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338 R. Haller-Dintelmann and J. Rehberg

8. Concluding remarks

Remark 8.1. Under the additional Assumption 6.4, Theorem 5.4 implies maximal

parabolic regularity for −∇·μ∇ on H

−1,q

for every q ∈ [2, ∞[, as in the 2-d case.

Besides, the question arises whether the limitation for the exponents, caused

by the localization procedure, is principal in nature or may be overcome when

applying alternative ideas and techniques (cf. Theorem 4.3). We suggest that the

latter is the case.

Remark 8.2. Equations of type (1.1)/(1.2) may be treated in an analogous way, if

under the time derivative a suitable superposition operator is present, see [40] for

details.

Remark 8.3. In the semilinear case, it turns out that one can achieve satisfactory

results without Assumption 6.4, at least when the nonlinear term on the right-hand

side depends only on the function itself and not on its gradient.

Remark 8.4. Let us explicitly mention that Assumption 6.4 is not always fulﬁlled

in the 3-d case. First, there is the classical counterexample of Meyers, see [48], a

simpler (and somewhat more striking) one is constructed in [25], see also [26]. The

point, however, is that not the mixed boundary conditions are the obstruction but

a somewhat ‘irregular’ behavior of the coeﬃcient function μ in the inner of the

domain. If one is confronted with this, spaces with weight may be the way out.

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Robert Haller-Dintelmann

Technische Universit¨at Darmstadt

Fachbereich Mathematik

Schlossgartenstr. 7

D-64298 Darmstadt, Germany

e-mail: haller@mathematik.tu-darmstadt.de

Joachim Rehberg

Weierstrass Institute for Applied Analysis and Stochastics

Mohrenstr. 39

D-10117 Berlin, Germany

e-mail: rehberg@wias-berlin.de

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 343–355

 2011 Springer Basel AG

On the Relation Between the Large Time

Behavior of the Stokes Semigroup and

the Decay of Steady Stokes Flow at Inﬁnity

Toshiaki Hishida

DedicatedtoProfessorHerbertAmannonhis70th birthday

Abstract. Let e

−tA

be the Stokes semigroup over an unbounded domain Ω.

For construction of the Navier-Stokes ﬂow globally in time, it is crucial to

derive L

-L

decay estimate (1.4) for ∇e

−tA

;thus,givenΩ,weneedtoask

which (q, r) admits (1.4). The present paper provides a principle which inter-

prets how this question is related to spatial decay properties of steady Stokes

ﬂow in the domain Ω under consideration.

Mathematics Subject Classiﬁcation (2000). Primary 35Q30; Secondary 76D07.

Keywords. Stokes semigroup, L

-L

estimate, steady Stokes ﬂow.

1. Introduction

Let Ω be an unbounded domain in R

(n ≥ 2) with smooth boundary ∂Ω. Consider

the Stokes system for a viscous incompressible ﬂuid that occupies the domain Ω

subject to the non-slip boundary condition on ∂Ω. This paper makes it clear that

the temporal decay property for t →∞of the solution to the initial value problem

⎧

⎪

⎨

⎪

⎩

∂

u − Δu + ∇p =0 (x ∈ Ω,t>0),

div u =0 (x ∈ Ω,t≥ 0),

u =0 (x ∈ ∂Ω,t>0),

u(·, 0) = u

(x ∈ Ω),

(1.1)

Supported in part by Grant-in-Aid for Scientiﬁc Research, No. 19540170, from the Japan Society

for the Promotion of Science. The paper is in ﬁnal form and no version of it will be published

elsewhere.

344 T. Hishida

is closely related to the spatial decay property for |x|→∞of the solution to the

steady problem

−Δv + ∇π =divF, div v =0 (x ∈ Ω); v|

∂Ω

=0. (1.2)

Here, u(x, t)=(u

,...,u

)

and v(x)=(v

,...,v

)

are the velocity ﬁelds;

p(x, t)andπ(x) are the pressures.

Since the celebrated paper by Kato [28] for the whole space problem, it is

well known that L

-L

estimates (q ≤ r)

u(t)

≤ Ct

−(n/q−n/r)/2

u



(t>0), (1.3)

∇u(t)

≤ Ct

−(n/q−n/r)/2−1/2

u



(t>0), (1.4)

of the solution (the Stokes semigroup) to (1.1) are important tools for construction

of global strong solutions to the Navier-Stokes system with small initial data in

(Ω) (the smallness is redundant for the case n = 2). Estimates (1.3) and (1.4)

play the same role as fractional powers of the Stokes operator, which were em-

ployed by Fujita and Kato [19] and Giga and Miyakawa [23] when Ω is a bounded

domain. So far, besides those cases – bounded domain, the whole space –, the

above-mentioned Navier-Stokes theory (n ≥ 3) has been established (or essen-

tially available) for the following physically relevant unbounded domains:

(1) half-space ([44], [30]);

(2) exterior domain ([27]);

(3) aperture domain with ﬂux condition ([25], [33]);

(4) perturbed half-space ([34], [35]);

(5) inﬁnite layer ([1]);

(6) asymptotically ﬂat layer ([2], [3]);

(7) inﬁnite cylinder ([40]).

The last three cases (5)–(7) are not kept in mind in this paper because the solution

to (1.1) decays exponentially and algebraic decay properties (1.3)–(1.4) do not

make sense as optimal decay.

In view of the Navier-Stokes nonlinearity u ·∇u, whether or not one can

prove the global existence theorem as above depends crucially on the answer to

the following question: Which (q, r) admits the gradient estimate (1.4)? The case

q = r = n is particularly important because the initial data are taken from L

(Ω).

On the other hand, it is also interesting to ask optimal decay at space inﬁnity

(expected in general) of the solution to (1.2) with F ∈ C

∞

(Ω)

n×n

. The purpose

of this paper is to show a principle which interprets the relation between both

questions when we do not restrict ourselves to the speciﬁed domains (1)–(4) above

but consider general unbounded domains which satisfy the reasonable hypothesis

(H1) (see Section 2). Roughly speaking (see Theorem 2.3 to be precise), it is proved

that if (1.4) holds for q ≤ r<q

, then the problem (1.2) possesses a solution of

class v ∈ L

(Ω)

,s>p

,foreachF ∈ C

∞

(Ω)

n×n

,where1/p

=1− 1/n −1/q

;

thus, (1.4) for larger r implies better summability of the steady ﬂow at inﬁnity.

Indeed the opposite implication is hopeless, but the best possible range of (q,r)

Large Time Behavior of the Stokes Semigroup 345

which admits (1.4) is suggested as long as we know the optimal rate of decay

of generic steady ﬂows in the domain Ω under consideration (as for an example

related to this idea, see (iii) in the ﬁnal section).

The idea of the proof is very simple. We consider the unsteady problem

⎧

⎨

⎩

∂

v − Δv + ∇π =divF (t)(x ∈ Ω,t∈ R),

div v =0 (x ∈ Ω,t∈ R),

v =0 (x ∈ ∂Ω,t∈ R),

(1.5)

and look for a solution which remains bounded for t →−∞in a suitable sense

so that the uniqueness of solutions is ensured. We have time-periodic solutions in

mind as typical examples, and steady solution can be regarded as a special case

of them. So the problem is to derive L

-summability of the solution v(t)atspace

inﬁnity by use of (1.4) and we wish to ﬁnd the summabilty exponent s as small

as possible.

For the unbounded domains (1), (3) and (4) mentioned above, we know (1.4)

for 1 <q≤ r<∞ ([44], [33], [34], [35]), so that our result implies the summability

v ∈ L

(Ω)

,s>n/(n − 1), for (1.2). As for the exterior problem, estimate (1.4)

for 1 <q≤ r ≤ n was proved by Iwashita [27], Dan and Shibata [12] (n =2)

and Maremonti and Solonnikov [36]. Indeed this is fortunately enough to solve

the Navier-Stokes system globally in time, but it is interesting to ask whether the

restriction r ≤ n is optimal or not. As a corollary of our result (see Corollary 2.4),

it turns out that (1.4) for 1 <q≤ r<q

with some q

>nis impossible since we

know the optimal rate of decay of generic steady ﬂows in exterior domains (Lemma

4.1). This fact was ﬁrst pointed out by Maremonti and Solonnikov [36]. Although

the spatial decay property of exterior ﬂows is the point in [36] as well, their proof

is diﬀerent from ours explained above. In fact, they multiply the equation (1.1)

by the decaying solution of

−Δv + ∇π =0, div v =0 (x ∈ Ω); v|

∂Ω

= b ∈ C

(∂Ω)

and estimate v, u(t) by using (1.4) with r>nto accomplish |v, u

| ≤ Cu



n/2

for all u

∈ C

∞

0,σ

(Ω), yielding v ∈ L

n/(n−2)

(Ω)

, a contradiction. Our strategy in

the present paper is merely to estimate the solution v(t) of (1.5) and thus the

proof is considerably easier.

This paper consists of ﬁve sections. In Section 2 we provide our results. Sec-

tion 3 is devoted to the proof of the main theorem. Section 4 studies the exterior

problem to show the corollary. We conclude the paper with some remarks in the

ﬁnal section.

2. Results

To begin with, we ﬁx notation. Let Ω be a domain in R

.TheclassC

∞

(Ω) consists

of all C

∞

-functions whose supports are compact in Ω. Set C

∞

(Ω) = {f = g|

; g ∈

∞

)}.For1≤ q ≤∞we denote by L

(Ω) the usual Lebesgue space with norm

·

q,Ω

;wesimplywrite·

= ·

q,Ω

when Ω is a given unbounded domain in

346 T. Hishida

(1.1), (1.2). Let C

∞

0,σ

(Ω) be the class of all vector ﬁelds u =(u

,...,u

)

which

satisfy u

∈ C

∞

(Ω) (1 ≤ j ≤ n) and div u =0.For1<q<∞ we deﬁne L

(Ω)

by the completion of C

∞

0,σ

(Ω) in the space L

(Ω)

. Note that the space L

(Ω)

may involve an additional side condition for some Ω; for instance, vanishing ﬂux

condition through an aperture is hidden in the case of aperture domains, see [18],

[39].

Given an unbounded domain Ω in (1.1) with smooth boundary ∂Ω, as usual,

the ﬁrst step is to establish the Helmholtz decomposition

(Ω)

= L

(Ω) ⊕ G

(Ω),G

(Ω) = {∇p ∈ L

(Ω)

; p ∈ L

loc

(Ω)}. (2.1)

For example, we refer to [38], [42] for exterior domains, and to [18], [39] for aperture

domains; also, see [20, Section III.1] and the references therein. When q =2,we

have (2.1) for any domain even if assuming no regularity on ∂Ω, where L

loc

(Ω)

should be replaced by L

loc

(Ω) in G

(Ω). However, that is not always the case for

q =2.Infact,whichq admits the decomposition (2.1) depends on the geometry

of the domain, see [6], [37]. Assuming (2.1) for some q,wedenotebyP = P

the projection operator from L

(Ω)

onto L

(Ω) along G

(Ω). Then the Stokes

operator is deﬁned by



(A)={u ∈ W

2,q

(Ω)

∩ L

(Ω); u|

∂Ω

=0},

Au = −P Δu,

(2.2)

where W

2,q

(Ω) is the usual L

-Sobolev space of second order.

We make the following assumption on the Stokes operator.

(H1) There is q

∈ (2, ∞] such that the following properties hold: Set



n−q

(2 <q

<n),

∞ (n ≤ q

≤∞),





1(r

= ∞),

−1

< ∞).

Then, for every q ∈ (r



), the decomposition (2.1) holds and the oper-

ator −A generates a bounded analytic semigroup {e

−tA

}

t≥0

on L

(Ω).

Furthermore, u(t)=e

−tA

with u

∈ C

∞

0,σ

(Ω) satisﬁes

(1.3) for r



<q≤ r<r

; (1.4) for r



<q≤ r<q

Remark 2.1. Note that r



always holds. It is also natural to assume (1.3) for

r<r

when assuming (1.4) for r<q

from the viewpoint of embedding relation.

Remark 2.2. When we think of the duality relation between existence and unique-

ness for the Neumann problem, it seems to be reasonable to assume (2.1) for

q ∈ (r



)withsomer

; and then, we have the relation P

∗

= P

q/(q−1)

.When

the boundary ∂Ω is unbounded, this exponent r

depends on the geometry of ∂Ω

near inﬁnity, no matter how smooth it is ([6], [37]).

Our main theorem reads as follows.

Large Time Behavior of the Stokes Semigroup 347

Theorem 2.3. Suppose (H1) and deﬁne the exponent p

=1−

−

. (2.3)

Then, for each F ∈ C

∞

(Ω)

n×n

, the problem (1.2) necessarily admits a solution v,

which is of class

v ∈ L

(Ω), ∀s ∈ (p

). (2.4)

Here, we note the relation r



on account of q

> 2.

Let us next consider the exterior problem. We know that (1.4) holds for

1 <q≤ r ≤ n ([27], [12], [36]). In order to make the linear theory complete, we

wish to ask whether or not the restriction r ≤ n is optimal. The following corollary

tells us that the answer is positive.

Corollary 2.4. Let Ω be an exterior domain in R

,n≥ 2, with smooth boundary

∂Ω. Then it is impossible to have (1.4) for 1 <q≤ r<q

with some q

>n.

Remark 2.5. When r>nfor the exterior problem, we have slower decay estimate

∇e

−tA



≤ Ct

−

u



(t ≥ 1),

for 1 <q≤ r, see [27], [12], [36].

3. Proof of Theorem 2.3

This section studies the unsteady problem (1.5) in order to prove Theorem 2.3.

Concerning the external force, we may consider very smooth one; say, we make

the following assumption.

(H2) F (t) ∈ C

∞

(Ω)

n×n

for each t ∈ R and sup

t∈R

F (t)

∞

< ∞.Thereisa

compact set K ⊂ R

such that Supp F (t) ⊂ K for all t ∈ R.Furthermore,

there is α ∈ (0, 1) such that div F ∈ C

loc

(R; L

∞

(Ω)

Under the condition (H1) we rewrite (1.5) as

+ Av = P (div F (t)) (t ∈ R) (3.1)

in L

(Ω),r∈ (r



). We say that v(t) is a solution to (3.1) if it is of class

v ∈ C

(R; L

(Ω)) ∩ C(R; D

(A))

and satisﬁes (3.1) in L

(Ω) for some r ∈ (r



). If this solution enjoys the addi-

tional condition

v(t) ∈ L

(Ω), ∀t ∈ R;sup

t∈R

v(t)

< ∞ (3.2)

for some s ∈ (r



,r), then it must be of the form

v(t)=



−∞

−(t−τ )A

P (div F (τ)) dτ. (3.3)