3.1 A priori estimates
Multiplying (1.2) by
u
→
, (1.5) by
B
→
, (1.6) by
E
→
and adding the result to equations (1.3) and (1.4), we get the total energy conservation law
(3.1)
∂
t
(
1
2
ϱ

u
→

2
+
ϱ
e
+
E
r
+
1
2
(

B
→

2
+

E
→

2
)
)
+
div
x
(
(
ϱ
E
+
E
r
)
u
→
+
(
p
+
p
r
)
u
→
+
E
→
×
B
→
)
=
0
.
Introducing the entropy
s
of the fluid by the Gibbs law
ϑ
d
s
=
d
e
+
p
d
(
1
ϱ
)
and denoting by
S
r
:=
4
3
a
T
r
3
the radiative entropy, equation (1.4) is rewritten as
(3.2)
∂
t
S
r
+
div
x
(
S
r
u
→
)
=

σ
a
E
r

a
ϑ
4
T
r
.
The internal energy equation is
∂
t
(
ϱ
e
)
+
div
x
(
ϱ
e
u
→
)
+
p
div
x
u
→

ν
ϱ

u
→

2
=

σ
a
(
a
ϑ
4

E
r
)
,
and by dividing it by
ϑ
, we get the entropy equation for matter
(3.3)
∂
t
(
ϱ
s
)
+
div
x
(
ϱ
s
u
→
)

ν
ϑ

u
→

2
=

σ
a
a
ϑ
4

E
r
ϑ
.
So adding (3.3) and (3.2), we obtain
(3.4)
∂
t
(
ϱ
s
+
S
r
)
+
div
x
(
(
ϱ
s
+
S
r
)
u
→
)
=
a
σ
a
ϑ
T
r
(
ϑ

T
r
)
2
(
ϑ
+
T
r
)
(
ϑ
2
+
T
r
2
)
+
ν
ϑ

u
→

2
.
By subtracting (3.4) from (3.1) and using the conservation of mass, we get
∂
t
(
1
2
ϱ

u
→

2
+
H
ϑ
¯
(
ϱ
,
ϑ
)

(
ϱ

ϱ
¯
)
∂
ϱ
H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)

H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)
+
H
r
,
ϑ
¯
(
T
r
)
+
1
2
(

B
→

B
→
¯

2
+

E
→

2
)
)
=
div
x
(
(
ϱ
E
+
E
r
)
u
→
+
(
p
+
p
r
)
u
→
+
ϑ
¯
(
ϱ
s
+
S
r
)
u
→
)

ϑ
¯
a
σ
a
ϑ
T
r
(
ϑ

T
r
)
2
(
ϑ
+
T
r
)
(
ϑ
2
+
T
r
2
)

ν
ϑ

u
→

2
.
By introducing the Helmholtz functions
H
ϑ
¯
(
ϱ
,
ϑ
)
:=
ϱ
(
e

ϑ
¯
s
)
and
H
r
,
ϑ
¯
(
T
r
)
:=
E
r

ϑ
¯
S
r
,
we check that the quantities
H
ϑ
¯
(
ϱ
,
ϑ
)

(
ϱ

ϱ
¯
)
∂
ϱ
H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)

H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)
and
H
r
,
ϑ
¯
(
T
r
)

H
r
,
ϑ
¯
(
T
¯
r
)
are nonnegative and strictly coercive functions reaching zero minima at the equilibrium state
(
ϱ
¯
,
ϑ
¯
,
E
¯
r
)
.
Lemma 3.1.
Let
ϱ
¯
and
ϑ
¯
=
T
¯
r
be given positive constants. Let
O
1
and
O
2
be the sets defined by
𝒪
1
:=
{
(
ϱ
,
ϑ
)
∈
ℝ
2
:
ϱ
¯
2
<
ϱ
<
2
ϱ
¯
,
ϑ
¯
2
<
ϑ
<
2
ϑ
¯
}
,
𝒪
2
:=
{
T
r
∈
ℝ
:
T
¯
r
2
<
T
r
<
2
T
¯
r
}
.
Then there exist positive constants
C
1
,
2
(
ϱ
¯
,
ϑ
¯
)
and
C
3
,
4
(
T
¯
r
)
such that
(3.5)
C
1
(

ϱ

ϱ
¯

2
+

ϑ

ϑ
¯

2
)
≤
H
ϑ
¯
(
ϱ
,
ϑ
)

(
ϱ

ϱ
¯
)
∂
ϱ
H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)

H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)
≤
C
2
(

ϱ

ϱ
¯

2
+

ϑ

ϑ
¯

2
)
for all
(
ϱ
,
ϑ
)
∈
O
1
, and
C
3

T
r

T
¯
r

2
≤
H
r
,
ϑ
¯
(
T
r
)

H
r
,
ϑ
¯
(
T
¯
r
)
≤
C
4

T
r

T
¯
r

2
for all
T
r
∈
O
2
.
Proof.
The first assertion is proved in [8], and we only sketch the proof for convenience. According to the decomposition
ϱ
→
H
ϑ
¯
(
ϱ
,
ϑ
)

(
ϱ

ϱ
¯
)
∂
ϱ
H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)

H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)
=
ℱ
(
ϱ
)
+
𝒢
(
ϱ
)
,
where
ℱ
(
ϱ
)
=
H
ϑ
¯
(
ϱ
,
ϑ
¯
)

(
ϱ

ϱ
¯
)
∂
ϱ
H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)

H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)
and
𝒢
(
ϱ
)
=
H
ϑ
¯
(
ϱ
,
ϑ
)

H
ϑ
¯
(
ϱ
,
ϑ
¯
)
,
one checks that
ℱ
is strictly convex and reaches a zero minimum at
ϱ
¯
, while
𝒢
is strictly decreasing for
ϑ
<
ϑ
¯
and strictly increasing for
ϑ
>
ϑ
¯
, according to the standard thermodynamic stability properties, see [8]. Computing the derivatives of
H
ϑ
¯
leads directly to estimate (3.5).
The second assertion follows from the properties of
x
↦
H
r
,
ϑ
¯
(
x
)

H
r
,
ϑ
¯
(
T
r
)
=
a
x
3
(
x

4
3
ϑ
¯
)
+
a
3
ϑ
¯
4
.
∎
Using the previous entropy properties, we have the following energy estimate.
Proposition 3.2.
Let the assumptions of Theorem 2.1 be satisfied with
V
=
(
ϱ
,
u
→
,
ϑ
,
E
r
,
B
→
,
E
→
)
𝑎𝑛𝑑
V
¯
=
(
ϱ
¯
,
0
,
ϑ
¯
,
E
r
¯
,
B
→
¯
,
0
)
.
Consider a solution
(
ϱ
,
u
→
,
ϑ
,
E
r
,
B
→
,
E
→
)
of system (1.1)–(1.3) on
[
0
,
t
]
, for some
t
>
0
. Then, for a constant
C
0
>
0
, one gets
(3.6)
∥
V
(
t
)

V
¯
∥
L
2
2
+
∫
0
t
∥
u
→
(
τ
)
∥
L
2
2
d
τ
≤
C
0
∥
V
0

V
¯
∥
L
2
2
.
Proof.
We define
η
(
t
,
x
)
=
H
ϑ
¯
(
ϱ
,
ϑ
)

(
ϱ

ϱ
¯
)
∂
ϱ
H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)

H
ϑ
¯
(
ϱ
¯
,
ϑ
¯
)
+
H
r
,
ϑ
¯
(
T
r
)
,
multiply (3.4) by
ϑ
¯
, and subtract the result to (3.1). By integrating over
[
0
,
t
]
×
ℝ
3
, we find
∫
ℝ
3
1
2
ϱ

u
→

2
+
η
(
t
,
x
)
+
1
2

B
→

B
→
¯

2
+
1
2

E
→

2
d
x
+
∫
0
t
∫
ℝ
3
ϑ
¯
ϑ
ν

u
→

2
≤
∫
ℝ
3
1
2
ϱ
0

u
→
0

2
(
t
)
+
η
(
0
,
x
)
+
1
2

B
→
0

B
→
¯

2
+
1
2

E
→
0

2
d
x
.
Applying Lemma 3.1 yields (3.6). ∎
By defining, for any
d
≥
3
, the auxiliary quantities
E
(
t
)
:=
sup
0
≤
τ
≤
t
∥
(
ϱ

ϱ
¯
,
u
→
,
B
→

B
→
¯
,
E
→
)
(
τ
)
∥
W
1
,
∞
,
F
(
t
)
:=
sup
0
≤
τ
≤
t
∥
(
V

V
¯
)
(
τ
)
∥
H
d
,
I
2
(
t
)
:=
∫
0
t
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
2
𝑑
τ
and
D
2
(
t
)
:=
∫
0
t
(
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
(
τ
)
∥
H
d
2
+
∥
E
→
(
τ
)
∥
H
d

1
2
+
∥
∂
x
B
→
(
τ
)
∥
H
d

2
2
)
𝑑
τ
,
we can bound the spatial derivatives as follows.
Proposition 3.3.
Assume that the hypotheses of Theorem 2.1 are satisfied. Then, for
C
0
>
0
, we have
∥
∂
x
V
(
t
)
∥
H
d

1
2
+
∫
0
t
∥
∂
x
u
→
(
τ
)
∥
H
d

1
2
d
τ
≤
C
0
∥
∂
x
V
0
∥
H
d

1
2
+
C
0
(
E
(
t
)
D
(
t
)
2
+
F
(
t
)
I
(
t
)
D
(
t
)
)
.
Proof.
By rewriting system (1.1)–(1.6) in the form
(3.7)
{
∂
t
ϱ
+
u
→
⋅
∇
x
ϱ
+
ϱ
div
x
u
→
=
0
,
∂
t
u
→
+
(
u
→
⋅
∇
x
)
u
→
+
p
ϱ
ϱ
∇
x
ϱ
+
p
ϑ
ϱ
∇
x
ϑ
+
1
3
a
ϱ
∇
x
E
r
+
E
→
+
u
→
×
B
→
¯
+
ν
u
→
=

u
→
×
(
B
→

B
→
¯
)
,
∂
t
ϑ
+
(
u
→
⋅
∇
x
)
ϑ
+
ϑ
p
ϑ
ϱ
C
v
div
x
u
→
=

σ
a
ϱ
C
v
(
a
ϑ
4

E
r
)
,
∂
t
E
r
+
(
u
→
⋅
∇
x
)
E
r
+
4
3
E
r
div
x
u
→
=

σ
a
(
E
r

a
ϑ
4
)
,
∂
t
B
→
+
curl
x
E
→
=
0
,
∂
t
E
→

curl
x
B
→

ϱ
¯
u
→
=
(
ϱ

ϱ
¯
)
u
→
,
and applying
∂
x
ℓ
to this system, we get
∂
t
(
∂
x
ℓ
ϱ
)
+
(
u
→
⋅
∇
x
)
∂
x
ℓ
ϱ
+
ϱ
div
x
∂
x
ℓ
u
→
=
F
1
ℓ
,
∂
t
(
∂
x
ℓ
u
→
)
+
(
u
→
⋅
∇
x
)
∂
x
ℓ
u
→
+
p
ϱ
ϱ
∇
x
∂
x
ℓ
ϱ
+
p
ϑ
ϱ
∇
x
∂
x
ℓ
ϑ
+
1
3
a
ϱ
∇
x
∂
x
ℓ
E
r
+
∂
x
ℓ
E
→
+
∂
x
ℓ
u
→
×
B
→
¯
+
ν
∂
x
ℓ
u
→
=

∂
x
ℓ
[
u
→
×
(
B
→

B
→
¯
)
]
+
F
2
ℓ
,
∂
t
(
∂
x
ℓ
ϑ
)
+
(
u
→
⋅
∇
x
)
∂
x
ℓ
ϑ
+
ϑ
p
ϑ
ϱ
C
v
div
x
∂
x
ℓ
u
→
=

∂
x
ℓ
[
σ
a
ϱ
C
v
(
a
ϑ
4

E
r
)
]
+
F
3
ℓ
,
∂
t
(
∂
x
ℓ
E
r
)
+
(
u
→
⋅
∇
x
)
∂
x
ℓ
E
r
+
4
3
E
r
div
x
∂
x
ℓ
u
→
=

∂
x
ℓ
[
σ
a
(
E
r

a
ϑ
4
)
]
+
F
4
ℓ
,
∂
t
(
∂
x
ℓ
B
→
)
+
curl
x
∂
x
ℓ
E
→
=
0
,
∂
t
(
∂
x
ℓ
E
→
)

curl
x
∂
x
ℓ
B
→

ϱ
¯
∂
x
ℓ
u
→
=
∂
x
ℓ
[
(
ϱ

ϱ
¯
)
u
→
]
,
where
F
1
ℓ
:=

[
∂
x
ℓ
,
u
→
⋅
∇
x
]
u
→

[
∂
x
ℓ
,
ϱ
div
x
]
u
→
,
F
2
ℓ
:=

[
∂
x
ℓ
,
u
→
⋅
∇
x
]
u
→

[
∂
x
ℓ
,
p
ϱ
ϱ
∇
x
]
ϱ

[
∂
x
ℓ
,
p
ϑ
ϱ
∇
x
]
ϑ

[
∂
x
ℓ
,
1
3
a
ϱ
∇
x
]
E
r
,
F
3
ℓ
:=

[
∂
x
ℓ
,
u
→
⋅
∇
x
]
ϑ

[
∂
x
ℓ
,
ϑ
p
ϑ
ϱ
C
v
div
x
]
u
→
,
F
4
ℓ
:=

[
∂
x
ℓ
,
u
→
⋅
∇
x
]
E
r

[
∂
x
ℓ
,
4
3
E
r
div
x
]
u
→
.
Then, by taking the scalar product of each of the previous equations, respectively, by
p
ϱ
ϱ
2
∂
x
ℓ
ϱ
,
∂
x
ℓ
u
→
,
C
v
ϑ
∂
x
ℓ
ϑ
,
1
4
a
ϱ
E
r
∂
x
ℓ
E
r
,
∂
x
ℓ
B
→
and
∂
x
ℓ
E
→
,
and adding the resulting equations, we get
(3.8)
∂
t
ℰ
ℓ
+
div
x
ℱ
→
ℓ
+
ν
(
∂
x
ℓ
u
→
)
2
=
ℛ
ℓ
+
𝒮
ℓ
,
where
ℰ
ℓ
:=
1
2
(
∂
x
ℓ
u
→
)
2
+
1
2
p
ϱ
ϱ
(
∂
x
ℓ
ϱ
)
2
+
1
2
C
v
ϑ
(
∂
x
ℓ
ϑ
)
2
+
1
2
1
4
a
ϱ
E
r
(
∂
x
ℓ
E
r
)
2
+
1
2
(
∂
x
ℓ
E
→
)
2
+
1
2
(
∂
x
ℓ
B
→
)
2
,
ℱ
→
ℓ
:=
(
p
ϱ
ϱ
∂
x
ℓ
ϱ
+
p
ϑ
ϱ
∂
x
ℓ
ϑ
+
1
3
a
ϱ
∂
x
ℓ
E
r
)
∂
x
ℓ
u
→
+
1
2
(
(
∂
x
ℓ
u
→
)
2
+
p
ϱ
ϱ
(
∂
x
ℓ
ϱ
)
2
+
C
v
ϑ
(
∂
x
ℓ
ϑ
)
2
+
1
4
a
ϱ
E
r
(
∂
x
ℓ
E
r
)
2
)
u
→
,
ℛ
ℓ
:=
1
2
[
p
ϱ
ϱ
2
]
t
(
∂
x
ℓ
ϱ
)
2
+
1
2
[
C
v
ϑ
]
t
(
∂
x
ℓ
ϑ
)
2
+
1
2
[
1
4
a
ϱ
E
r
]
t
(
∂
x
ℓ
E
r
)
2
+
1
2
div
x
(
p
ϱ
ϱ
2
u
→
)
(
∂
x
ℓ
ϱ
)
2
+
1
2
div
x
u
→
(
∂
x
ℓ
u
→
)
2
+
1
2
div
x
(
C
v
ϑ
u
→
)
(
∂
x
ℓ
ϑ
)
2
+
1
2
div
x
(
1
4
a
ϱ
E
r
u
→
)
(
∂
x
ℓ
E
r
)
2
+
∇
x
(
p
ϱ
ϱ
)
∂
x
ℓ
ϱ
∂
x
ℓ
u
→
+
∇
x
(
p
ϑ
ϱ
)
∂
x
ℓ
ϑ
∂
x
ℓ
u
→
+
∇
x
(
1
3
a
ϱ
)
∂
x
ℓ
E
r
∂
x
ℓ
u
→
+
p
ϱ
ϱ
2
∂
x
ℓ
ϱ
F
1
ℓ
+
∂
x
ℓ
u
→
F
2
ℓ
+
C
v
ϑ
∂
x
ℓ
ϑ
F
3
ℓ
+
∂
x
ℓ
E
r
F
4
ℓ
+
ϱ
¯
∂
x
ℓ
E
→
⋅
∂
x
ℓ
u
→
,
𝒮
ℓ
:=

∂
x
ℓ
u
→
⋅
∂
x
ℓ
[
u
→
×
(
B
→

B
→
¯
)
]

C
v
ϑ
∂
x
ℓ
ϑ
∂
x
ℓ
[
σ
a
ϱ
C
v
(
a
ϑ
4

E
r
)
]

1
4
a
ϱ
E
r
∂
x
ℓ
E
r
∂
x
ℓ
[
σ
a
(
E
r

a
ϑ
4
)
]
+
∂
x
ℓ
E
→
∂
x
ℓ
[
(
ϱ

ϱ
¯
)
u
→
]
.
By integrating (3.8) on space, one gets
∂
t
∫
ℝ
3
ℰ
ℓ
𝑑
x
+
∥
∂
x
ℓ
u
→
∥
L
2
2
≤
∫
ℝ
3
(

ℛ
ℓ

+

𝒮
ℓ

)
𝑑
x
.
Integrating now with respect to
t
and summing on
ℓ
, with

ℓ

≤
d
, yields
∥
∂
x
V
(
t
)
∥
H
d

1
2
+
∫
0
t
∥
∂
x
u
→
(
τ
)
∥
H
d

1
2
d
τ
≤
C
0
∥
∂
x
V
0
∥
H
d

1
2
+
C
0
∑

ℓ

=
1
d
∫
ℝ
3
(

ℛ
ℓ

+

𝒮
ℓ

)
d
x
.
By observing that

∂
t
ϱ

≤
C

∂
x
ϱ

,

∂
t
ϑ

≤
C
(

∂
x
ϱ

+

∂
x
ϑ

+

∂
x
E
r


Δ
ϑ

)
and

∂
t
E
r

≤
C
(

∂
x
ϱ

+

∂
x
ϑ

+

∂
x
E
r

)
,
and using the commutator estimates (see the Mosertype calculus inequalities in [16])
∥
(
F
1
ℓ
,
F
2
ℓ
,
F
3
ℓ
,
F
4
ℓ
)
∥
L
2
≤
∥
∂
x
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
2
2
,
we see that

ℛ
ℓ

≤
C
(
∥
∂
x
ϱ
∥
L
∞
+
∥
∂
x
u
→
∥
L
∞
+
∥
∂
x
ϑ
∥
L
∞
+
∥
∂
x
E
r
∥
L
∞
)
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
2
2
.
Then integrating with respect to time gives
∫
0
t

ℛ
ℓ
(
τ
)

d
τ
≤
C
sup
0
≤
τ
≤
t
{
∥
∂
x
ϱ
∥
L
∞
+
∥
∂
x
u
→
∥
L
∞
+
∥
∂
x
ϑ
∥
L
∞
+
∥
∂
x
E
r
∥
L
∞
}
∫
0
t
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
2
2
d
τ
≤
C
E
(
t
)
D
2
(
t
)
for any

ℓ

≤
d
. Similarly, we estimate

𝒮
ℓ

≤
C
∥
∂
x
ℓ
u
→
∥
L
2
2
∥
∂
x
ℓ
[
u
→
×
(
B
→

B
→
¯
)
]
∥
L
2
2
+
C
∥
∂
x
ℓ
ϑ
∥
L
2
2
∥
∂
x
ℓ
[
σ
a
ϱ
C
v
(
a
ϑ
4

E
r
)
]
∥
L
2
2
+
C
∥
∂
x
ℓ
E
r
∥
L
2
2
∥
∂
x
ℓ
[
σ
a
(
E
r

a
ϑ
4
)
]
∥
L
2
2
+
C
∥
∂
x
ℓ
E
→
∥
L
2
2
∥
∂
x
ℓ
[
(
ϱ

ϱ
¯
)
u
→
]
∥
L
2
2
.
Then we get

𝒮
ℓ

≤
C
∥
B
→

B
→
¯
∥
L
∞
∥
∂
x
ℓ
u
→
∥
L
2
2
+
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
2
∥
∂
x
ℓ
(
B
→
,
E
→
)
∥
L
∞
+
C
(
∥
∂
x
ϱ
∥
L
∞
+
∥
∂
x
u
→
∥
L
∞
+
∥
∂
x
ϑ
∥
L
∞
+
∥
∂
x
E
r
∥
L
∞
)
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
2
2
.
Then integrating with respect to time yields
∫
0
t

𝒮
ℓ
(
τ
)

d
τ
≤
C
sup
0
≤
τ
≤
t
∥
(
B
→

B
→
¯
)
(
τ
)
∥
L
∞
∫
0
t
∥
∂
x
ℓ
u
→
(
τ
)
∥
L
2
2
d
τ
+
C
sup
0
≤
τ
≤
t
∥
∂
x
ℓ
(
B
→
,
E
→
)
(
τ
)
∥
L
2
×
∫
0
t
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
(
τ
)
∥
L
∞
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
(
τ
)
∥
L
2
d
τ
+
C
sup
0
≤
τ
≤
t
{
∥
∂
x
ϱ
∥
L
∞
+
∥
∂
x
u
→
∥
L
∞
+
∥
∂
x
ϑ
∥
L
∞
+
∥
∂
x
E
r
∥
L
∞
(
τ
)
}
∫
0
t
∥
∂
x
ℓ
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
2
2
d
τ
≤
C
(
E
(
t
)
D
2
(
t
)
+
F
(
t
)
I
(
t
)
D
(
t
)
)
for any

ℓ

≤
d
. ∎
The above results, together with (3.6), allow us to derive the following energy bound.
Corollary 3.4.
Assume that the assumptions of Proposition 3.2 are satisfied. Then
(3.9)
∥
(
V

V
¯
)
(
t
)
∥
H
d
2
+
∫
0
t
∥
u
→
(
τ
)
∥
H
d
2
d
τ
≤
C
∥
(
V

V
¯
)
(
0
)
∥
H
d
2
+
C
(
E
(
t
)
D
(
t
)
2
+
F
(
t
)
I
(
t
)
D
(
t
)
)
.
Our goal is now to derive bounds for the integrals in the right and lefthand sides of equation (3.9). For this purpose we adapt the results of Ueda, Wang and Kawashima [20].
Lemma 3.5.
Under the assumptions of Theorem 2.1, and supposing that
d
≥
3
, we have the following estimate for any
ε
>
0
:
∫
0
t
(
∥
(
ϱ

ϱ
¯
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
(
τ
)
∥
H
d
2
+
∥
E
→
(
τ
)
∥
H
d

1
2
)
𝑑
τ
(3.10)
≤
ε
∫
0
t
∥
∂
x
B
→
(
τ
)
∥
H
d

2
2
d
τ
+
C
ε
{
∥
V
0

V
¯
∥
H
d

1
2
+
E
(
t
)
D
(
t
)
2
+
F
(
t
)
I
(
t
)
D
(
t
)
}
.
Proof.
We linearize the principal part of system (1.1)–(1.3) as follows:
(3.11)
∂
t
ϱ
+
ϱ
¯
div
x
u
→
=
g
1
,
(3.12)
∂
t
u
→
+
a
¯
1
∇
x
ϱ
+
a
¯
2
∇
x
ϑ
+
a
¯
3
∇
x
E
r
+
E
→
+
u
→
×
B
→
¯
+
ν
u
→
=
g
2
,
(3.13)
∂
t
ϑ
+
b
¯
1
div
x
u
→
+
b
¯
2
(
ϑ

ϑ
¯
)
=
g
3
,
(3.14)
∂
t
E
r
+
c
¯
1
div
x
u
→
+
c
¯
3
(
E
r

E
¯
r
)
=
g
4
,
(3.15)
∂
t
B
→
+
curl
x
E
→
=
0
,
(3.16)
∂
t
E
→

curl
x
B
→

ϱ
¯
u
→
=
g
5
,
with coefficients
a
1
(
ϱ
,
ϑ
)
=
p
ϱ
ϱ
,
a
2
(
ϱ
,
ϑ
)
=
p
ϑ
ϱ
,
a
3
(
ϱ
,
ϑ
)
=
1
3
ϱ
,
a
¯
j
=
a
j
(
ϱ
¯
,
ϑ
¯
)
,
b
1
(
ϱ
,
ϑ
)
=
ϑ
p
ϑ
ϱ
C
v
,
b
2
(
ϱ
,
ϑ
,
E
r
)
=
a
σ
a
ϱ
C
v
(
ϑ
2
+
ϑ
¯
2
)
(
ϑ
+
ϑ
¯
)
,
b
3
(
ϱ
,
ϑ
,
E
r
)
=
a
σ
a
ϱ
C
v
,
b
¯
j
=
b
j
(
ϱ
¯
,
ϑ
¯
)
,
c
1
(
ϱ
,
ϑ
,
E
r
)
=
4
3
E
r
,
c
2
(
ϱ
,
ϑ
,
E
r
)
=
a
σ
a
(
ϑ
2
+
ϑ
¯
2
)
(
ϑ
+
ϑ
¯
)
,
c
3
(
ϱ
,
ϑ
,
E
r
)
=
σ
a
,
c
¯
j
=
c
j
(
ϱ
¯
,
ϑ
¯
)
,
and sources
g
1
:=

{
u
→
⋅
∇
x
ϱ
+
(
ϱ

ϱ
¯
)
div
x
u
→
}
,
g
2
:=

{
(
u
→
⋅
∇
x
)
u
→
+
(
a
1

a
¯
1
)
∇
x
ϱ
+
(
a
2

a
¯
2
)
∇
x
ϑ
+
(
a
3

a
¯
3
)
∇
x
E
r
+
u
→
×
(
B
→

B
→
¯
)
}
,
g
3
:=

{
(
u
→
⋅
∇
x
)
ϑ
+
(
b
1

b
¯
1
)
div
x
u
→
+
(
b
2

b
¯
2
)
(
ϑ

ϑ
¯
)
+
b
3
(
E
r

E
¯
r
)
}
,
g
4
:=

{
(
u
→
⋅
∇
x
)
E
r
+
(
c
¯
1

c
1
)
div
x
u
→
+
c
2
(
ϑ

ϑ
¯
)
+
(
c
3

c
¯
3
)
(
E
r

E
¯
r
)
}
and
g
5
=
(
ϱ

ϱ
¯
)
u
→
.
By multiplying (3.11) by

a
¯
1
div
x
u
→
, (3.12) by
a
¯
1
∇
x
ϱ
+
a
¯
2
∇
x
ϑ
+
a
¯
3
∇
x
E
r
+
E
→
, (3.13) by

a
¯
2
div
x
u
→
+
ϑ

ϑ
¯
, (3.14) by

a
¯
3
div
x
u
→
+
E
r

E
¯
r
, (3.15) by
1
, (3.16) by
u
→
and summing up, we get
a
¯
1
(
∇
x
ϱ
u
→
t

ϱ
t
div
x
u
→
)
+
a
¯
2
(
∇
x
ϑ
u
→
t

ϑ
t
div
x
u
→
)
+
a
¯
3
(
∇
x
E
r
u
→
t

(
E
r
)
t
div
x
u
→
)
+
E
→
u
→
t
+
E
→
t
u
→
+
{
1
2
[
(
ϑ

ϑ
¯
)
2
+
(
E
r

E
¯
r
)
2
]
}
t
+
(
a
¯
1
∇
x
ϱ
+
a
¯
2
∇
x
ϑ
+
a
¯
3
∇
x
E
r
+
E
→
)
2
+
(
a
¯
1
∇
x
ϱ
+
a
¯
2
∇
x
ϑ
+
a
¯
3
∇
x
E
r
+
E
→
)
(
u
→
×
B
→
¯
+
ν
u
→
)
+
b
¯
2
(
ϑ

ϑ
¯
)
2
+
c
¯
3
(
E
r

E
¯
r
)
2
+
b
¯
1
(
ϑ

ϑ
¯
)
div
x
u
→
+
c
¯
1
(
E
r

E
¯
r
)
div
x
u
→
+
(
a
¯
3
c
¯
2

a
¯
2
b
¯
2
)
(
ϑ

ϑ
¯
)
div
x
u
→
(3.17)
+
(
a
¯
2
b
¯
3

a
¯
3
c
¯
3
)
(
E
r

E
¯
r
)
div
x
u
→

u
→
curl
x
B
→

ϱ
¯
u
→
2

(
div
x
u
→
)
2
[
a
¯
1
+
a
¯
2
+
a
¯
3
]
=
G
1
0
,
where
G
1
0
:=

a
¯
1
g
1
div
x
u
→
+
[
a
¯
1
∇
x
ϱ
+
a
¯
2
∇
x
ϑ
+
a
¯
3
∇
x
E
r
+
E
→
]
g
2

[
a
¯
2
+
ϑ

ϑ
¯
]
div
x
u
→
g
3

[
a
¯
3
+
E
r

E
¯
r
]
div
x
u
→
g
4
+
g
5
u
→
.
By rearranging the lefthand side of (3.17), we get
(3.18)
{
H
1
0
}
t
+
div
x
F
→
1
0
+
D
1
0
=
M
1
0
+
G
1
0
,
where
H
1
0
=

[
a
¯
1
(
ϱ

ϱ
¯
)
+
a
¯
2
(
ϑ

ϑ
¯
)
+
a
¯
3
(
E
r

E
¯
r
)
]
div
x
u
→
+
E
→
⋅
u
→
+
1
2
[
(
ϑ

ϑ
¯
)
2
+
(
E
r

E
¯
r
)
2
]
,
F
→
1
0
=
[
a
¯
1
(
ϱ

ϱ
¯
)
+
a
¯
2
(
ϑ

ϑ
¯
)
+
a
¯
3
(
E
r

E
¯
r
)
]
u
→
t

2
[
a
¯
1
(
ϱ

ϱ
¯
)
+
a
¯
2
(
ϑ

ϑ
¯
)
+
a
¯
3
(
E
r

E
¯
r
)
]
E
→
+
(
a
¯
3
c
¯
2

a
¯
2
b
¯
2
+
b
¯
1
)
(
ϑ

ϑ
¯
)
u
→
+
(
a
¯
2
b
¯
3

a
¯
3
c
¯
3
+
c
¯
1
)
(
E
r

E
¯
r
)
u
→
,
D
1
0
=
a
¯
1
2

∇
x
ϱ

2
+
a
¯
2
2

∇
x
ϑ

2
+
a
¯
3
2

∇
x
E
r

2
+

E
→

2
+
2
a
¯
1
(
ϱ

ϱ
¯
)
2
+
b
¯
2
(
ϑ

ϑ
¯
)
2
+
c
¯
3
(
E
r

E
¯
r
)
2
,
M
1
0
=

{
2
a
¯
1
a
¯
2
∇
x
ϱ
⋅
∇
x
ϑ
+
2
a
¯
1
a
¯
3
∇
x
ϱ
⋅
∇
x
E
r
+
2
a
¯
2
a
¯
3
∇
x
ϑ
⋅
∇
x
E
r
+
2
a
¯
2
(
ϱ

ϱ
¯
)
(
ϑ

ϑ
¯
)
+
2
a
¯
2
(
ϱ

ϱ
¯
)
(
E
r

E
¯
r
)
+
(
a
¯
1
∇
x
ϱ
+
a
¯
2
∇
x
ϑ
+
a
¯
3
∇
x
E
r
+
E
→
)
(
u
→
×
B
→
¯
+
ν
u
→
)

u
→
curl
x
B
→

ϱ
¯
u
→
2

(
div
x
u
→
)
2
[
a
¯
1
+
a
¯
2
+
a
¯
3
]

(
a
¯
3
c
¯
2

a
¯
2
b
¯
2
+
b
¯
1
)
∇
x
ϑ
⋅
u
→

(
a
¯
2
b
¯
3

a
¯
3
c
¯
3
+
c
¯
1
)
∇
x
E
r
⋅
u
→
}
.
Integrating (3.18) over space and using Young’s inequality yields
d
d
t
∫
ℝ
3
H
1
0
𝑑
x
+
C
(
∥
ϱ
∥
L
2
2
+
∥
∇
x
ϑ
∥
L
2
2
+
∥
∇
x
E
r
∥
L
2
2
+
∥
E
→
∥
L
2
2
+
∥
ϱ

ϱ
¯
∥
L
2
2
)
≤
ε
∥
∂
x
B
→
∥
L
2
2
+
C
ε
(
∥
u
→
∥
H
1
2
+
∥
ϑ

ϑ
¯
∥
H
1
2
+
∥
E
r

E
¯
r
∥
H
1
2
)
+
∫
ℝ
3

G
1
0

d
x
.
In fact, in the same way one obtains estimates for the derivatives of
V
. Namely, applying
∂
x
ℓ
to system (3.11)–(3.16), we get
(3.19)
{
H
1
ℓ
}
t
+
div
x
F
→
1
ℓ
+
D
1
ℓ
=
M
1
ℓ
+
G
1
ℓ
,
where
H
1
ℓ
=

[
a
¯
1
∂
x
ℓ
(
ϱ

ϱ
¯
)
+
a
¯
2
∂
x
ℓ
(
ϑ

ϑ
¯
)
+
a
¯
3
∂
x
ℓ
(
E
r

E
¯
r
)
]
div
x
∂
x
ℓ
u
→
+
∂
x
ℓ
E
→
⋅
∂
x
ℓ
u
→
+
1
2
[
(
∂
x
ℓ
ϑ
)
2
+
(
∂
x
ℓ
E
r
)
2
]
,
F
→
1
ℓ
=
[
a
¯
1
∂
x
ℓ
(
ϱ

ϱ
¯
)
+
a
¯
2
∂
x
ℓ
(
ϑ

ϑ
¯
)
+
a
¯
3
∂
x
ℓ
(
E
r

E
¯
r
)
]
u
→
t
+
(
a
¯
3
c
¯
2

a
¯
2
b
¯
2
+
b
¯
1
)
∂
x
ℓ
ϑ
∂
x
ℓ
u
→
+
(
a
¯
2
b
¯
3

a
¯
3
c
¯
3
+
c
¯
1
)
∂
x
ℓ
E
r
∂
x
ℓ
u
→

2
[
a
¯
1
∂
x
ℓ
(
ϱ

ϱ
¯
)
+
a
¯
2
∂
x
ℓ
(
ϑ

ϑ
¯
)
+
a
¯
3
∂
x
ℓ
(
E
r

E
¯
r
)
]
∂
x
ℓ
E
→
+
∂
x
ℓ
u
→
×
∂
x
ℓ
(
B
→

B
→
¯
)
,
D
1
ℓ
=
a
¯
1
2

∇
x
∂
x
ℓ
ϱ

2
+
a
¯
2
2

∂
x
ℓ
∇
x
ϑ

2
+
a
¯
3
2

∂
x
ℓ
∇
x
E
r

2
+

∂
x
ℓ
E
→

2
+
2
a
¯
1
(
∂
x
ℓ
(
ϱ

ϱ
¯
)
)
2
+
b
¯
2
(
∂
x
ℓ
ϑ
)
2
+
c
¯
3
(
∂
x
ℓ
E
r
)
2
,
M
1
ℓ
=

{
2
a
¯
1
a
¯
2
∇
x
∂
x
ℓ
ϱ
⋅
∇
x
∂
x
ℓ
ϑ
+
2
a
¯
1
a
¯
3
∇
x
∂
x
ℓ
ϱ
⋅
∇
x
∂
x
ℓ
E
r
+
2
a
¯
2
a
¯
3
∇
x
∂
x
ℓ
ϑ
⋅
∇
x
∂
x
ℓ
E
r
+
2
a
¯
2
∂
x
ℓ
(
ϱ

ϱ
¯
)
∂
x
ℓ
(
ϑ

ϑ
¯
)
+
2
a
¯
2
∂
x
ℓ
(
ϱ

ϱ
¯
)
∂
x
ℓ
(
E
r

E
¯
r
)
+
(
a
¯
1
∇
x
∂
x
ℓ
ϱ
+
a
¯
2
∇
x
∂
x
ℓ
ϑ
+
a
¯
3
∇
x
∂
x
ℓ
E
r
+
∂
x
ℓ
E
→
)
(
∂
x
ℓ
u
→
×
B
→
¯
+
ν
∂
x
ℓ
u
→
)

(
a
¯
3
c
¯
2

a
¯
2
b
¯
2
+
b
¯
1
)
∇
x
∂
x
ℓ
ϑ
⋅
∂
x
ℓ
u
→

(
a
¯
2
b
¯
3

a
¯
3
c
¯
3
+
c
¯
1
)
∇
x
∂
x
ℓ
E
r
⋅
∂
x
ℓ
u
→

curl
x
∂
x
ℓ
u
→
∂
x
ℓ
(
B
→

B
→
¯
)

ϱ
¯
(
∂
x
ℓ
u
→
)
2

(
div
x
∂
x
ℓ
u
→
)
2
[
a
¯
1
+
a
¯
2
+
a
¯
3
]
}
,
G
1
ℓ
=

a
¯
1
∂
x
ℓ
g
1
div
x
∂
x
ℓ
u
→
+
[
a
¯
1
∇
x
∂
x
ℓ
ϱ
+
a
¯
2
∇
x
∂
x
ℓ
ϑ
+
a
¯
3
∇
x
∂
x
ℓ
E
r
+
∂
x
ℓ
E
→
]
∂
x
ℓ
g
2

a
¯
2
∂
x
ℓ
g
3
div
x
∂
x
ℓ
u
→

a
¯
3
∂
x
ℓ
g
4
div
x
∂
x
ℓ
u
→
+
∂
x
ℓ
g
5
∂
x
ℓ
u
→
+
∂
x
ℓ
g
3
∂
x
ℓ
ϑ
+
∂
x
ℓ
g
4
∂
x
ℓ
E
r
.
Integrating (3.19) over space and time yields
∫
ℝ
3
H
1
ℓ
(
t
)
𝑑
x

∫
ℝ
3
H
1
ℓ
(
0
)
𝑑
x
+
C
∫
0
t
(
∥
∇
x
∂
x
ℓ
ϱ
∥
L
2
2
+
∥
∇
x
∂
x
ℓ
ϑ
∥
L
2
2
+
∥
∇
x
∂
x
ℓ
E
r
∥
L
2
2
+
∥
∂
x
ℓ
E
→
∥
L
2
2
)
𝑑
τ
+
C
∫
0
t
(
∥
∂
x
ℓ
(
ϱ

ϱ
¯
)
∥
L
2
2
+
∥
∂
x
ℓ
(
ϑ

ϑ
¯
)
∥
L
2
2
+
∥
∂
x
ℓ
(
E
r

E
¯
r
)
∥
L
2
2
)
𝑑
τ
(3.20)
≤
ε
∫
0
t
∥
∂
x
ℓ
(
B
→

B
→
¯
)
∥
L
2
2
d
τ
+
C
ε
∫
0
t
(
∥
∂
x
ℓ
u
→
∥
H
1
2
+
∥
∂
x
ℓ
(
ϑ

ϑ
¯
)
∥
H
1
2
+
∥
∂
x
ℓ
(
E
r

E
¯
r
)
∥
H
1
2
)
d
τ
+
∫
0
t
∫
ℝ
3

G
1
ℓ

d
x
d
τ
.
By observing that

∫
ℝ
3
H
1
ℓ
(
t
)
𝑑
x

≤
C
(
∥
∂
x
ℓ
(
ϱ

ϱ
¯
)
∥
L
2
2
+
∥
∂
x
ℓ
(
ϑ

ϑ
¯
)
∥
L
2
2
+
∥
∂
x
ℓ
(
E
r

E
¯
r
)
∥
L
2
2
+
∥
∂
x
ℓ
u
→
∥
H
1
2
)
,
and summing (3.20) on
ℓ
for
1
≤
ℓ
≤
d

1
, we get
∫
0
t
(
∥
(
ϱ

ϱ
¯
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
(
τ
)
∥
H
d
2
+
∥
E
→
(
τ
)
∥
H
d

1
2
)
d
τ
≤
C
ε
∥
(
V

V
¯
)
(
0
)
∥
H
d
2
+
ε
∫
0
t
∥
∂
x
B
→
(
τ
)
∥
H
d

2
2
d
τ
+
C
ε
(
E
(
t
)
D
2
(
t
)
+
F
(
t
)
I
(
t
)
D
(
t
)
)
+
∑

ℓ

=
1
d

1
∫
0
t
∫
ℝ
3

G
1
ℓ
(
τ
)

d
x
d
τ
,
where we used Corollary 3.4.
Let us estimate the last integral in (3.20). We have
(3.21)
{
∥
∂
x
ℓ
g
1
∥
L
2
≤
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
+
1
(
ϱ
,
u
→
)
∥
L
2
,
∥
∂
x
ℓ
g
2
∥
L
2
≤
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
+
1
(
ϱ
,
u
→
)
∥
L
2
+
C
∥
B
→

B
→
¯
∥
L
∞
∥
∂
x
ℓ
u
→
∥
L
2
+
C
∥
∂
x
ℓ
(
B
→

B
→
¯
)
∥
L
2
∥
u
→
∥
L
∞
,
∥
∂
x
ℓ
g
3
∥
L
2
≤
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
+
1
(
ϱ
,
u
→
)
∥
L
2
+
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
+
2
(
ϑ
,
E
r
)
∥
L
2
,
∥
∂
x
ℓ
g
4
∥
L
2
≤
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
+
1
(
ϱ
,
u
→
)
∥
L
2
+
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
+
2
(
ϑ
,
E
r
)
∥
L
2
,
∥
∂
x
ℓ
g
5
∥
L
2
≤
C
∥
(
ϱ

ϱ
¯
,
u
→
,
ϑ

ϑ
¯
,
E
r

E
¯
r
)
∥
L
∞
∥
∂
x
ℓ
(
ϱ
,
u
→
)
∥
L
2
for
1
≤

ℓ

≤
d

1
. Then
∫
0
t
∫
ℝ
3

G
1
ℓ
(
τ
)

d
x
d
τ
≤
C
∥
∂
x
ℓ
+
1
u
→
∥
L
2
∥
∂
x
ℓ
g
1
∥
L
2
+
C
(
∥
∂
x
ℓ
+
1
ϱ
∥
L
2
+
∥
∂
x
ℓ
+
1
ϑ
∥
L
2
+
∥
∂
x
ℓ
+
1
E
r
∥
L
2
+
∥
∂
x
ℓ
E
→
∥
L
2
)
∥
∂
x
ℓ
g
2
∥
L
2
+
C
∥
∂
x
ℓ
+
1
u
→
∥
L
2
∥
∂
x
ℓ
g
3
∥
L
2
+
C
∥
∂
x
ℓ
+
1
u
→
∥
L
2
∥
∂
x
ℓ
g
4
∥
L
2
+
C
∥
∂
x
ℓ
u
→
∥
L
2
∥
∂
x
ℓ
g
5
∥
L
2
.
Plugging bounds (3.21) into the last inequality gives
∑

ℓ

=
1
d

1
∫
0
t
∫
ℝ
3

G
1
ℓ
(
τ
)

d
x
d
τ
≤
C
E
(
t
)
D
2
(
t
)
,
which completes the proof of Lemma 3.5. ∎
Finally, we check from [20, Lemma 4.4] that the following result for the Maxwell’s system holds true for our system with a similar proof.
Lemma 3.6.
Under the assumptions of Theorem 2.1, and supposing that
d
≥
3
, for any
ε
>
0
, the following estimate (here, we set
V
=
(
ϱ
,
u
→
,
ϑ
,
E
r
,
B
→
,
E
→
)
T
) holds:
(3.22)
∫
0
t
∥
∂
x
B
→
(
τ
)
∥
H
s

2
2
d
τ
≤
C
∥
V
0

V
¯
∥
H
s

1
2
+
C
∫
0
t
∥
∂
x
E
→
(
τ
)
∥
H
s

2
2
d
τ
+
C
(
E
(
t
)
D
(
t
)
2
+
F
(
t
)
I
(
t
)
D
(
t
)
)
.
Proof.
By applying
∂
x
ℓ
to (1.5) and (1.6), multiplying, respectively, by

curl
x
∂
x
ℓ
B
→
and
curl
x
∂
x
ℓ
E
→
, and adding the resulting equations, we get

(
∂
x
ℓ
E
→
⋅
curl
x
∂
x
ℓ
B
→
)
t
+

curl
x
∂
x
ℓ
B
→

2

div
x
(
∂
x
ℓ
E
→
×
∂
x
ℓ
B
→
t
)
=
M
2
ℓ
+
G
2
ℓ
,
where
M
2
ℓ
=

ϱ
¯
∂
x
ℓ
u
→
⋅
curl
x
∂
x
ℓ
B
→
+

curl
x
∂
x
ℓ
E
→

2
and
G
2
ℓ
=

∂
x
ℓ
(
(
ϱ

ϱ
¯
)
u
→
)
⋅
curl
x
∂
x
ℓ
B
→
.
Integrating in space gives

d
d
t
∫
ℝ
3
∂
x
ℓ
E
→
⋅
curl
x
∂
x
ℓ
B
→
d
x
+
C
∥
curl
x
∂
x
ℓ
B
→
∥
L
2
2
≤
∥
curl
x
∂
x
ℓ
E
→
∥