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258 G.P. Galdi and M. Kyed

with c

independent of R, and so, the previous inequality implies

|α|=2

ψ

v

2,Ω

≤ c



|x|



2,Ω

+ 

∇v

|x|



2,Ω

+ ψ

(Δv)

2,Ω

where c

is independent of R. If we use the assumption v ∈ D

1,2

(Ω) ∩ L

(Ω)

in this relation, along with a Hardy-type inequality (see for example [7, Theorem

II.5.1]) and the fact that Δv ∈ L

(Ω

), we deduce

|α|=2

ψ

v

2,Ω

≤ c

, (3.12)

where c

is independent of R. The desired property for D

v then follows by letting

R →∞in (3.12). 

In the next lemma, we establish L

(Ω)-summability of the pressure. More

precisely, we have:

Lemma 3.2. Let f ∈ L

(Ω)

∩L

(Ω)

, v

∗

∈ W

(∂Ω)

,andlet(v, p) ∈ D

1,2

(Ω)

∩

(Ω)

×L

loc

(Ω) be a corresponding solution to (1.1).Thenp+c ∈ L

(Ω) for some

constant c ∈ R.

Proof. Standard regularity theory for elliptic systems again yields p ∈ W

1,2

loc

(Ω).

Consequently, by Sobolev embedding, we have p ∈ L

loc

(Ω). We therefore only need

to show p + c ∈ L

(Ω

)forsomeρ>0andc ∈ R.

Let ρ>diam(R

\Ω) and ψ ∈ C

∞

; R) be a “cut-oﬀ” function with ψ =0

on B

and ψ =1onR

\ B

2ρ

.Moreover,let

σ(x):=





∂ B

2ρ

v · n dx



∇E, E(x):=

4π|x|

. (3.13)

Since



2ρ

∇ψ · (v + σ) dx =



2ρ

div

ψ(v + σ)



∂ B

2ρ

v ·n dx +



∂ B

2ρ

σ · n dx =0,

there exists (see [7, Theorem III.3.2]) a ﬁeld

H ∈ W

3,2

), supp H ⊂ B

2ρ

, div H = ∇ψ · (v + σ). (3.14)

Put

w = ψv + ψσ −H, π = ψp.

Using the fact that e

×x ·∇σ − e

×σ = 0, we ﬁnd that



Δw −∇π +Ta



×x ·∇w − e

×w



= ψf + G +Reψv·∇v in R

div w =0 inR

(3.15)

Leray Solution around Rotating Obstacle 259

where G ∈ L

) with supp(G) ⊂ B

2ρ

. Taking divergence on both sides in (3.15)

yields

−Δπ =div

ψf

+divG +Rediv

ψv·∇v

in R

(3.16)

in the sense of distributions. We now observe that we can write f as follows (again

in the sense of distributions)

f =divF, F ∈ L

(Ω). (3.17)

In fact, it is enough to choose F

= ∇E ∗ f

,where{f

}

∞

k=1

⊂ C

∞

(Ω) converges

to f in L

(Ω), and then pass to the limit k →∞, in the sense of distributions. We

can express, again in the sense of distributions,

ψf = ψ div F =div[ψF] − F ·∇ψ.

Thus, introducing

G := G − F ·∇ψ,

F := ψF, and

f := div

from (3.16) we have

−Δπ =div

f +div

G +Rediv

ψv·∇v

in R

, (3.18)

where

f =div

F ∈ L

), and

G ∈ L

) with supp(

G) ⊂ B

2ρ

Consider now the three separate equations

−Δπ

=div

f in R

, (3.19)

−Δπ

=div

G in R

, (3.20)

−Δπ

=Rediv

ψv·∇v

in R

, (3.21)

with respect to unknowns π

,π

. Using the Riesz transformations,

: L

) → L

), ∀q>1, R

(u):=F

−1



|ξ|

F(u)



where F denotes the Fourier transformation, we ﬁnd that

:= F

−1



iξ

|ξ|

)



= F

−1



−ξ

|ξ|

)



= −R

◦ R

(

) (3.22)

is a solution to (3.19) with π

∈ L

). Moreover, since clearly

G ∈ L

), we

can use the Riesz potential

I : L

) → L

), I(u):=F

−1



|ξ|

F(u)



to obtain a solution

:= F

−1



iξ

|ξ|







= i R

◦ I





(3.23)

260 G.P. Galdi and M. Kyed

to (3.20) with π

∈ L

). Similarly, putting h := Re ψv·∇v,wehaveh ∈ L

)

and obtain by

:= F

−1



iξ

|ξ|

F(h

)



= i R

◦ I(h

) (3.24)

a solution to (3.21) with π

∈ L

). We furthermore conclude that

∂

= F

−1



−ξ

|ξ|

)



= −R

◦ R

(

) ∈ L

), (3.25)

∂

= F

−1



−ξ

|ξ|







= −R

◦ R





∈ L

), (3.26)

∂

= F

−1



−ξ

|ξ|

F(h

)



= −R

◦ R

) ∈ L

), (3.27)

for k =1, 2, 3. We therefore deduce that

¯π(x):=π

(x)+π

(x) (3.28)

is a solution to (3.18) with ¯π ∈ L

)and∇¯π ∈ L

). Since also π satisﬁes

the same equation, it follows that Z := ∇(¯π − π)isharmonicinR

,sothat,by

the mean-value theorem, we have for each ﬁxed x ∈ R

Z(x)=



(x)

∇(¯π − π)dy =:



(R)+I

(R)



. (3.29)

By the H¨older inequality we ﬁnd

(R)|≤∇¯π

≤ c

R. (3.30)

Moreover, from Lemma 3.1, we have v ∈ D

2,2

(Ω). Thus, Δw ∈ L

), and from

(3.15)

we infer

∇π

(1 + |x|)

∈ L

Therefore, by Schwarz inequality,

(R)|≤c

R ∇π/(1 + |y|)

≤ c

. (3.31)

Combining (3.29)–(3.31) and letting R →∞, we ﬁnd Z(x) = 0 for all x ∈ R

Hence, ¯π = π+c,forsomeconstantc, which concludes the proof of the lemma. 

In the next lemma, we show that a weak solution satisfying the energy in-

equality and a solution decaying like

|x|

must coincide under a suitable smallness

condition. The proof follows essentially that of the main theorem in [6].

Lemma 3.3. Let f ∈ L

(Ω)

∩L

(Ω)

,andv

∗

∈ W

(∂Ω)

. Moreover, let (v, p) ∈

1,2

(Ω)

∩L

(Ω)

×L

loc

(Ω) be a solution to (1.1) that satisﬁes the energy inequality

(1.4).If(w, π) ∈ D

1,2

(Ω)

∩L

(Ω)

×L

(Ω) is another solution to (1.1) and [[ w]]

8Re

,then(w, π)=(v, p). In this case, (v, p) satisﬁes the energy equality (1.5).

Leray Solution around Rotating Obstacle 261

Proof. By standard regularity theory for elliptic systems, we have (v,p), (w, π) ∈

2,2

loc

(Ω) × W

1,2

loc

(Ω). We can thus multiply (1.1)

with w and integrate over Ω

(R>diam(R

\ Ω)). By partial integration, we then obtain

−



∇v : ∇w dx +



∂ B

(∇v · n) · w dS −



∂ B

p (w · n)dS

− Re



(v ·∇v) · w dx +Ta





×x ·∇v − e

×v



·w dx

= −



∂Ω



(∇v − pI) · n



· w dS +



f · w dx.

(3.32)

Analogously, by switching the roles of v and w,weget

−



∇v : ∇w dx +



∂ B

(∇w · n) ·v dS −



∂ B

π (v · n)dS

−Re



(w ·∇w) · v dx +Ta





×x ·∇w − e

×w



·v dx

= −



∂Ω



(∇w − pI) · n



· v dS +



f ·v dx.

(3.33)

We shall now examine the integrals over ∂ B

in (3.32) and (3.33) in the

limit as R →∞. For this purpose, we utilize Lemma 3.2 and obtain p ∈ L

(Ω

)

for some ρ>0. Consequently, we can ﬁnd a sequence {R

}

∞

n=1

⊂ [ρ, ∞]sothat

lim

n→∞

= ∞ and

lim

n→∞



∂ B

|p|

+ |∇v|

+ |v|

+ |π|

+ |∇w|

+ |w|

=0. (3.34)

We conclude that



∂ B

(∇v · n) · w dS

≤ c

[[ w]]



∂ B

|∇v|

≤ c

[[ w]]





∂ B

|∇v|



→ 0asn →∞

(3.35)

and



∂ B

p (w · n)dS

≤ c

[[ w]]



∂ B

|p|

≤ c

[[ w]]





∂ B

|p|



→ 0asn →∞.

(3.36)

262 G.P. Galdi and M. Kyed

Furthermore,



∂ B

(∇w · n) · v dS

≤ c





∂ B

|∇w|







∂ B

|v|



= c





∂ B

|∇w|







∂ B

|v|



→ 0asn →∞

(3.37)

and



∂ B

π (v · n)dS

≤ c





∂ B

|π|







∂ B

|v|



→ 0asn →∞.

(3.38)

We now turn our attention to the limits as R

→∞of the integrals over Ω

in (3.32) and (3.33). We begin to observe that, by using a Hardy-type inequality

(see for example [7, Theorem II.5.1]), we ﬁnd



|(v ·∇v) · w|dx ≤ [[ w]]





|∇v|







|v|

(1 + |x|)



< ∞. (3.39)

Consequently,

lim

n→∞



(v ·∇v) · w dx =



(v ·∇v) ·w dx. (3.40)

Similarly, we have



|(w ·∇w) · v|dx ≤ [[ w]]





|∇w|







|v|

(1 + |x|)



< ∞

and thus

lim

n→∞



(w ·∇w) ·v dx =



(w ·∇w) · v dx. (3.41)

Concerning the integrals involving the data f, we observe that they are both well

deﬁned, in the sense of Lebesgue, because f ∈ L

(Ω) and w, v ∈ L

(Ω). We thus

ﬁnd

lim

n→∞



f ·v dx =



f · v dx. (3.42)

and

lim

n→∞



f · w dx =



f · w dx. (3.43)

Leray Solution around Rotating Obstacle 263

Now put u := v − w.Then





×x ·∇u − e

×u



· u dx =





×x ·∇u



· u dx



∂ B

|u|

×x) ·n dS =0,

(3.44)

where the last equality holds since n =

|x|

on ∂ B

. By the same argument, we

also have





×x ·∇v − e

×v



·v dx =



∂Ω

∗

×x) · n dS

(3.45)

and





×x ·∇w − e

×w



·w dx =



∂Ω

∗

×x) · n dS.

(3.46)

It follows from (3.44), (3.45), and (3.46) that





×x ·∇v − e

×v



·w dx +





×x ·∇w − e

×w



· v dx



∂Ω

∗

×x) · n dS.

(3.47)

Adding together (3.32) and (3.33), utilizing (3.47), and ﬁnally letting n →∞,

we ﬁnd that

− 2



∇v : ∇w dx

=Re





(v ·∇v) ·w dx +



(w ·∇w) · v dx





f · v dx −



∂Ω



(∇v − pI) · n



·v

∗



f · w dx −



∂Ω



(∇w − πI) · n



· v

∗

− Ta



∂Ω

∗

×x) · n dS.

(3.48)

We can now write



|∇u|

dx =



|∇v|

dx +



|∇w|

dx − 2



∇v : ∇w dx. (3.49)

By assumption, (v, p) satisﬁes the energy inequality



|∇v|

dx ≤−



f · v dx +



∂Ω



(∇v − pI) · n



· v

∗

−



∂Ω

∗

·n dS +



∂Ω

∗

×x ·n dS.

(3.50)

264 G.P. Galdi and M. Kyed

From the decay properties of (w, π), it is easy to verify that (w, π) satisﬁes the

energy equality



|∇w|

dx = −



f ·w dx +



∂Ω



(∇w − πI) · n



· v

∗

−



∂Ω

∗

· n dS +



∂Ω

∗

×x ·n dS.

(3.51)

Combining now (3.48), (3.49), (3.50), and (3.51), we obtain



|∇u|

dx ≤ Re





(v ·∇v) · w dx +



(w ·∇w) · v dx



−Re



∂Ω

∗

·n dS.

Next, we observe that



(u ·∇u) · w dx −



(w ·∇u) ·u dx



(v ·∇v) · w dx +



(w ·∇w) · v dx −



∂Ω

∗

· n dS.

By an argument similar to (3.39) and (3.40), all integrals above are well deﬁned

and ﬁnite. We can now conclude that



|∇u|

dx ≤ Re





(u ·∇u) · w dx −



(w ·∇u) · u dx



and thus estimate, using again the Hardy-type inequality, this time in form



|u|

|x|

dx ≤ 4



|∇u|

dx,

valid for all ﬁelds vanishing at the boundary ∂Ω,



|∇u|

dx ≤ 2Re [[w]]





|u|

1+|x|

|∇u|dx



≤ 2Re [[w]]





|u|

(1 + |x|)







|∇u|



≤ 8Re [[w]]



|∇u|

dx.

(3.52)

We ﬁnally conclude that u =0when8Re[[w]]

< 1. 

4. Main theorem

Our main theorem follows as a consequence of Lemma 3.3 and the fact that a

solution (w, π) with the in Lemma 3.3 required properties exists, provided the

data are suitably restricted [10].

Leray Solution around Rotating Obstacle 265

Theorem 4.1. Let Ω ⊂ R

be an exterior domain of class C

and Re , Ta ∈ (0,B],

for some B>0.Supposev

∗

∈ W

(∂Ω)

and f =divF ,with

F :=



[[ F ]]

+[[f ]]

+ [[div div F ]]



< ∞. (4.1)

Then, there is a constant M

= M

(Ω,B) > 0 such that if



F + v

∗



(∂Ω)



, (4.2)

then a weak solution (v, p) ∈ D

1,2

(Ω)

∩L

(Ω)

×L

loc

(Ω) to (1.1) that satisﬁes the

energy inequality (1.4), that is, a Leray solution, also satisﬁes, for some constant

c ∈ R,

|v|

2,2

+[[v]]

+[[∇v]]

+[[p + c]]

+[[∇p]]

3,Ω

R ≤ C



F + v

∗



(∂Ω)



(4.3)

where C

= C

(Ω,B,R). Moreover, (v, p) satisﬁes the energy equality (1.5).Fi-

nally, (v, p) is unique (up to addition of a constant to p) in the class of weak

solutions satisfying (1.4).

Proof. The existence of a solution (w, π) satisfying the properties stated for (v, p)

has been established in [10, Theorem 2.1 and Remark 2.1] in the case v

∗

≡ 0.

Moreover, in [9] the methods from [10] have been further developed to also consider

this more general case. Now, from (4.3) – written with w and π in place of v and

p – and from (4.2), it follows that, if M

is taken “suﬃciently small”, we ﬁnd, in

particular, [[w]]

8Re

. Therefore, the stated properties for (v, p)atoncefollow

from the uniqueness Lemma 3.3. 

Remark 4.2. The properties satisﬁed by the Leray solution (v, p) in Theorem 4.1

imply that (v, p) is, in fact, physically reasonable in the sense of Finn [5].

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Giovanni P. Galdi

Department of Mechanical Engineering

and Materials Science

University of Pittsburgh

USA

e-mail: galdi@pitt.edu

Mads Kyed

Institut f¨ur Mathematik

RWTH-Aachen

Germany

e-mail: kyed@instmath.rwth-aachen.de

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 267–274

 2011 Springer Basel AG

A Remark on Maximal Regularity

of the Stokes Equations

Matthias Geissert and Horst Heck

Dedicated to Prof. Herbert Amann on the occasion of his 70th birthday

Abstract. Assuming that the Helmholtz decomposition exists in L

(Ω)

is proved that the Stokes equation has maximal L

-regularity in L

(Ω) for

s ∈ [min{q, q



}, max{q, q



}]. Here Ω ⊂ R

is an (ε, ∞) domain with uniform

-boundary.

Mathematics Subject Classiﬁcation (2000). 35Q30.

Keywords. Stokes equations, maximal regularity.

1. Introduction and main result

For the understanding of nonlinear parabolic equations the property of maximal

-regularity has been proved to be very useful. For equations considered in a

Hilbert space the question whether we have maximal L

-regularity estimates re-

duces to the question whether the associated operator generates an analytic C

semigroup. But the question is more diﬃcult to answer in a general Banach space

setting. See, e.g., [Ama95], [Ama97], [DHP03], [KW01] and the references therein.

In this paper we study the property of maximal L

-regularity for the Stokes

equations that are given by

∂

u − Δu + ∇p = f in Ω × (0,T)

div u =0 inΩ×(0,T)

u =0 on∂Ω × (0,T)

u(·, 0) = 0 in Ω

(1.1)

in a possibly unbounded domain Ω ⊂ R

.Hereu denotes the velocity of the ﬂuid

and p the pressure. We are going to study this set of equations in the L

-setting,

where q =2.