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168 B. Ducomet and

S. Neˇcasov´a

Historically, the weak formulation of the problem has been introduced and

studied by Judakov see [39] and after that by many authors: Desjardins and Este-

ban [6, 7], Hoﬀmann and Starovoitov [29, 30], San Martin, Starovoitov, Tucsnak

[34], Serre [35], Galdi [23], among others.

Concerning the problem of the existence of collisions, let us mention that in

the case of compressible ﬂuids there is a result obtained by E. Feireisl.

In [16], E. Feireisl considered a rigid sphere surrounded by a compressible

viscous ﬂuid inside a cavity. He constructed a solution to the subsequent system

in which the sphere sticks to the ceiling of the cavity without falling down. On the

other hand, in the incompressible case, Hesla [26] and Hillairet [27] proved a no-

collision result when there is only one body in a bounded two-dimensional cavity.

Later the result was extended to the three-dimensional situation by Hillairet and

Takahashi [28].

Starovoitov in [38] showed that collisions, if any, must occur with zero rela-

tive translational velocity if the boundaries of the rigid objects are smooth and the

gradient of the underlying velocity ﬁeld is square integrable – a hypothesis satis-

ﬁed by any Newtonian ﬂuid ﬂow of ﬁnite energy. The possibility or impossibility

of collisions in a viscous ﬂuids is related to the properties of the velocity gradi-

ent. A simple argument reveals that the velocity gradient must become singular

(unbounded) at the contact point, since otherwise the streamlines would be well

deﬁned, in particular, they could never meet each other.

Indeed Starovoitov [38] showed that collisions of two or more rigid objects

areimpossibleif:

• the physical domain Ω ⊂ R

as well as the rigid objects in its interior have

boundaries of class C

1,1

;

• the pth power of the velocity gradient is integrable, with p ≥ 4.

Inspired by work of Starovoitov, Feireisl et al. [17] have considered the motion

of several rigid bodies in a non-Newtonian ﬂuid of power-law type (see Chapter 1

in M´alek et al. [33] for details), where the viscous stress tensor S depends on the

symmetric part D[u],

D[u]=∇

u + ∇

of the gradient of the velocity ﬁeld u in the following way:

S = S[ D[u]], S : R

3×3

sym

→ R

3×3

sym

is continuous, (1.1)



S[M] −S[N]





M −N



> 0 for all M = N, (1.2)

and

|M|

≤ S[M]:M ≤ c

(1 + |M|

) for a certain p ≥ 4 (1.3)

and they showed not only the existence of a weak solution but also that collisions

cannot occur in such viscous ﬂuids.

The question how the smoothness of boundary has inﬂuence on the existence

of collisions was investigated in the work of Gerard-Varet and Hillairet [24].

On the Motion of Several Rigid Bodies 169

The present work is an extension of that of Feireisl et al. (see [17]) to the case

where selfgravitating forces are present and two diﬀerent possibilities of penaliza-

tion technique are used.We have to use a slightly diﬀerent approximation scheme

by means of artiﬁcial viscosity terms to handle the selfgravitating force. Moreover,

we have to assume the regularity of boundary C

2,ν

, ν ∈ (0, 1) to get the strong

solution of the regularized continuity equation.

2. Formulation of the problem

2.1. Bodies and motions

A rigid body can be identiﬁed with a connected compact subset

B of the Euclidean

space R

. The motion is represented as a mapping η :(0,T) × R

→ R

,which

deﬁnes an isometry

η(t, ·):R

→ R

(2.1)

for any time t ∈ (0,T).

We adopt the Eulerian (spatial) description of motion, where the coordinate

system is attached to a ﬁxed region of the physical space currently occupied by the

ﬂuid. The position x and the time t ∈ (0,T) play the role of independent variables.

The mappings η(t, ·) are isometries satisfying the following identity

η(t, x)=X

(t)+O(t)(x − X

(0)),

where X

; the position of the center of mass at a time t and O(t) is a matrix satis-

fying O

O = I. Moreover, the translation and angular velocities can be expressed

respectively by

= U

(2.2)

and



O(t)



(t)=Q(t). (2.3)

The solid velocity in the Eulerian coordinate system can be written as

(t, x)=

∂η

∂t

(t, η

−1

(t, x)) = U

(t)+Q(t)(x −X

(t)).

The total force F

acting on the body B

is a sum of the gravitation force and

the contact force

(t)=



(t)

dx +



∂B

(t)

Sn dσ,

where S is the Cauchy stress, g

= G∇



j=i

|x−y|

is the gravitation force

and

(t)=η(t,

). Under the assumption of the continuity of stress, the balance

of linear momentum for body B

is expressed by Newton’s second law

(t)=



(t)

dx =



(t)

dx +



∂B

(t)

Sn dσ, (2.4)

where m is the total mass of the body.

170 B. Ducomet and

S. Neˇcasov´a

The matrix Q is skew-symmetric, therefore it can be represented by a vector

ω such that

Q(t)(x − X

)=ω(t) × (x − X

Moreover the balance of angular momentum reads

(Jω)=



(t)

(x − X

) × u

dx (2.5)



∂B

(t)

(x − X

) × Sn dσ +



(t)

(x − X

) × g

dx,

where J is the inertia tensor

Ja · b =



(t)

(a × (x −X

)) · (b × (x − X

)) dx.

2.2. The ﬂuid motion

The ﬂuid is completely determined by the density ρ

,andthevelocityu

.The

standard mass and momentum balance equations are the following:

∂

+div(ρ

)=0,

divu

=0,

∂

(ρ

)+div(ρ

⊗u

)+∇P =divS + ρ

G∇



|x−y|

dy,

⎫

⎪

⎬

⎪

⎭

in Ω

, (2.6)

where Ω

=Ω\

i=1

, P is the pressure and S is the viscous stress tensor.

We also consider a no-slip boundary condition for velocity

=0on∂Ω. (2.7)

We will introduce the notation

Q := I × Ω,

= {(t, x)|t ∈ I,x ∈

(t)},

;

i=1

:= Q \ Q

and deﬁne the quantities

ρ(t, x)=

⎧

⎨

⎩

(t, x)onQ

0onR

\ Ω,

and

u(t, x)=

⎧

⎨

⎩

(t, x)onQ

0onR

\ Ω.

We will restrict ourselves to the particular case where the density is noncon-

stant only on the solid part and constant on the ﬂuid part to get local estimates

on the pressure.

On the Motion of Several Rigid Bodies 171

3. Weak formulation

We begin with a description of the initial position of a set of rigid bodies. We

assume that the initial position of the rigid bodies is determined through a family

of domains

⊂ R

,i=1,...,n,

each of them being diﬀeomorphic to the unit ball in R

.Inaddition,inorderto

facilitate the analysis, the boundaries of all the rigid bodies are assumed to be

regular, more speciﬁcally, there exists δ

> 0 such that for any x ∈ ∂B

,thereare

two closed balls B

int

, B

ext

of radius δ

such that

x ∈ B

int

∩ B

ext

, B

int

⊂ B

, B

ext

⊂ R

\ B

. (3.1)

Similarly, the underlying physical space Ω ⊂ R

, occupied by the ﬂuid containing

the rigid bodies, is supposed to be a domain such that for any x ∈ ∂Ω, there are

two closed balls B

int

, B

ext

of radius δ

such that

x ∈ B

int

∩ B

ext

, B

int

⊂ Ω, B

ext

⊂ R

\ Ω. (3.2)

The motion η

associated to the body B

is a mapping

= η

(t, x),t∈ [0,T), x ∈ R

,η

(t, ·):R

→ R

together with the initial condition

(0, x)=x for all x ∈ R

,i=1,...,n,

that is an isomorphism R

→ R

Accordingly, the position of the body B

at a time t is given by the formula

(t)=η

(t, B

),i=1,...,n.

We proceed with the deﬁnition of a weak solution for ﬂuid structure interac-

tion, which was introduced by Judakov [39] based on the Eulerian reference system

and a class of test functions depending on the position of the speciﬁed rigid bod-

ies (see Desjardins and Esteban [6, 7], Galdi [22], [23], Hoﬀmann and Starovoitov

[29], San Martin et al. [34], Serre [35], among others) however we will use a slightly

diﬀerent deﬁnition in the last paragraph.

The mass density  = (t, x) and the velocity ﬁeld u = u(t, x)atatime

t ∈ (0,T) and the spatial position x ∈ Ω satisfy the integral identity





ρ∂

φ + ρu ·∇



dx dt = −



φ dx, φ ∈ C

([0,T) ×

Ω), (3.3)





ρu · ∂

ϕ + ρu ⊗ u : ∇

[ϕ] − S : D[ϕ]



dx dt

= −



ρG∇



|x −y|

dy · ϕ dx dt −



· ϕ dx dt,

(3.4)

ϕ ∈ C

([0,T) ×

Ω),ϕ(t, ·) ∈R(t), (3.5)

172 B. Ducomet and

S. Neˇcasov´a

R(t)={φ ∈ C

(

Ω) | div Φ = 0 in Ω,φ= 0 on a neighborhood of ∂Ω, (3.6)

D[Φ] = 0 on a neighborhood of ∪

i=1

(t)},

where



ρG∇





|x −y|



ϕdxdt=



ρG∇

Fdxdt,

with

F =



i=j



|x −y|

dy +



|x −y|



Finally, we require the velocity ﬁeld u to be compatible with the motion of the

bodies. As the mappings η

(t, i) are isometries on R

, they can be written in the

form

(t, x)=x

(t)+O

(t)x.

Accordingly, we require the velocity ﬁeld u to be compatible with the family of

motions {η

,...,η

} if

u(t, x)=u

(t, x)=U

(t)+Q

(t)(x−x

(t)) for a.a. x ∈

(t),i=1,...,n (3.7)

for a.a. t ∈ [0,T), where

= U





= Q

a.a. on (0,T). (3.8)

We introduce now some notation:

• Given a collection



i=1,...,m

of open relatively compact connected subsets in

Ω, we denote by

d[∪



]:=inf{ inf

i,j=1,...,m

dist(B

, B

), inf

i=1,...,m

dist(B

,∂Ω)}

the minimal interdistance “solids-boundary”.

• The signed distance to the boundary is

(x)=dist[x; R

\S] − dist [x,S].

Given a subset S ⊂ R

, and α>0, we denote

[S]

= db

−1

((−∞, −α)), ]S[

= db

−1

((−∞,α)). (3.9)

In the sense of J.A. San Martin, M. Tucsnak and V. Starovoitov, [S]

is the α-

neighborhood of S and ]S[

is the α-kernel of O.

V := {φ ∈C

∞

(Ω), such that div(φ)=0}.

1. V

stands for the closure of V in W

1,p

(Ω) and V

for the closure of V in

s,2

(Ω). For simplicity V = V

= V

2. Given B ⊂ Ω, we write

(B)={v ∈ V

with D(v)=0 in B},

(B)={v ∈ V

with D(v)=0 in B}.

On the Motion of Several Rigid Bodies 173

3. Given a subset B of Ω, P

{B} is the orthogonal projection of V

onto

([B]

We will introduce the energy inequality (EI) of the problem as follows:





ρ|u|

) dx −



Gρ(t

, x)Fdx





S : D[u] dx dt

≤





ρ|u|

) dx −



Gρ(t

, x)Fdx



Problem P. Let the initial distribution of the density and the velocity ﬁeld be de-

termined through given functions ρ

, u

, respectively. The initial position of the

rigid bodies is B

⊂ Ω, i =1,...,m. We say that a family ρ, u, η

, i =1,...,m,

represent a variational solution of problem (P) on a time interval (0,T) if the

following conditions are satisﬁed:

• The density ρ is a non-negative bounded function, the velocity ﬁeld u belongs

to the space L

∞

(0,T; L

(Ω; R

)) ∩ L

(0,T; W

1,p

(Ω; R

)), and they satisfy

energy inequality (EI) for t

=0anda.a.t

∈ (0,T).

• The continuity equation holds on (0,T) ×R

provided ρ and u are extended

to be zero outside Ω.

• The momentum equation (the integral identity) holds for any admissible test

function w ∈R(t).

• The mappings η

, i =1,...,m are aﬃne isometries of R

compatible with

the velocity ﬁeld u in the sense of compatibility conditions.

4. Main result I

Let us formulate one of our main existence results.

Theorem 4.1. Let the initial position of the rigid bodies be given through a family

of open sets

⊂ Ω ⊂ R

, B

diﬀeomorphic to the unit ball for i =1,...,n,

where both ∂B

, i =1,...,n,and∂Ω belong to the regularity class speciﬁed in

(3.1), (3.2). In addition, suppose that

dist[B

, B

] > 0 for i = j, dist[B

\ Ω] > 0 for any i =1,...,n

and we assume that the boundaries of Ω and B

belong to C

2,ν

, ν ∈ (0, 1).Fur-

thermore, 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







=const> 0 in Ω \∪

i=1



on S

, where 

∈ L

∞

(Ω), ess inf



> 0,i=1,...,n,

while

∈ L

(Ω; R

), div

=0in D



(Ω), D[u

]=0in D



; R

3×3

) for i =1,...,n.

174 B. Ducomet and

S. Neˇcasov´a

Then there exist a density function ,

 ∈ C([0,T]; L

(Ω)), 0 < ess inf

(t, ·) ≤ ess sup

(t, ·) < ∞ for all t ∈ [0,T],

a family of isometries {η

(t, ·)}

i=1

, η

(0, ·)=I, and a velocity ﬁeld u,

u ∈ C

weak

([0,T]; L

(Ω; R

)) ∩L

(0,T; W

1,p

(Ω; R

)),

compatible with {η

}

i=1

in the sense speciﬁed in (3.7), (3.8), such that , u satisfy

the integral identity (3.3) for any test function φ ∈ C

([0,T)×R

), and the integral

identity (3.4) for any ϕ satisfying (3.5), (3.6).

5. Approximate problem

We will construct weak solutions based on a three-level approximation scheme

which consists in solving the system of equations

∂

 +div

([u]

)=dΔρ, (5.1)

∂

(u)+div

(u ⊗ [u]

)+∇

P + d∇ρ∇u =div

([μ

]

S)+∇

F − χ

u, (5.2)

∂

+div

(μ

[u]

)=0, (5.3)

div

u =0, (5.4)

where

[u]

= σ

∗ u (spatial convolution) for 0 <δ<δ

with

(x)=



|x|



, (5.5)

σ ∈D(−1, 1),σ(z) > 0for − 1 <z<1,σ(z)=σ(−z),



−1

σ(z)dz =1.

As Ω is bounded, we can assume that Ω ⊂ [−L, L]

for a certain L>0and

consider system (5.1–5.4) on the spatial torus

T =[(−L, L)|

{−L,L}

]

meaning that all quantities are assumed to be spatially periodic with period 2L.

System (5.1)–(5.4) is supplemented with the initial conditions

(0, ·)=

0,δ

= 

i=1



,δ

, (5.6)

where



∈D(B

),

,δ

(x) = 0 whenever dist[x, ∂B

] <δ,i=1,...,n. (5.7)

Similarly, we prescribe

μ(0, ·)=μ

0,ε

=1+

i=1

, (5.8)

On the Motion of Several Rigid Bodies 175

where



∈D(B

),μ

(x) = 0 whenever dist[x, ∂B

] <δ,

(x) > 0forx ∈ B

, dist[x, ∂B

] >δ

i =1,...,n. (5.9)

We take

χ, χ ∈D(T ),χ>0onT\Ω,χ=0inΩ. (5.10)

Finally we add the homogeneous Neumann boundary condition

∇ρ ·n =0 on∂Ω ∪

;

i=1

∂B

, (5.11)

and we assume that

0,δ

∈ C

2,ν

(Ω).

(5.12)

The parameters ε, d and δ are small positive numbers. The ε-dependent

initial distribution of the “viscosity” μ can be easily identiﬁed as the penalization

introduced by Hoﬀmann and Starovoitov [29] and San Martin et al. [34], where

the rigid bodies are replaced by a ﬂuid of high viscosity becoming singular for

ε → 0. Here, the extra parameter δ>0 has been introduced to keep the density

constant in the approximate ﬂuid region in order to construct the local pressure.

Moreover, we regularized the continuity equation by an artiﬁcial viscosity term to

get the estimation of the selfgravitating forces. We will also introduce in the last

section another possibility of penalization which was introduced by Bost et al. [2]

and gives the main idea of the proof. As for ε>0, δ>0andd>0 ﬁxed, we report

the following existence result that can be proved in a standard way through the

solution of the continuity equation through L

-L

regularity (see for more details

[18], Lemma 3.1) and by means of a standard monotonicity argument (see M´alek

et al. [33]) which was extended to nonhomogeneous ﬂuids by Frehse et al. [20],

[21].

Proposition 5.1. Suppose that p ≥ 4. Let the initial distribution of , μ be given

through (5.6)–(5.9),withﬁxedε>0, δ>0, d>0. Moreover, assume that

u(0, ·)=u

∈T, u

∈ L

(T ; R

), div

=0in D



(T ; R

), (5.13)

and χ

∈ C

∞

(T ),whereχ

is determined by (5.10) and let us assume 5.12.

Then problem (5.1)–(5.4), supplemented with the initial data (5.6)–(5.9),

(5.12) and additional boundary conditions (5.11), possesses a (weak) solution ,

μ, u belonging to the class

μ ∈ C

1,2

([0,T] ×T),

 ∈ C([0,T]; C

2,ν

(T )),∂

 ∈ C([0,T]; C

0,ν

(T )),

u ∈ C

weak

([0,T]; L

(T ; R

)) ∩L

(0,T; W

1,p

(T ; R

)).

176 B. Ducomet and

S. Neˇcasov´a

In addition, the solution satisﬁes the energy inequality



|u|

(τ)dx +



[μ

]

S : ∇

u dx dt +



|u|

dx dt ≤



|u|

(s)dx

(5.14)

for a.a. 0 ≤ s<τ ≤ T including s =0.

Let us remark that, in the weak formulation, equation (5.2) is represented by

the integral identity



u · ∂

ϕ + (u ⊗[u]

):∇

ϕ dx dt +



δ∇ρ∇u ϕdxdt



[μ

]

S : D[ϕ]dx dt −



ρG ·∇





ρ(t, y)

|x − y|



ϕ dx dt

−



u · ϕ dx dt −





0,δ

· ϕ(0, ·)dx

(5.15)

to be satisﬁed for any test function ϕ ∈D([0,T) ×T; R

), such that div

ϕ =0.

Remark. From [18] [Chapter 3, Lemma 3.1] it follows that in the level of approx-

imation δ,  the density satisﬁes the following bounds:

ρ ∈ L

(0,T; W

2,2

T ) ∩ C(0,T; W

1,2

(T )),∂

ρ ∈ L

((0,T) ×T) (5.16)

and also through a bootstrap argument we have

ρ

C([0,T ];W

2−2/p,p

(T ))

+ ρ

(0,T ;W

2,p

(T ))

(5.17)

+∂

ρ

((0,T )×T )

≤ cρ



2−2/p,p

(T )

ρ

C([0,T ];C

2,ν

(

T ))

≤ c, (5.18)

∂

ρ

C([0,T ];C

0,ν

(

T ))

≤ c. (5.19)

The term with selfgraviting force is estimated by the following way:



ρG∇





ρ(t, y)

|x −y|



φdxdt ≤ cρ

ρ

u

(T )

for more details see [8, 9].

6. Artiﬁcial viscosity limit

Our ﬁrst task is to identify the limit problem resulting from (5.1)–(5.9) for ﬁxed

δ>0andε → 0. To this end, let us denote by {

,μ

, u

}

ε>0

the associated family

of approximate solutions, the existence of which is guaranteed by Proposition 5.1.

As already pointed out, there are two major issues to be addressed, namely the

strong (pointwise) convergence of the velocity ﬁelds, and strong convergence of the

velocity gradients in order to pass to the limit in the non-linearity of the stress

tensor. The ﬁrst goal is accomplished basically in the same way as in [34] and

repeated in [17], so we will give only the main ideas of the proof. Note that in the

On the Motion of Several Rigid Bodies 177

present setting, the analysis is considerably simpliﬁed thanks to the no-collision

result by Starovoitov [38].

Let us start with the following stability result for solutions to the transport

equation established in [16, Proposition 5.1]:

Proposition 6.1. Let v

= v

(t, x) be a family of vector ﬁelds such that

}

∞

n=1

is bounded in L

(0,T; W

1,∞

; R

)).

Let η

(t, ·):R

→ R

be the solution operator associated to the family of charac-

teristic curves generated by v

,speciﬁcally,

∂

∂t

(t, x)=v

(t, η

(t, x)),η

(0,x)=x for all x ∈ R

Then, at least for a suitable subsequence,

→ v weakly-(*) in L

(0,T; W

1,∞

(Ω; R

)),

(t, ·) → η(t, ·) in C

loc

) uniformly for t ∈ [0,T],

where η istheuniquesolutionof

∂

∂t

η(t, x)=v(t, η(t, x)),η(0,x)=x, x ∈ R

If, in addition,

→ S,

then

(t, S

) ≡ S

(t)

→ S(t) ≡ η(t, S)

which means that

(t)

→ db

S(t)

in C

loc

) uniformly with respect to t ∈ [0,T].

The energy inequality (5.14), together with the coercivity hypothesis (1.3),

that {u

}

ε>0

give us that {u



} is a bounded sequence in L

(0,T; W

1,p

(T ; R

)).

Consequently, passing to a suitable subsequence as the case may be, we can assume

→ u weakly in L

(0,T; W

1,p

(T ; R

))

where the limit velocity ﬁeld satisﬁes div

u = 0 a.a. on (0,T) ×T. Accordingly,

the regularized sequence {[u

]

}

ε>0

satisﬁes

]

→ [u]

weakly-(*) in L

(0,T; W

1,∞

(T ; R

)), div

[u]

=0. (6.1)

Moreover, using hypothesis (5.10) combined with (5.14), we get

u = 0 a.a. in the set (0,T) × (T\Ω).

As Ω is regular, this yields

∂Ω

= 0; whence u ∈ L

(0,T; V

1,p

Seeing that {

}

ε>0

solves the modiﬁed transport equation (5.1), we can use

Proposition 6.1 and (6.1) to deduce that



→  in C([0,T] ×T), (6.2)