Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift
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178 B. Ducomet and
ˇ
S. Neˇcasov´a
where
inf
x∈T
0,δ
≤ inf
x∈T
ε
(t, x) ≤ sup
x∈T
ε
(t, x) ≤ sup
x∈T
0,δ
,
in particular,
inf
x∈T
0,δ
≤ inf
x∈T
(t, x) ≤ sup
x∈T
(t, x) ≤ sup
x∈T
0,δ
. (6.3)
Consequently, employing once more the energy inequality (5.14), we conclude
that
u
ε
→ u in L
∞
(0,T; L
2
(T ; R
3
)).
Clearly, the limit density satisfies the equation of continuity
∂
t
+div
x
([u]
δ
)=dΔ in (0,T) ×R
3
(6.4)
provided that has been extended to be
f
outside Ω. Moreover, in accordance
with Proposition 6.1 and hypothesis (5.7),
=
f
on the set
(0,T) × Ω
\∪
t∈[0,T ]
∪
n
i=1
η(t, [B
i
]
δ
), (6.5)
where η solves
∂
t
η(t, x)=[u]
δ
(t, η(t, x)),η(0,x)=x (6.6)
and [B
i
]
δ
denotes the δ-kernel introduced in Section 3.
6.1. Identifying the position of the rigid bodies
In order to identify the position of the rigid bodies, we proceed through several
steps. We will give only the main points since we apply a technique similar to that
used in the work of Feireisl et al., see [17].
• Step 1: We have to prove that
D[u]=0a.a.ontheset ∪
t∈[0,T ]
∪
n
i=1
]η(t, [B
i
]
ω
)[
δ
for any ω>δ, (6.7)
where η is determined by (6.6), and the symbols ] ·[, [·] are specified in (3.9).
Note that the kernels [B
i
]
ω
as well as their images η(t, [B
i
]
ω
) are non-empty
connected open sets when 0 <δ<ω<δ
0
/2, where δ
0
has been introduced
in hypothesis (3.1).
• Step 2: In accordance with (6.7), the limit velocity u coincides with a rigid
velocity field u
B
i
on the δ-neighborhood of each of the sets η(t, [B
i
]
ω
), ω>δ,
i =1,...,n; in particular, we deduce that
u(t, x)=u
B
i
(t, x)=[u]
δ
(t, x)fort ∈ [0,T],
x ∈ η(t, [B
i
]
δ
),i=1,...,n.
(6.8)
Note that rigid velocity fields coincide with their regularization, here
[u
B
i
]
δ
= u
B
i
.
On the Motion of Several Rigid Bodies 179
• Step 3: Letting ε → 0 in the momentum equation (5.15) we deduce that
T
0
Ω
u · ∂
t
ϕ + (u ⊗ [u]
δ
):∇
x
ϕ dx dt + d
T
0
Ω
∇ρ∇u ϕ dx dt (6.9)
=
T
0
Ω
S : D[ϕ]dx dt −
T
0
T
∇
x
F · ϕ dx dt −
T
0,δ
u
0
·ϕ(0, ·)dx
for any test function ϕ ∈ C
1
([0,T) × Ω),
ϕ(t, ·) ∈ [RM](t),
where
[RM](t)={φ ∈ C
1
(Ω) | div
x
φ =0inΩ,φ= 0 on a neighborhood of ∂Ω,
D[φ] = 0 on a neighborhood of ∪
n
i=1
B
i
(t)},
with
[B
i
(t)=η
i
(t, B
i
),i=1,...,n.
Indeed, because the η
i
are isometries, it implies that
]η
i
(t, [B
i
]
δ
)[
δ
= η
i
(t, B
i
),i=1,...,n.
Consequently, [μ
ε
]
δ
converges uniformly locally to 1 in the complement of
∪
n
i=1
B
i
(t) for any t ∈ [0,T]. It yields (6.9), in which the bar denotes weak limits:
u
ε
⊗ [u
ε
]
δ
→ u ⊗[u]
δ
weakly in L
2
(0,T; L
2
(Ω; R
3
)),
∇ρ
→∇ρ strongly in L
2
(Q),
∇ρ
∇u
→ ρ∇u in D
((0,T) × Ω)
(6.10)
and from a monotonicity argument together with results of Frehse et al. [20, 21]
it follows that
S
ε
→ S weakly in L
p
((0,T) ×Ω; R
3×3
),
1
p
+
1
p
=1. (6.11)
6.2. Convergence of the selfgravitating force
From the strong convergence of density in C(0,T,C
2,ν
)andthe∇ρ in L
2
((0,T) ×
Ω) there follows the weak * convergence in L
2
(0,T,L
∞
(Ω)) of the term
−
T
0
Ω
Gρ∇
R
3
ρ(t, y)
|x −y|
dy
ϕ.
180 B. Ducomet and
ˇ
S. Neˇcasov´a
6.3. Pointwise convergence of the velocities
Our aim is to identify the weak limit in (6.10); more specifically, we show that
u
ε
→ u in L
2
(0,T; L
2
(Ω; R
3
)). (6.12)
Note that the main difficulty here is the possible existence of oscillations of the
velocity fields in time.
We know from the result of Starovoitov [38, Theorem 3.1], that collisions
between two rigid objects are eliminated, because the fluid is incompressible, and
that the velocity gradients are bounded in the Lebesgue space L
p
,withp ≥ 4.
Although originally stated for only one body in a bounded domain, it is easy to
see that this result extends directly to the case of several bodies. Here we will use
the terminology introduced in Section 3,
d(∪
n
i=1
B
i
(t)) = d(t) > 0 uniformly for t ∈ [0,T], (6.13)
and, in agreement with Proposition 8.1,
d(∪
n
i=1
B
ε
i
(t)) = d
ε
→ d in C[0,T], (6.14)
wherewehavesetB
ε
i
(t)=η
ε
(t, B
i
).
The absence of contacts facilitates considerably the proof of compactness of
the velocity fields that can be carried over by means of the same method as in [34].
To begin with, as
B
ε
i
(t)
b
→ B
i
(t) uniformly with respect to t ∈ [0,T],i=1,...,n,
we have, for any fixed σ>0,
B
i
(t) ⊂]B
ε
i
(t)[
σ
, B
ε
i
(t) ⊂]B
i
(t)[
σ
, for all t ∈ [0,T],i=1,...,n,
and all ε<ε
0
(σ) small enough.
Lemma 6.1. Given a family of smooth open sets {B
i
}
n
i=1
⊂ Ω, 0 <k<1/2,there
exists a function h :(0,σ
0
) → R
+
,withh(σ) → 0 when σ → 0, such that, for
arbitrary v ∈ V
1,p
:
#
#
#
v−P
k
∪
n
i=1
]B
i
[
σ
v
#
#
#
W
1,k
(Ω;R
3
)
≤ c
D(v)
L
2
(∪
n
i=1
B
i
;R
3×3
)
+h(σ)v
W
1,p
(Ω;R
3
)
(6.15)
with an absolute constant c<∞. Moreover, h and c are independent of the position
of B
i
inside Ω as long as d[∪
n
i=1
B
i
] > 2σ
0
.
Proof. See [17].
At this stage, we use a local-in-time Lions-Aubin argument in order to show
the following:
Lemma 6.2. For al l σ>0 sufficiently small, and 0 <k<1/2, we have
lim
ε→0
T
0
Ω
ε
u
ε
·P
k
∪
n
i=1
]B
i
(t)[
σ
[u
ε
]dx dt =
T
0
Ω
u·P
k
∪
n
i=1
]B
i
(t)[
σ
[u]dx dt.
Proof. See [17].
On the Motion of Several Rigid Bodies 181
Combining Lemmas 6.1, 6.2 we deduce
lim
ε→0
T
0
Ω
ε
|u
ε
|
2
dx dt =
T
0
Ω
ρ|u|
2
dx dt (6.16)
yielding the desired conclusion (6.12).
6.4. Compactness of the velocity gradients
Our ultimate goal is to establish strong convergence of the velocity gradients in
the “fluid” part of the cylinder (0,T)×Ω. To this end, we consider equation (5.15)
on the set I ×A,whereI ⊂ (0,T)isanintervalandA ⊂ Ω is a ball. In accordance
with (6.5), we may assume that =
f
in I × A.Inparticular,wehave
T
0
Ω
f
u
ε
· ∂
t
ϕ +(
f
u
ε
⊗ [u
ε
]
δ
−S[u
ε
]) : ∇
x
ϕ dx dt
= −
T
0
Ω
ρ
f
∇
R
3
ρ
f
|x − y|
ϕdxdt
(6.17)
for any test function
ϕ ∈D(I × A; R
3
), div
x
ϕ =0.
At this stage, the problem must be localized by separating the fluid part from
the rigid bodies. To this end, we introduce a “local” pressure
p = p
reg
+ ∂
t
p
harm
, (6.18)
where p
reg
enjoys the same regularity properties as the sum of the convective and
viscous terms, while p
harm
is a harmonic function. The basic idea of the concept of
local pressure was developed by Wolf [44, Theorem 2.6]. A similar global result was
proved by H. Koch and V. Solonnikov [32]. Note, however, that our construction
gives a result different from that of Wolf [44]. In particular, the regular part is
given in the form of Riesz transforms suitable for application to problems with
non-standard growth conditions.
Lemma 6.3. Let I =(T
1
,T
2
) be a time interval and A ⊂ R
3
a domain with
regular C
2+μ
boundary. Assume that U ∈ L
∞
(I; L
2
(A; R
3
)), div
x
U =0,and
T ∈ L
q
(I × A, R
3×3
), 1 <q<2, satisfy the integral identity
I
A
U · ∂
t
ϕ + T : ∇
x
ϕ
dx dt = 0 (6.19)
for all ϕ ∈D(I × A; R
3
), div
x
ϕ =0.
Then there exist two functions
p
reg
∈ L
q
(I × A),
p
harm
∈ L
∞
(I; L
q
(Ω)), Δ
x
p
harm
=0in D
(I ×A),
B
p
harm
(t, ·)dx =0
satisfying
I
A
U ·∂
t
ϕ + T : ∇
x
ϕ
dx dt =
I
A
p
reg
div
x
ϕ + p
harm
∂
t
div
x
ϕ
dx dt
182 B. Ducomet and
ˇ
S. Neˇcasov´a
for any ϕ ∈D(I ×A; R
3
). In addition,
p
reg
L
q
((0,T )×A)
≤ c(q)T
L
q
(I×A;R
3
)
, (6.20)
p
harm
L
∞
(I;L
q
(Ω))
≤ c(q, I, A)
T
L
q
(I×A;R
3
)
+ U
L
∞
(0,T ;L
2
(Ω;R
3
))
. (6.21)
Proof. See [17].
Accordingly, for any ε>0, there exist two scalar functions p
ε
reg
, p
ε
harm
such
that
p
ε
reg
∈ L
p
(I; L
p
(A)),p
ε
harm
∈ L
∞
(I; L
p
(A)), are uniformly bounded (6.22)
and
T
0
Ω
1
(
f
u
ε
+ ∇
x
p
ε
harm
) · ∂
t
ϕ
2
dxdxdt
+
T
0
Ω
1
f
u
ε
⊗ [u
ε
]
δ
−S[u
ε
]+p
ε
reg
I
∇
x
ϕ
2
dxdxdt
+
T
0
Ω
1
ρ
f
R
3
ρ
f
|x −y|
: ∇
x
ϕ
2
dx dt =0
(6.23)
for any test function ϕ ∈D(I × A; R
3
).
Moreover,
Δp
harm
=0,
A
p
ε
harm
(t, ·) dx =0, ∀t ∈ I.
Consequently, the standard elliptic theory implies that p
ε
harm
is uniformly bounded
in L
∞
(I; W
1,2
loc
(A)). The standard Lions-Aubin argument yields
f
u
ε
+ ∇
x
p
ε
harm
→
f
u + ∇
x
p
harm
in L
2
(I; L
2
(A
; R
3
)), (6.24)
for arbitrary A
⊂⊂ A, where p
harm
is a harmonic function in x.
On the other hand, by virtue of (6.12), the velocity field {u
ε
}
ε>0
is precom-
pact in L
2
(0,T; L
2
(Ω; R
3
)); whence we are allowed to conclude that
∇
x
p
ε
harm
→∇
x
p
harm
in L
2
(I; W
1,2
(A
; R
3
)). (6.25)
As the argument is valid for any A
, letting ε → 0 in (6.23) we get
T
0
Ω
1
(
f
u + ∇
x
p
harm
) · ∂
t
ϕ +(
f
u ⊗ [u]
δ
−S[u]+p
reg
I):∇
x
ϕ
2
dx dt =0
(6.26)
for any test function ϕ ∈D(I × A; R
3
), where
S[u
ε
] → S[u]weaklyinL
p
(0,T; L
p
(Ω; R
3×3
sym
)),
and
p
ε
reg
→ p
reg
weakly in L
p
(I × A).
Finally, taking
φ
ε
= ψ(t)r(x)(
f
u
ε
+ ∇
x
p
ε
harm
),ψ∈D(I),r∈D(A),
On the Motion of Several Rigid Bodies 183
and
φ = ψ(t)r(x)(
f
u + ∇
x
p
harm
)
as a test function in (6.23), and (6.26), respectively, and letting ε → 0 we deduce
the desired conclusion
lim
ε→0
T
0
Ω
ψr S[u
ε
]:∇
x
[u
ε
]dx dt =
T
0
Ω
ψr S[u]:∇
x
[u]dx dt.
This yields, by means of the standard monotonicity argument,
S[u
ε
] → S[u]a.e.inI × A. (6.27)
Indeed, we have
T
0
Ω
f
u
ε
+ ∇
x
p
ε
harm
· ∂
t
φ
ε
dx dt =
T
0
Ω
1
2
"
"
"
f
u
ε
+ ∇
x
p
ε
harm
"
"
"
2
r∂
t
ψ dx dt
→
T
0
Ω
1
2
"
"
"
f
u + ∇
x
p
harm
"
"
"
2
r∂
t
ψ dx dt =
T
0
Ω
f
u + ∇
x
p
harm
·∂
t
φ dx dt,
while
T
0
Ω
p
ε
reg
div
x
φ
ε
dx dt =
T
0
Ω
ψp
ε
reg
∇
x
r · (
f
u
ε
+ ∇
x
p
ε
harm
)dx dt
→
T
0
Ω
ψp
reg
∇
x
r · (
f
u + ∇
x
p
harm
)dx dt =
T
0
Ω
p
reg
div
x
φ dx dt
and
T
0
Ω
ρ
f
R
3
ρ
f
|x −y|
: ∇
x
φ
dt →
T
0
Ω
ρ
f
R
3
ρ
f
|x − y|
: ∇
x
φ dt
as ε → 0.
6.5. Conclusion
In accordance with the results obtained in the preceding two sections, relation
(6.9) reduces to
T
0
Ω
u ·∂
t
ϕ + (u ⊗[u]
δ
):∇
x
ϕ dx dt +
T
0
Ω
d∇ρ∇u dx dt (6.28)
=
T
0
Ω
S[u]:D[ϕ]dx dt −
T
0
T
∇
x
F · ϕ dx dt −
T
0,δ
u
0
·ϕ(0, ·)dx
for any test function ϕ ∈ C
1
([0,T) × Ω),
ϕ(t, ·) ∈ [RM](t),
where
[RM](t)={φ ∈ C
1
(Ω) | div
x
φ =0inΩ,φ= 0 on a neighborhood of ∂Ω,
D[φ] = 0 on a neighborhood of ∪
n
i=1
B
i
(t)},
with
B
i
(t)=η
i
(t, B
i
),i=1,...,n.
184 B. Ducomet and
ˇ
S. Neˇcasov´a
Moreover, the limit solution satisfies the energy inequality
Ω
1
2
|u|
2
(τ)dx +
τ
s
Ω
S : D(u)dx dt
≤
T
1
2
|u|
2
(s)dx +
τ
s
Ω
∇
x
F · u dx dt
(6.29)
for any τ and a.a. s ∈ (0,T) including s =0.
7. The passage to the limit for d → 0 and δ → 0
Our final goal is to let d → 0, δ → 0 in the system of equations
∂
t
+div
x
([u]
δ
)=dΔρ in (0,T) × R
3
, (7.1)
(6.4), (6.22) as well as in the associated family of isometries {η
i
}
n
i=1
.
First we are passing with d → 0. Let us denote by {
d
, u
d
, {η
d
i
}
n
i=1
}
d>0
the
corresponding solutions, the applying the results from [18] we get
d∇ρ
d
∇u
d
→ 0inL
1
((0,T) × Ω),
dΔρ
d
→ 0inL
2
(0,T; W
−1,2
(Ω)).
Let us denote by {
δ
, u
δ
, {η
δ
i
}
n
i=1
}
δ>0
the corresponding solutions constructed in
the previous section.
To begin with, the theory of transport equations developed by DiPerna and
P.-L. Lions [10] can be used in order to show that
δ
→ in C([0,T]; L
1
(Ω)). (7.2)
In order to see this, observe first that the initial data
B
i
,δ
in (5.6) can be taken
in such a way that
B
i
,δ
L
∞
(Ω)
≤ c,
f
+
B
i
,δ
→
B
i
as δ → 0inL
1
(Ω),i=1,...,n,
where {
B
i
}
n
i=1
are the initial distributions of the mass of the rigid bodies in
Theorem 4.1.
In addition, by virtue of the energy inequality (6.29), we have
u
δ
→ u weakly in L
p
(0,T; W
1,p
(Ω; R
3
))
where both u
δ
as well as the limit velocity u are solenoidal. In particular, the
continuity equation (7.1) reduces to a transport equation
∂
t
+ u ·∇
x
=0,
for which the abstract theory developed by DiPerna and P.-L. Lions [10]
yields (7.2).
The rest of the convergence proof can be done repeating step by step the
arguments of the preceding section.
On the Motion of Several Rigid Bodies 185
8. An alternative proof
8.1. Definition of weak solution II
We give hereafter another formulation of a weak solution.
Given initial conditions H
0i
= χ
B
i
(0)
, ρ
0
= ρ
B
i
H
0i
+ ρ
f
(1 − H
0i
)andu =
u
0
∈RM(0), find (x, t) → (ρ(x, t),u(x, t),H
i
(x, t)) such that
u ∈ L
∞
(0,T,L
2
(Ω)) ∩ L
p
(0,T,W
1,p
(Ω)),
H
i
,ρ∈ C(0,T,L
q
(Ω)), for all 1 <q<∞,
<
(8.1)
T
0
Ω
ρu · ∂
t
ϕ + ρu ⊗ u : ∇
x
[ϕ] − S : D[ϕ]
dxdt
= −
T
0
Ω
1
ρG∇
x
!
i
R
3
ρ
|x −y|
dy · ϕ
2
dxdt −
Ω
ρ
0
u
0
· ϕ dx
(8.2)
ϕ ∈ C
1
([0,T) ×
¯
Ω),ϕ(t, ·) ∈R(t), (8.3)
T
0
Ω
ρ∂
t
φ + ρu ·∇
x
φ
dxdt = −
Ω
ρ
0
φdx, φ ∈ C
1
([0,T) ×
¯
Ω), (8.4)
T
0
Ω
H
i
∂φ
∂t
+ H
i
u∇φ
dxdt +
Ω
H
0i
φ(0)dx =0. (8.5)
Here we have again used that ρ = ρ
B
i
H
i
+ ρ
f
(1 − H
i
).
Theorem 8.1. Let the initial position of the rigid bodies be given through a family
of open sets
B
i
⊂ Ω ⊂ R
3
, B
i
diffeomorphic to the unit ball for i =1,...,n,
where both ∂[B
i
], i =1,...,n,and∂Ω belong to the regularity class specified in
(3.1), (3.2). In addition, suppose that
dist[B
i
, B
j
] > 0 for i = j, dist[B
i
,R
3
\ Ω] > 0 for any i =1,...,n
and we assume that ∂Ω and ∂B
i
, i =1,...,n belong to C
2,ν
. Furthermore, let the
viscous stress tensor S satisfy hypotheses (1.1)–(1.3),withp ≥ 4.
Finally, let the initial distribution of the density be given as
0
=
f
=const> 0 in Ω \∪
n
i=1
B
i
,
B
i
on B
i
, where
B
i
∈ L
∞
(Ω), ess inf
B
i
B
i
> 0,i=1,...,n,
while
u
0
∈ L
2
(Ω; R
3
), div
x
u
0
=0in D
(Ω), D[u
0
]=0in D
(B
i
; R
3×3
) for i =1,...,n.
Then there exist a density function ,
∈ C([0,T]; L
1
(Ω)), 0 < ess inf
Ω
(t, ·) ≤ ess sup
Ω
(t, ·) < ∞ for all t ∈ [0,T],
a family of isometries {η
i
(t, ·)}
n
i=1
, η
i
(0, ·)=I, and a velocity field u,
u ∈ C
weak
([0,T]; L
2
(Ω; R
3
)) ∩L
p
(0,T; W
1,p
0
(Ω; R
3
)),
186 B. Ducomet and
ˇ
S. Neˇcasov´a
compatible with {η
i
}
n
i=1
in the sense specified in (3.7), (3.8), such that , u satisfy
the integral identity (8.3) for any test function ϕ ∈ C
1
([0,T)×R
3
), and the integral
identity (8.4) for any ϕ satisfying (8.5).
We introduce an − δ scheme; for the penalization part see [2] or [19].
∂
t
ε
+div
x
(
ε
[u]
δ
)=dΔρ
, (8.6)
∂
t
(
ε
u
ε
)+div
x
(
ε
u
ε
⊗ [u]
δ
)+∇
x
P
+ d∇ρ
∇u
=div
x
([μ]S
ε
)+
ε
∇
x
F
ε
− χ
ε
u
ε
+
1
ρ
ε
H
i
ε
(u
ε
−
˜
u
ε
),
(8.7)
∂
t
H
i
ε
+div
x
(
˜
u
ε
H
i
ε
)=0, (8.8)
div
x
u
ε
=0, (8.9)
˜
u
ε
=
1
M
ε
Ω
ρ
ε
u
ε
H
i
ε
dx +
J
−1
ε
Ω
ρ
ε
(r
ε
×u
ε
)H
i
ε
dx
× R (8.10)
where
[u]
δ
= σ
δ
∗ u (spatial convolution), 0 <δ<δ
0
,
with
σ
δ
(x)=
1
δ
3
σ
|x|
δ
, (8.11)
σ ∈D(−1, 1),σ(z) > 0for − 1 <z<1,σ(z)=σ(−z),
1
−1
σ(z)dz =1,
M
ε
=
Ω
ρ
ε
H
i
ε
dx, |B
i
(t)| = |B
i
(0)|,
since ˜u is divergence free and H
i
vanishes on ∂Ω.
The inertia tensor is defined by
J
ε
=
Ω
ρ
ε
H
i
ε
(r
2
ε
I −r
ε
×r
ε
)dx
with r = x − x
G
= x −
Ω
ρ
ε
H
i
ε
.Fora ∈ R
3
−{0},
a
T
J
ε
a =
Ω
ρ
ε
|r
ε
× a|
2
dx ≥ min(ρ
B
i
,ρ
f
)
B
i
(t)
|r
ε
×a|
2
dx.
Moreover, we supplemented the system with the initial conditions
ρ(0,.)=ρ
0,δ
= ρ
f
+
n
!
i=1
ρ
B
i
,δ
,
where
ρ
B
i
∈D(B
i
),ρ
B
i
,δ
= 0 whenever dist[x, ∂B
i
] <δ,i=1,...,n.
Finally, we add the homogeneous Neumann boundary condition
∇ρ · n =0on∂Ω ∪
;
n
i=1
∂B
i
,
and we assume the following regularity of data
ρ
0,δ
∈ C
2,ν
(Ω).
On the Motion of Several Rigid Bodies 187
8.2. limit
For fixed δ we identify the limit problem for → 0. We will show, in this limit,
the strong convergence of u
ε
in L
2
of u and the strong convergence of H
i
in
C(0,T; L
q
(Ω)) for all q ≥ 1. The other part (strong convergence of the gradient of
the velocity field) is the same as in the previous proof.
Proposition 8.1. Let ξ be a rigid velocity field, i.e., such that ξ(x)=V + ω ×r(x)
for some constant vectors V ∈ R
3
and ω ∈ R
3
.Then if (˜u
ε
) is defined by (8.10)
the identity
Ω
ρ
H
i
(u
− ˜u
ε
) ·ξdx =0
is satisfied.
Proof. See [2].
8.2.1. Estimates for transport equations and momentum equations. Using stan-
dard estimates for transport equations, we get
ρ
ε
,H
i
ε
∈ L
∞
(0,T,L
∞
(Ω)).
Moreover
ρ
min
:= min(ρ
s
,ρ
f
) ≤ ρ
ε
(x, t) ≤ max(ρ
s
,ρ
f
),
H
i
ε
∈{0, 1} a.e. x ∈ Ω.
Multiplying the momentum equation by u
ε
we get
T
∂
t
(
ε
u
ε
)+div
x
(
ε
u
ε
⊗ [u
ε
]
δ
)
u
ε
dx
=
T
div
x
([μ]S
ε
)+
ε
∇
R
3
ρ
ε
|x −y|
dy
u
ε
dx
+
T
−χ
ε
u
ε
+
1
ρ
ε
H
ε
i
(u
ε
−
˜
u
ε
)
u
ε
dx.
(8.12)
Since
T
ρ
ε
H
i
ε
(u
ε
−
˜
u
ε
) ·u
ε
dx =0,
H
i
ε
= H
i
ε
,
applying again the results from [18] Chapter 3, Lemma 3.1 we have that the density
satisfies the bounds
ρ
ε
∈ L
2
(0,T; W
2,2
(T )) ∩ C(0,T; W
1,2
(T )),∂
t
ρ
ε
∈ L
2
((0,T) ×T),
and also through a bootstrap argument we have
ρ
ε
C([0,T ];W
2−2/p,p
(T ))
+ ρ
ε
L
p
(0,T ;W
2,p
(T ))
+|∂
t
ρ
ε
L
p
((0,T )×T )
≤ cρ
ε,0
W
2−2/p,p
(T )
,
ρ
ε
C([0,T ];C
2,ν
(
¯
T ))
≤ c
∂
t
ρ
ε
C([0,T ];C
0,ν
(
¯
T ))
≤ c.