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158 W. Desch and S.-O. Londen

Finally, since for all F, G, u

the operator T

[F,G,u

]

is a strict contraction

with Lipschitz constant

< 1onL

([0,T] ×Ω; D((−A)

)) (with the norm |||·|||),

and since T

[F,G,u

]

v depends Lipschitz on F, G, u

by Part (3), the standard

contraction arguments yield Part (4). 

We are now ready to ﬁnish the proofs of the main results:

Proof of Theorem 3.6. We may assume without loss of generality that δ

,δ

≤ θ.

(If any δ

is greater that θ, it may be replaced by θ.) Obviously, the unique solution

of (1.1) in L

([0,T]×Ω; D((−A)

)) is exactly the unique ﬁxed point of T

[F,G,u

]

constructed in Lemma 6.2. 

Proof of Theorem 3.7. Let u be the solution of (1.1), thus, with f = N

u ∈

([0,T] × Ω; D((−A)



)) and g = N

u ∈ L

([0,T] × Ω; L

(B; l

)) we have that

u solves (5.1). Let v be deﬁned by (3.13). Now, ζ,η,δ

,δ

, satisfy the conditions

of Proposition 5.2, which yields immediately the required additional regularity

results. 

Proof of Corollary 3.8. To prove Part (1), choose η such that

−

<η<1,

and such that the conditions (3.9), (3.10), (3.11), and (3.12) from Theorem 3.7 are

satisﬁed. Then D

v ∈ L

([0,T]; L

(Ω; D((−A)

))). Notice that q<

1−pη

,sothat

we infer from [9, p. 421] that v ∈ L

([0,T]; L

(Ω; D((−A)

))).

To prove Part (2), put μ+

= η. Consequently conditions (3.9), (3.10), (3.11),

and (3.12) from Theorem 3.7 hold. Then D

v ∈ L

([0,T]; L

(Ω; D((−A)

))). Then

by [9, p. 421] we infer that v ∈ h

η−p

−1

0→0

([0,T]; L

(Ω; D((−A)

))). 

Proof of Theorem 3.11. For i =1, 2, let u

0,i

1,i

]

be the solution of (1.1)

with u

replaced by u

0,i

,etc.Letv

0,i

1,i

]

be deﬁned by (3.13) with the

obvious modiﬁcations. From Lemma 6.2, Part (4) we have a Lipschitz estimate

u

0,1

1,1

]

− u

0,2

1,2

]



([0,T ]×Ω;D((−A)

))

≤ Md with

d =

u

0,1

− u

0,2



(Ω;D((−A)

))

+ u

1,1

−u

1,2



(Ω;D((−A)

))

+ μ

ΔF



([0,T ]×Ω;R)

+ μ

ΔG



([0,T ]×Ω;R)

Now let f

= N

0,i

1,i

]

and g

= N

0,i

1,i

]

. By Lemma 6.1(1) we

have

f

−f



([0,T ]×Ω;D((−A)



))

≤ M

Md, g

− g



([0,T ]×Ω;L

(B,l

))

≤ M

Md.

The diﬀerence v = v

0,1

1,1

]

− v

0,2

1,2

]

solves (5.1) with u

replaced

by u

0,1

− u

0,2

, etc. Proposition 5.2 yields now

v

0,1

1,1

]

− v

0,2

1,2

]



([0,T ]×Ω;D((−A)

))

≤ Md

with a suitable constant M. 

Stochastic Integral Equations 159

7. Maximal regularity considerations

In this section, we consider the case that B = R

and A =Δ:W

2,p

) →

), the Laplacian in L

). In this case, a maximum regularity result can

be proved. To keep the paper at a reasonable size we concentrate on the stochastic

part and conﬁne ourselves to the equation

u(t, ω, x)=Δ



(t − s)u(s, ω, x) ds

∞

k=1



(t − s)G

(s, ω, u(s, ω, x)) dw

(7.1)

and the linear equation

u(t, ω, x)=Δ



(t − s)u(s, ω, x) ds +

∞

k=1



(t −s)g

(s, ω, x) dw

. (7.2)

Notice that various results on maximal regularity with respect to determin-

istic forcing functions (see, e.g., [39]) and to initial data (e.g., [9]) are available.

These could be combined with the results given here and adapted to the semilinear

case.

For (7.2) we obtain

Proposition 7.1 ([13], Theorem 4.14). For a positive integer n,letΔ:W

2,p

) →

) be the Laplacian with 1 <p<∞. Suppose that the probability space Ω and

the Wiener processes w

satisfy Hypothesis 3.2.LetT>0, β>

, α ∈ (0, 2),and

g ∈ L

([0,T] × Ω; L

)).

(a) Then there exists a unique function u ∈ L

([0,T] × Ω,L

)) such that for

almost all t ∈ [0,T],



(t − s)u(s) ds ∈ W

2,p

)

and (7.2) holds.

(b) Moreover, if ζ ∈ [0, 1] is such that

αζ +

≤ β, (7.3)

then u ∈ L

([0,T] × Ω, D((−Δ)

)),and

u

([0,T ]×Ω,D((−Δ)

))

≤ Mg

([0,T ]×Ω,l

)

(7.4)

with a constant M dependent on n, T, p, α, β, ζ.

small by taking suﬃciently small T .

Proof. Of course, if strict inequality holds in (7.3), then the assertions above are

just a special case of Proposition 5.2 with A =Δ,u

= u

=0,andη =0.Butfor

such A and η, the assertion of Lemma 5.8 holds also if equality holds in (7.3), with

160 W. Desch and S.-O. Londen

the only exception that

T,Ito

cannot be made small by taking small T .See[12,

Theorem 1.2]. (To prove this, the general estimates from Lemma 5.4 are replaced

by a more sophisticated analysis of the resolvent kernel for the Laplacian, using

the heat kernel and its self-similarity properties. This has been done for the heat

equation by Krylov in [20], and generalized to the case of integral equations in

[12].) Once Lemma 5.8 is established, the proof continues exactly as in Section 5.

More details can be found in [13]. 

Since M in (7.4) cannot be controlled simply by taking short time intervals,

we need a more sophisticated Lipschitz condition. (For the heat equation, compare

[21, Assumption 5.6].)

Hypothesis 7.2. There exists some θ ∈ (0, 1) such that

G :[0,T] × Ω ×D((−Δ)

) → L

; l

)

[G(t, ω, u)](x):=



(t, ω, u)(x)



∞

k=1

satisﬁes the following assumptions:

(a) For ﬁxed u ∈D((−Δ)

), the function G(·, ·,u) is measurable from [0,T] × Ω

into L

; l

(b) For each >0, there exists a constant M

() > 0, such that for all t ∈ [0,T],

and all u

∈D((−Δ)

) the following Lipschitz estimate holds:

G(t, ω, u

) − G(t, ω, u

)

)

(7.5)

≤



u

− u



D((−Δ)

)

+ M

()

u

− u



)

1/p

for ω ∈ Ωa.e..



G(t, ω, 0)

)

dt dP

1/p

= M

G,0

< ∞. (7.6)

Theorem 7.3. Let the probability space (Ω, F, P) and the Wiener processes





∞

k=1

be as in Hypothesis 3.2.Letp ∈ [2, ∞),andΔ be the Laplacian on L

).Let

α ∈ (0, 2), β>

,andT>0. Assume that G :[0,T]×Ω ×D((−Δ)

) → L

; l

)

satisﬁes Hypothesis 7.2 with suitable θ ∈ (0, 1), such that

αθ +

= β. (7.7)

Then there exists a unique function u ∈ L

([0,T] × Ω; D((−Δ)

)) such that for

almost all t ∈ [0,T]



(t −s)u(s, ω, ·) ds ∈D(Δ) for a.e. ω ∈ Ω,

and (7.1) is satisﬁed for almost all ω ∈ Ω.

Stochastic Integral Equations 161

Proof. We reﬁne the contraction argument from Section 6. As in Lemma 6.1 we

deﬁne for v ∈ L

([0,T] × Ω; D((−Δ)

))

(v):



[0,T] × Ω → L

; l

t × ω → G(t, ω, v(t, ω)).

For g ∈ L

([0,T] × Ω; L

)) we deﬁne Sg := u ∈ L

([0,T] × Ω; D((−Δ)

)),

where u is the solution of (7.2) according to Proposition 7.1 with forcing function

g. As in Section 6 the desired solution u is a ﬁxed point of the operator T := S◦N

which maps L

([0,T] × Ω; D((−Δ)

)) into itself.

By (7.4) for ζ = 0 and for ζ = θ we infer

Sg

([0,T ]×Ω;L

))

≤ M

(T )g

([0,T ]×Ω;L

))

Sg

([0,T ]×Ω;D((−Δ)

))

≤ M

g

([0,T ]×Ω;L

))

with ﬁxed M

, while M

(T ) can be made arbitrarily small by taking T suﬃciently

small. We ﬁx >0 such that M

<

and choose the corresponding M

()ac-

cordingtoHypothesis7.2.OnL

([0,T]×Ω; D((−Δ)

)) we introduce the following

equivalent norm

v

([0,T ]×Ω;D((−Δ)

)),equiv





v(t, ω)

D((−Δ)

)

+ M

()v(t, ω)

)

dP(ω) dt.

With respect to this norm, the nonlinear operator

: L

([0,T] × Ω; D((−Δ)

)) → L

([0,T] × Ω; L

; l

))

has Lipschitz constant 1 by Hypothesis 7.2. On the other hand

Sg

([0,T ]×Ω;D((−Δ)

)),equiv

≤ (

+ M

()

(T )

)

1/p

g

([0,T ]×Ω;L

))

We infer that T is Lipschitz on L

([0,T] × Ω; D((−Δ)

)) with respect to the

equivalent norm ·

([0,T ]×Ω;D((−Δ)

)),equiv

,andifT is suﬃciently small, so that

(

+ M

()

(T )

)

1/p

, then the Lipschitz constant of T is less than

. We can now proceed as in Lemma 6.2 (2) to construct an equivalent norm on

([0,T] ×Ω; D((−A)

)) which makes T a strict contraction also for large T . 

8. Krylov’s approach versus B-space valued stochastic integration

At the center of the study of stochastic integral equations in Banach spaces is the

problem of deﬁning and estimating stochastic integrals, in particular stochastic

convolutions, in Banach spaces. Krylov’s approach, which is used in this paper,

is elementary in the sense that stochastic integrals are taken pointwise, so they

are classical Ito-integrals of scalar-valued processes. The Burkholder-Davis-Gundy

inequality provides the step from L

-estimates to L

. Of course, this can only

162 W. Desch and S.-O. Londen

be done for suﬃciently “nice” integrands. The ﬁnal step is to extend the results

obtained for smooth initial data and elementary forcing terms to more general

-data by a completion argument.

On the other hand, the recent progress on stochastic integration in Banach

spaces provides a convenient tool to handle stochastic convolutions directly in the

Banach space. In [4] stochastic convolution is developed to the point where semi-

linear stochastic diﬀerential equations can be treated in M-type 2 UMD spaces.

(This covers our case X = L

.) In fact, the key is a generalized version of the

Burkholder-Davis-Gundy inequality. Further developments of stochastic integra-

tion can be found in [26].

In [13] we compared our linear results with those obtained in [14], [37]. In the

context of the present paper it appears interesting to make a similar brief com-

parison concerning semilinear equations. Note that the aim of the present paper

is to treat fractional diﬀerential equations and not only the diﬀerential equation

case α = β = γ = 1. To our knowledge, no results are available for stochastic

fractional integral equations using abstract integration methods in spaces other

than Hilbert spaces. However, there are several papers dealing with diﬀerential

equations, in settings that are quite more general than ours. For instance, nonau-

tonomous problems ([38]) and local Lipschitz conditions ([4], [27]) can be treated.

And, as stated above, the abstract methods are not conﬁned to the space L

With α = β = γ = 1 our equation (1.1) reduces to the stochastic nonlinear

diﬀerential equation

du(t)=Au(t) dt + G(t, ω, u(t)) dW

+ F (t, ω, u(t)) dt. (8.1)

It is this case, where we can compare our results to the results obtained by

the abstract integration theory. Note that in abstract notation, W

is a cylin-

drical Wiener process in a separable Hilbert space H and that, for ﬁxed u, G ∈

([0,T] ×Ω; γ(H, L

(B))) where γ(H, L

(B)) denotes the space of γ-radonifying

operators H → L

(B). This is equivalent to writing the stochastic forcing in

Krylov’s notation

G =

∞

k=1

with (for ﬁxed u) {G

}

∞

k=1

∈ L

([0,T] × Ω; L

(B,l

)) (use, e.g., [37, Proposi-

tion 3.2.3]).

Possibly the results which can be most easily compared with ours are those

of [4], which are formulated for any type 2 UMD space. Rewritten to our notation

(and reduced to the globally Lipschitzian case, and L

instead of general X), the

essential conditions in [4] read:

• There are constants  ∈ (−1, 0), ν ∈ (0, 1], ϑ ≥ 0, and

0 ≤ θ

<θ

+ p

−1

< min



 +1,

−ϑ



, (8.2)

such that the following conditions hold:

Stochastic Integral Equations 163

• G :[0,T] × Ω ×D((−A)

) → L

(B; l

) satisﬁes a Lipschitz condition

(−A)

−ϑ

G(t, ω, u

) −(−A)

−ϑ

G(t, ω, u

)

(B;l

)

≤ Ku

−u



D((−A)

)

u

− u



1−ν

D((−A)

)

for almost all ω ∈ Ω, and all t ∈ [0,T], u

∈D((−A)

• F :[0,T] × Ω ×D((−A)

) →D((−A)



) satisﬁes a Lipschitz condition

F (t, ω, u

) − F (t, ω, u

)

D((−A)



)

≤ Ku

− u



D((−A)

)

u

− u



1−ν

D((−A)

)

for almost all ω ∈ Ω, and all t ∈ [0,T], u

∈D((−A)

• u

∈ L

(Ω, F

, D((−A)

• In addition, suitable conditions for measurability and linear growth of F and

G in u are given.

With these conditions, (8.1) admits a unique mild solution

u ∈ L

([0,T] × Ω; D((−A)

)) ∩ L

(Ω; C([0,T]; D((−A)

))).

This assertion is stronger than our Theorem 3.6 since it ensures that trajectories

are continuous a.s. in the some space D((−A)

), while our theorem only provides

asolutioninL

([0,T] ×Ω; D((−A)

)). Accordingly, the conditions on θ

will not

have a counterpart in Theorem 3.6. The Lipschitz condition on G can be compared

to Hypothesis 3.5 if we identify θ

above with our θ and put ϑ =0,and(with

some abuse) forget about the role of θ

. The Lipschitz condition on F is more

general than Hypothesis 3.4, since it allows for negative . We expect that it might

be a minor technical task to sharpen our arguments to match this situation. With

α = β = γ = 1, our conditions (3.5), (3.6) and (3.7) can be rewritten in the form

< 1+, θ

,θ

+ δ

The conditions on θ

and  are exactly what is left from (8.2) if we forget about θ

Our Theorem 3.6 allows for u

∈D((−A)

)whereδ

>θ

−

. In [4] slightly more

regular u

∈D((−A)

) is required, with the payoﬀ that solutions are continuous

at least as functions with values in D((−A)

). The continuity assertion may be

(with some caution) compared to our Corollary 3.8 (2), if we identify θ

from [4]

with our ζ. Then the conditions of our Corollary 3.8 require μ ∈ (0, 1 −

)such

that

+ μ + θ

< min



1+,



,μ+ θ

<δ

Such μ can be found if (8.2) holds. Our corollary states that in this case u ∈

0→0

([0,T]; L

(Ω; D((−A)

))).

Acknowledgement

We wish to thank an anonymous referee for his careful reading of our paper and

helpful suggestions, in particular for drawing our attention to important references.

164 W. Desch and S.-O. Londen

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Wolfgang Desch

Institut f¨ur Mathematik und Wissenschaftliches Rechnen

Karl-Franzens-Universit¨at Graz

Heinrichstrasse 36

A-8010 Graz, Austria

e-mail: georg.desch@uni-graz.at

Stig-Olof Londen

Department of Mathematics and Systems Analysis

Helsinki University of Technology

FI-02015 TKK, Finland

e-mail: stig-olof.londen@tkk.fi

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 167–192

 2011 Springer Basel AG

On the Motion of Several Rigid Bodies

in an Incompressible Viscous Fluid under

the Inﬂuence of Selfgravitating Forces

Bernard Ducomet and

S´arka Neˇcasov´a

Abstract. The global existence of weak solutions is proved for the problem of

the motion of several rigid bodies in non-newtonian ﬂuid of power-law with

selfgraviting forces.

Mathematics Subject Classiﬁcation (2000). 35Q35.

Keywords. Existence of weak solutions, motion of several rigid bodies, non-

Newtonian ﬂuid, selfgravitating forces.

1. Introduction

In the last decade, a large activity has been devoted to the study of motions of

rigid or elastic bodies in a ﬂuid. The motion of particles in a viscous liquid has

become an important goal of applied research. The presence of particles aﬀects the

ﬂow of the liquid, and the ﬂuid, in turn, aﬀects the motion of particles, so that

the problem of determining the ﬂow characteristics is a strongly coupled one.

There exists now a lot of numerical studies and simulations concerning this

system and its theoretical aspects. We deal here with the problem of several rigid

bodies embedded into a viscous ﬂuid.The ﬂuid and rigid bodies are contained in

a ﬁxed open bounded set of R

. We will suppose that this ﬂuid is non-Newtonian

with a suﬃciently high viscosity and also we consider selfgraviting forces. We are

interested in the existence of weak solutions and also in no existence of collisions

in ﬁnite time. We will apply two diﬀerent techniques of penalization. The ﬁrst one

was introduced by Conca, San Martin, Tucsnak and Starovoitov [34, 5] and the

second one by Bost, Cottet and Maitre [2]. We want to compare eﬃciency of both

methods.