
,
T
):(23)
γ
-
1
m
t
ε
,
N
×
m
ε
,
N
+
m
t
ε
,
N
+
a
Λ
2
α
m
ε
,
N
+
l
m
ε
,
N
,
ω
ε
,
N
+
m
ε
,
N
2
-
1
ε
m
ε
,
N
=
0
ρ
ω
t
t
ε
,
N
-
ω
x
x
ε
,
N
-
λ
m
1
ε
,
N
m
3
ε
,
N
x
=
0with initial and boundary conditions,(24)
ω
ε
,
N
·
,
0
=
ω
N
·
,
0
,
ω
t
ε
,
N
·
,
0
=
ω
t
N
·
,
0
,
m
ε
,
N
·
,
0
=
m
N
·
,
0
,
in
Ω
,
ω
ε
,
N
=
0
,
m
ε
,
N
x
,
t
=
m
ε
,
N
x
+
2
π
,
t
on
Σ
=
∂
Ω
×
0
,
T
,
∫
Ω
ω
N
x
,
0
g
i
x
d
x
=
∫
Ω
ω
0
x
g
i
x
d
x
,
∫
Ω
ω
t
N
x
,
0
g
i
x
d
x
=
∫
Ω
ω
1
x
g
i
x
d
x
,
∫
Ω
m
N
x
,
0
f
i
x
d
x
=
∫
Ω
m
0
x
f
i
x
d
x
.We are looking for approximate solutions(
m
ε
,
N
,
ω
ε
,
N
) to (23) under the form(25)
m
ε
,
N
=
∑
i
=
1
N
a
i
t
f
i
x
,
ω
ε
,
N
=
∑
i
=
1
N
b
i
t
g
i
x
.If we multiply each scalar equation of the first equation of (23) by f
i and the second by g
i and integrate in Ω we get to a system of ordinary differential equations in the unknown (
a
i
(
t
)
,
b
i
(
t
)
), i
=
1,2
,
…
,
N
. We observe that we can write the first equation in the form(26)
-
a
Λ
2
α
m
ε
,
N
-
l
m
ε
,
N
,
ω
ε
,
N
-
m
ε
,
N
2
-
1
ε
m
ε
,
N
=
A
m
ε
,
N
m
t
ε
,
Nwith(27)
A
=
1
γ
-
1
m
3
ε
,
N
-
γ
-
1
m
2
ε
,
N
-
γ
-
1
m
3
ε
,
N
1
γ
-
1
m
1
ε
,
N
γ
-
1
m
2
ε
,
N
-
γ
-
1
m
1
ε
,
N
1
.It is clear that the matrixA is invertible which implies the system of first-order ordinary differential equations is Lipschitz locally; then there exists a local solution to the problem that we can extend on [
0
,
T
] using a priori estimates. For this, we multiply the first equation of (23) by m
t
ε
,
N and the second by ω
t
ε
,
N; integrating in Ω, we obtain (by using Lemma 8)(28)
∫
Ω
m
t
ε
,
N
2
d
x
+
a
2
d
d
t
∫
Ω
Λ
α
m
ε
,
N
2
d
x
+
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
d
x
+
1
4
ε
d
d
t
∫
Ω
m
ε
,
N
2
-
1
2
d
x
=
0
ρ
2
d
d
t
∫
Ω
ω
t
ε
,
N
2
d
x
+
1
2
d
d
t
∫
Ω
ω
x
ε
,
N
2
d
x
-
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
=
0
.We find after summing(29)
∫
Ω
m
t
ε
,
N
2
d
x
+
a
2
d
d
t
∫
Ω
Λ
α
m
ε
,
N
2
d
x
+
1
4
ε
d
d
t
∫
Ω
m
ε
,
N
2
-
1
2
d
x
+
ρ
2
d
d
t
∫
Ω
ω
t
ε
,
N
2
d
x
+
1
2
d
d
t
∫
Ω
ω
x
ε
,
N
2
d
x
+
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
-
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
=
0
.On the other hand(30)
d
d
t
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
ω
x
ε
,
N
d
x
=
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
d
x
+
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
ω
x
,
t
ε
,
N
d
x
=
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
d
x
-
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
=
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
t
ω
x
ε
,
N
-
m
1
ε
,
N
m
3
ε
,
N
x
ω
t
ε
,
N
d
x
.Now integrating in time,(31)
∫
Q
m
t
ε
,
N
2
d
x
+
a
2
∫
Ω
Λ
α
m
ε
,
N
T
2
d
x
+
1
4
ε
∫
Ω
m
ε
,
N
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
,
N
T
2
d
x
+
1
2
∫
Ω
ω
x
ε
,
N
T
2
d
x
+
λ
∫
Ω
m
1
ε
,
N
m
3
ε
,
N
ω
x
ε
,
N
T
d
x
=
a
2
∫
Ω
Λ
α
m
N
0
2
d
x
+
1
4
ε
∫
Ω
m
N
0
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
N
0
2
d
x
+
1
2
∫
Ω
ω
x
N
0
2
d
x
+
λ
∫
Ω
m
1
N
m
3
N
ω
x
N
0
d
x
.Omitting superscripts, we obtain for allb
>
0,(32)
∫
Ω
m
1
m
3
ω
x
d
x
≤
∫
Ω
1
2
b
m
1
m
3
2
+
b
2
ω
x
2
d
x
≤
1
2
b
∫
Ω
m
4
d
x
+
b
2
∫
Ω
ω
x
2
d
x
=
1
2
b
∫
Ω
m
2
-
1
+
1
2
d
x
+
b
2
∫
Ω
ω
x
2
d
x
≤
1
b
∫
Ω
m
2
-
1
2
d
x
+
2
π
b
+
b
2
∫
Ω
ω
x
2
d
x
=
1
b
∫
Ω
m
2
-
1
2
d
x
+
2
π
b
+
b
2
ω
x
L
2
Ω
2
.Hence, taking into account (31),(33)
∫
Q
m
t
ε
,
N
2
d
x
+
a
2
∫
Ω
Λ
α
m
ε
,
N
T
2
d
x
+
1
4
ε
-
λ
b
∫
Ω
m
ε
,
N
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
,
N
T
2
d
x
-
2
λ
π
b
+
1
-
λ
b
2
∫
Ω
ω
x
ε
,
N
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
N
0
2
d
x
+
1
4
ε
+
λ
b
∫
Ω
m
N
0
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
N
0
2
d
x
+
2
λ
π
b
+
1
+
λ
b
2
∫
Ω
ω
x
N
0
2
d
xand hence for ε small enough (ε
<
1
/
16
λ
2) and b
=
1
/
2
λ one has(34)
a
2
∫
Ω
Λ
α
m
ε
,
N
T
2
d
x
+
∫
Q
m
t
ε
,
N
2
d
x
d
t
+
1
8
ε
∫
Ω
m
ε
,
N
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
,
N
T
2
d
x
+
1
4
∫
Ω
ω
x
ε
,
N
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
N
0
2
d
x
+
3
8
ε
∫
Ω
m
N
0
2
-
1
2
d
x
+
8
π
λ
2
+
ρ
2
∫
Ω
ω
t
N
0
2
d
x
+
3
4
∫
Ω
ω
x
N
0
2
d
x
.The right-hand side is uniformly bounded. Indeed, forα
∈
(
1
/
2
,
1
), H
α
(
Ω
)
↪
L
4
(
Ω
); furthermore (for constants C
1, C
2, and C independent of N)(35)
∫
Ω
m
N
0
2
-
1
2
d
x
=
∫
Ω
m
N
0
4
d
x
-
2
∫
Ω
m
N
0
2
d
x
+
2
π
≤
m
N
0
L
4
Ω
4
+
2
π
≤
C
1
m
0
N
H
α
Ω
4
+
C
2
,
∫
Ω
ω
x
N
0
2
d
x
=
∫
Ω
ω
x
N
0
-
ω
x
0
+
ω
x
0
2
d
x
≤
2
∫
Ω
ω
x
N
0
-
ω
x
0
2
d
x
+
2
∫
Ω
ω
x
0
2
d
x
≤
2
ω
N
0
-
ω
0
H
0
1
Ω
2
+
2
ω
0
H
0
1
Ω
2
≤
C
,thanks to the strong convergence ω
N
(
·
,
0
)
→
ω
0 in H
0
1
(
Ω
). For the other term (ω
t
N
(
0
)), the estimate can be carried out in an analogous way using the strong convergence ω
t
N
(
·
,
0
)
→
ω
1 in L
2
(
Ω
). Moreover, noting that (for a constant C independent of ε and N)(36)
∫
Ω
m
ε
,
N
2
d
x
=
∫
Ω
m
ε
,
N
2
-
1
+
1
d
x
≤
1
2
∫
Ω
m
ε
,
N
2
-
1
2
d
x
+
C
,therefore, for fixedε
>
0, we have(37)
m
ε
,
N
N
is  bounded  in
L
∞
0
,
T
;
H
α
Ω
,
m
t
ε
,
N
N
is  bounded  in
L
2
0
,
T
;
L
2
Ω
,
m
ε
,
N
2
-
1
N
is  bounded  in
L
∞
0
,
T
;
L
2
Ω
,
ω
ε
,
N
N
is  bounded  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
,
N
N
is  bounded  in
L
2
0
,
T
;
L
2
Ω
.Note that (37) is due to the Poincaré lemma. Now, from classical compactness results there exist two subsequences which we still denote by (
m
ε
,
N
) and (
ω
ε
,
N
) such that for fixed ε
>
0 and for any 1
<
p
<
∞
(38)
m
ε
,
N
⇀
m
ε
weakly  in
L
p
0
,
T
;
H
α
Ω
,
m
ε
,
N
⟶
m
ε
strongly  in
C
0
,
T
,
H
β
Ω
,
a.e.  for
0
≤
β
<
α
m
t
ε
,
N
⇀
m
t
ε
w
e
a
k
l
y
i
n
L
2
0
,
T
;
L
2
Ω
,
m
ε
,
N
2
-
1
⇀
ζ
weakly  in
L
p
0
,
T
;
L
2
Ω
,
ω
ε
,
N
⇀
ω
ε
weakly  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
,
N
⇀
ω
t
ε
weakly  in
L
2
0
,
T
;
L
2
Ω
,
ω
ε
,
N
⟶
ω
ε
strongly  in
L
2
Q
.Convergence (38) is due to Lemma 3 and thanks to Lemma 4 it can be shown that ζ
=
|
m
ε
|
2
-
1. Moreover from the Sobolev embedding (Lemma 5) H
α
(
Q
)
↪
L
4
(
Q
), the further compactness result follows:(39)
m
i
ε
,
N
m
j
ε
,
N
⟶
m
i
ε
m
j
ε
strongly  in
L
2
Q
.The above estimates allow us to pass to the limit asN goes to infinity and to get the desired result. Indeed consider the variational formulation of (23):(40)
γ
-
1
∫
Q
m
t
ε
,
N
×
m
ε
,
N
·
ϕ
d
x
d
t
+
∫
Q
m
t
ε
,
N
·
ϕ
d
x
d
t
+
a
∫
Q
Λ
α
m
ε
,
N
·
Λ
α
ϕ
d
x
d
t
+
∫
Q
l
m
ε
,
N
,
ω
ε
,
N
·
ϕ
d
x
d
t
+
∫
Q
m
ε
,
N
2
-
1
ε
m
ε
,
N
·
ϕ
d
x
d
t
=
0
-
ρ
∫
Q
ω
t
ε
,
N
ψ
t
d
x
d
t
+
∫
Q
ω
x
ε
,
N
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
ε
,
N
m
3
ε
,
N
ψ
x
d
x
d
t
=
0
,for any ϕ
∈
L
2
(
0
,
T
;
H
α
(
Ω
)
) and ψ
∈
H
0
1
(
Q
). Taking N
→
∞ in (40), we find(41)
γ
-
1
∫
Q
m
t
ε
×
m
ε
·
ϕ
d
x
d
t
+
∫
Q
m
t
ε
·
ϕ
d
x
d
t
+
a
∫
Q
Λ
α
m
ε
·
Λ
α
ϕ
d
x
d
t
+
∫
Q
l
m
ε
,
ω
ε
·
ϕ
d
x
d
t
+
∫
Q
m
ε
2
-
1
ε
m
ε
·
ϕ
d
x
d
t
=
0
,
-
ρ
∫
Q
ω
t
ε
ψ
t
d
x
d
t
+
∫
Q
ω
x
ε
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
ε
m
3
ε
ψ
x
d
x
d
t
=
0
,for any ϕ
∈
L
2
(
0
,
T
;
H
α
(
Ω
)
) and ψ
∈
H
0
1
(
Q
). We proved the following result.Proposition 9.
Givenm
0
∈
H
α
(
Ω
) such that |
m
0
|
=
1 a.e., ω
0
∈
H
0
1
(
Ω
), and ω
1
∈
L
2
(
Ω
), then there exists a solution m
ε to problem (19) in the sense of distributions. Moreover we have the following energy estimate:(42)
a
2
∫
Ω
Λ
α
m
ε
T
2
d
x
+
∫
Q
m
t
ε
T
2
d
x
d
t
+
1
8
ε
∫
Ω
m
ε
T
2
-
1
2
d
x
+
ρ
2
∫
Ω
ω
t
ε
T
2
d
x
+
1
4
∫
Ω
ω
x
ε
T
2
d
x
≤
a
2
∫
Ω
Λ
α
m
0
2
d
x
+
8
π
λ
2
+
ρ
2
∫
Ω
ω
1
2
d
x
+
1
4
∫
Ω
ω
x
0
2
d
x
.


## 4.2. Convergence of Approximate Solutions

Our aim here is to pass to the limit asε
→
0. For this, we will use estimate (42), from which we have the following:(43)
m
ε
ε
is  bounded  in
L
∞
0
,
T
;
H
α
Ω
,
m
t
ε
ε
is  bounded  in
L
2
0
,
T
;
L
2
Ω
,
m
ε
2
-
1
ε
is  bounded  in
L
∞
0
,
T
;
L
2
Ω
,
ω
ε
ε
is  bounded  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
ε
is  bounded  in
L
2
0
,
T
;
L
2
Ω
.Then there exist two subsequences which we still denote by(
m
ε
) and (
ω
ε
) such that for any 1
<
p
<
∞
(44)
m
ε
⇀
m
weakly  in
L
p
0
,
T
;
H
α
Ω
,
m
ε
⟶
m
strongly  in
C
0
,
T
,
H
β
Ω
,
a.e.  for
0
≤
β
<
α
m
t
ε
⇀
m
t
weakly  in
L
2
0
,
T
;
L
2
Ω
,
m
ε
2
-
1
⟶
0
strongly  in
L
p
0
,
T
;
L
2
Ω
,
a.e
.
ω
ε
⇀
ω
weakly  in
L
2
0
,
T
;
H
0
1
Ω
,
ω
t
ε
⇀
ω
t
weakly  in
L
2
0
,
T
;
L
2
Ω
,
ω
ε
⟶
ω
strongly  in
L
2
Q
.It can be shown from convergence (44) that m
=
1 a.e.Now in order to pass to the limitε
→
0 in (41), let φ
∈
C
∞
(
[
0
,
T
]
×
Ω
¯
), and let ϕ
=
m
ε
×
φ. We first recall the identity (
a
×
b
)
·
(
c
×
d
)
=
(
a
·
c
)
(
b
·
d
)
-
(
a
·
d
)
(
b
·
c
) for all a, b, c, and d in R
3.Asϕ is in L
2
(
0
,
T
;
H
α
(
Ω
)
), the following holds: (45)
γ
-
1
∫
Q
m
t
ε
·
m
ε
m
ε
·
φ
d
x
d
t
-
γ
-
1
∫
Q
m
t
ε
·
φ
m
ε
2
d
x
d
t
+
∫
Q
m
t
ε
×
m
ε
·
φ
d
x
d
t
+
a
∫
Q
Λ
α
m
ε
·
Λ
α
m
ε
×
φ
d
x
d
t
+
∫
Q
l
m
ε
,
ω
ε
×
m
ε
·
φ
d
x
d
t
=
0
-
ρ
∫
Q
ω
t
ε
ψ
t
d
x
d
t
+
∫
Q
ω
x
ε
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
ε
m
3
ε
ψ
x
d
x
d
t
=
0
.Note that for this choice we haveΛ
α
(
m
ε
×
φ
)
∈
L
2
(
Q
), indeed applying the multiplicative estimates (16) in Lemma 6 to m
ε and φ (for s
=
α, p
=
2, p
1
=
∞, p
2
=
2, p
3
=
2, and p
4
=
∞); we find for different constants C independent of ε:(46)
Λ
α
m
ε
×
φ
L
2
Q
≤
C
m
ε
L
∞
Q
φ
H
˙
α
Q
+
m
ε
H
˙
α
Q
φ
L
∞
Q
≤
C
;since 2
α
>
1 (1 here is the dimension) then H
α
(
Ω
)
↪
L
∞
(
Ω
) and consequently (
m
ε
)
ε is bounded in L
∞
(
Q
).Takingε
→
0 and following the idea introduced in [4] we have(47)
∫
Q
m
t
ε
·
m
ε
m
ε
·
φ
d
x
d
t
=
∫
Q
m
t
ε
·
m
ε
m
ε
·
φ
d
x
d
t
-
∫
Q
m
t
·
m
m
·
φ
d
x
d
t
=
∫
Q
m
t
ε
·
m
ε
m
ε
-
m
·
φ
d
x
d
t
+
∫
Q
m
ε
·
m
t
ε
-
m
t
m
·
φ
d
x
d
t
+
∫
Q
m
t
·
m
ε
-
m
m
·
φ
d
x
d
t
⟶
0
∫
Q
m
ε
2
m
t
ε
·
φ
d
x
d
t
=
∫
Q
m
ε
2
-
1
m
t
ε
·
φ
d
x
d
t
+
∫
Q
m
t
ε
·
φ
d
x
d
t
⟶
∫
Q
m
t
·
φ
d
x
d
t
.Now for the fourth term of the first equation, we introduce the commutator (see [4]):(48)
Γ
φ
m
≔
Λ
α
m
×
φ
-
φ
×
Λ
α
mLetI
ε
=
∫
Q
Λ
α
m
ε
·
Λ
α
(
m
ε
×
φ
)
d
x
d
t and I
=
∫
Q
Λ
α
m
·
Λ
α
(
m
×
φ
)
d
x
d
t. We will show that I
ε
→
I.First, note thatΓ
φ
(
m
)
∈
L
2
(
Q
). Indeed, applying (15) for p
1
=
1
/
1
-
α
,
p
2
=
2
/
2
α
-
1, p
3 and p
4 in (
2
,
+
∞
), we find (for different constants C)(49)
Γ
φ
m
L
2
Ω
≤
C
∇
φ
L
p
1
Ω
m
W
˙
α
-
1
,
p
2
Ω
+
φ
W
˙
α
,
p
3
Ω
m
L
p
4
Ω
≤
C
∇
φ
L
p
1
Ω
m
L
2
Ω
+
φ
W
˙
α
,
p
3
Ω
m
L
2
Ω
≤
C
m
L
2
Ω
, thanks to Lemma 7. Once again(50)
Γ
φ
m
ε
-
m
L
2
Ω
≤
C
∇
φ
L
p
1
Ω
m
ε
-
m
W
˙
α
-
1
,
p
2
Ω
+
φ
W
˙
α
,
p
3
Ω
m
ε
-
m
L
p
4
Ω
Γ
φ
m
ε
-
m
L
2
Q
≤
C
∇
φ
L
∞
0
,
T
;
L
p
1
Ω
m
ε
-
m
L
2
Q
+
φ
L
∞
0
,
T
;
W
˙
α
,
p
3
Ω
m
ε
-
m
L
2
0
,
T
;
H
β
Ω
,where the right-hand side of the last inequality tends to 0.Finally, we have(51)
I
ε
-
I
=
∫
Q
Λ
α
m
ε
·
Γ
φ
m
ε
d
x
d
t
-
∫
Q
Λ
α
m
·
Γ
φ
m
d
x
d
t
=
∫
Q
Λ
α
m
ε
·
Γ
φ
m
ε
-
m
d
x
d
t
+
∫
Q
Λ
α
m
ε
-
m
·
Γ
φ
m
d
x
d
t
≤
∫
Q
Λ
α
m
ε
·
Γ
φ
m
ε
-
m
d
x
d
t
+
∫
Q
Λ
α
m
ε
-
m
·
Γ
φ
m
d
x
d
t
≤
C
Γ
φ
m
ε
-
m
L
2
Q
+
∫
Q
Λ
α
m
ε
-
m
·
Γ
φ
m
d
x
d
t
⟶
0
.Therefore(52)
γ
-
1
∫
Q
m
t
·
φ
d
x
d
t
+
∫
Q
m
×
m
t
·
φ
d
x
d
t
=
a
∫
Q
Λ
α
m
·
Λ
α
m
×
φ
d
x
d
t
+
∫
Q
l
m
,
ω
×
m
·
φ
d
x
d
t
,
-
ρ
∫
Q
ω
t
ψ
t
d
x
d
t
+
∫
Q
ω
x
ψ
x
d
x
d
t
+
λ
∫
Q
m
1
m
3
ψ
x
d
x
d
t
=
0
.This being true for everyφ
∈
C
∞
(
[
0
,
T
]
×
Ω
¯
) and by a standard density argument, it is true for any φ in L
2
(
0
,
T
;
H
α
(
Ω
)
). Note that, from (42), one can easily get (14). Hence (
m
,
ω
) is a solution of problem (7)-(8)-(9) in the sense of Definition 1. The proof of Theorem 2 is complete.



---

*Source: 9238948-2016-10-24.xml*