DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Supplement 2011
Website: www.aimSciences.org
pp. 351–361
A LINEARIZED SYSTEM DESCRIBING STATIONARY
INCOMPRESSIBLE VISCOUS FLOW AROUND ROTATING AND
TRANSLATING BODIES: IMPROVED DECAY ESTIMATES OF
THE VELOCITY AND ITS GRADIENT
Paul Deuring
Univ Lille Nord de France, 59000 Lille, France
ULCO, LMPA, 62228 Calais cédex, France
Stanislav Kračmar and Šárka Nečasová
Department of Technical Mathematics
Czech Technical University
Karlovo nám. 13
12135 Prague 2, Czech Republic
Mathematical Institute
Academy of Sciences of the Czech Republic
Žitná 25
115 67 Prague 1, Czech Republic
Abstract. We consider a linearization of a model for stationary incompressible viscous flow past a rigid body performing a rotation and a translation.
Using a representation formula, we obtain pointwise decay bounds for the velocity and its gradient. This result improves estimates obtained by the authors
in a previous article.
1. Introduction. We consider the system of equations
−∆u(z) − (U + ω × z) · ∇u(z) + ω × u(z) + ∇π(z) = f (z), div u(z) = 0
(1)
3
for z ∈ R \D.
This system arises by linearization and normalization of a mathematical model
describing the stationary flow of a viscous incompressible fluid around a rigid body
moving at a constant velocity and rotating at a constant angular velocity. We
refer to [14] for more details on the physical background of (1). Here we only
indicate that D ⊂ R3 is an open bounded set describing the rigid body, the vector
U ∈ R3 \{0} represents the constant translational velocity of this body, and the
vector ω ∈ R3 \{0} its constant angular velocity. The given function f : R3 \D 7→ R3
describes a body force, and the unknowns u : R3 \D 7→ R3 and π : R3 \D 7→ R
correspond respectively to the normalized velocity and pressure field of the fluid.
We are interested only in the case U 6= 0, so we require this latter inequality to
hold. In addition, we assume the vectors U and ω to be parallel. This is a restriction
of generality only if U · ω = 0. Otherwise, it may be achieved by a suitable change
of variables that the relation U = σ · ω is valid for some σ ∈ R\{0}; see [18, Section
2000 Mathematics Subject Classification. Primary: 35Q30, 65N30; Secondary: 76D05.
Key words and phrases. viscous incompressible flow, rotating body, fundamental solution, decay, Navier-Stokes system.
351
352
PAUL DEURING, STANISLAV KRAČMAR AND ŠÁRKA NEČASOVÁ
1] for more details. By another transformation of variables, we may suppose there
is some τ > 0 with U = −τ · (1, 0, 0), hence ω = % · (1, 0, 0) for some % ∈ R\{0}. In
this way we end up with the following variant of equation (1):
L(u) + ∇π = f, div u = 0
in R3 \D,
(2)
where the differential operator L is defined by
L(u)(z) := −∆u(z) + τ · ∂1 u(z) − (ω × z) · ∇u(z) + ω × u(z)
for u ∈
2,1
Wloc
(U )3 ,
(3)
3
z ∈ U, U ⊂ R open.
According to Galdi, Silvestre [19, Theorem 3], if the velocity part u of a solution to
(2) verifies the relation
sup{|u(x)| · (1 + |x|) : x ∈ BSc 0 } < ∞
(4)
for some S0 > 0 (”physical reasonable solution”), then the following pointwise decay
estimate holds:
3/2
|u(y)| · |y| · 1 + τ · (|y| − y1 ) + |∇u(y)| · |y| · 1 + τ · (|y| − y1 )
(5)
≤ C · ku|∂Dk3/2, 2 + kf |BS k∞
5/2
+ sup{ |f (x)| · |x| · 1 + τ · (|x| − x1 )
: x ∈ BSc }
for y ∈ BSc , where S > 0 with D ⊂ BS . (See Section 2 for our notation.) The
constant C depends on D, S, τ and ω.
In [2], we could improve inequality (5) in some respects. For example, we showed
that a similar estimate as in (5) is valid even for weak solutions instead of physical
reasonable ones, and we used somewhat weaker norms of u and f in our decay
bound ([2, Theorem 5.3]). More precisely, we found that
3/2
|u(y)| · |y| · 1 + τ · (|y| − y1 ) + |∇u(y)| · |y| · 1 + τ · (|y| − y1 )
(6)
≤ C · k∇u|∂Dk1 + kπ|∂Dk1 + Cp · ku|∂Dk2−1/p, p + kf |BS k1
B
+ sup{ |f (x)| · |x|A · 1 + τ · (|x| − x1 )
: x ∈ BSc 1 }
for y ∈ BSc , where p ∈ (1, ∞) is linked to the regularity of u near ∂Ω, the parameters
S1 , S ∈ (0, ∞) are such that D ⊂ BS1 , S1 < S, and the numbers A, B ∈ R are
supposed to verify the relations A > 2, A + min{1, B} > 3 and A + B ≥ 7/2.
Examples for suitable values of A, B are A = 5/2, B = 1, and A = 2 + for some
∈ (0, ∞), B = 3/2. The constant C depends on S1 , S, A, B, τ and ω, whereas
Cp depends on D and p. Instead of (4), the function u is required to satisfy the
“nonlocal” decay conditions
u|BSc ∈ L6 (BSc ), ∇u|BSc ∈ L2 (BSc )9 , π|BSc ∈ L2 (BSc ),
(7)
which characterize our notion of weak solutions of (2).
On the other hand, an important feature of the theory in [19] is not included in
the results established in [2]. In fact, article [19] presents an estimate as in (5) for
physical reasonable solutions even of the nonlinear problem
L(u) + τ · (u · ∇)u + ∇π = f, div u = 0
in R3 \D,
(8)
under the conditions that the volume force f and the Dirichlet boundary data of
u are small in a suitable sense. In [16], it is shown that a decay estimate as in (5)
holds even for Leray solutions of (8) if the support of f is compact. No smallness
DECAY ESTIMATES
353
condition on the data is required in [16]. It should be noted that the notion of Leray
solution is more general than that of a weak solution as understood here and in [2].
In fact, the latter notion requires the pressure π to be L2 -integrable outside a ball
around Ω, whereas the former supposes only local integrability of π. We further
remark in this context that in [1] and [2], we mistakenly referred to [18] instead of
[19] when mentioning the decay results established by Galdi and Silvestre.
Our proof of (6) in [2] is largely based on the use of the fundamental solution of
(2) constructed by Guenther, Thomann [21]. This fundamental solution allows to
establish a representation formula for solutions to (2). Then the problem reduces to
estimating the terms of this formula. To this end, we derived suitable upper bounds
of the Guenther-Thomann solution and of its gradient ([2, (2.22)]), which then led
to inequality (6), via the representation formula just mentioned.
However, in [2] we did not derive any estimate of the second-order derivatives of
the fundamental solution in question. As a consequence, the term ku|∂Dk2−1/p, p
arises on the right-hand side of (6); c.f. the remarks preceding Corollary 1 below.
But by now, suitable estimates of these second derivatives have been established in
[3]. In the work at hand, we want to show how to exploit these estimates in order
to replace the term ku|∂Dk2−1/p, p on the right-hand side of (6) by the quantity
ku|∂Dk1 (Theorem 6). In doing this, we consider functions u that are somewhat
more general than the velocity part of a solution to (2). In fact, we admit vector
fields u whose divergence has compact support but does not necessarily vanish.
Our results may be interpreted in the sense that we derive pointwise decay bounds
of u and ∇u in terms of L(u) + ∇π, div u, u|∂D, ∇u|∂D and π|∂D, under the
assumption that the pair (u, π) satisfies (7) as well as some integrability conditions
near ∂D.
Concerning other previous articles besides [2], [3], [14], [16], [18], [19], [21] pertaining to equation (2) or (8) or to their time-dependent counterparts, we mention
[1], [4] – [13], [15], [17], [20], [22]– [29], [35] – [37]. Additional references may be
found in [14].
We further mention that a main idea in [17] – [19] consists in reducing a boundary
value problem related to (2) to the Oseen system in the whole space R3 . That latter
system may be handled by using a well-known Oseen fundamental solution, studied
in [30] for example. As remarked above, our approach is based on the GuentherThomann fundamental solution of (2). In fact, we proceed in a similar way as
Farwig, Hishida [8], [9], who considered the linear equation (2) and the nonlinear
one (8) in the case τ = 0 (flow around a body that rotates but does not perform
a translation). It turned out that a fundamental solution of (2) with τ = 0 may
be constructed in two steps: first a suitable rotational term is introduced into
the fundamental solution of the time-dependent Stokes system; then the function
obtained in this way is integrated with respect to time. With this fundamental
solution as starting point, Farwig and Hishida succeeded in exhibiting detailed
profiles of the flow in question, both in the linear ([9]) and in the nonlinear case
([8]). The profiles we obtained in [2] and [3] for the case τ 6= 0 are less elaborated.
This is due to the markedly more complicated structure of the Guenther-Thomann
fundamental solution compared to the function constructed by Farwig, Hishida.
But the results of the latter authors may serve as a guide for future research with
respect to the case τ 6= 0. Reference [6] discusses the situation arising when the
translational velocity is not parallel to the axis of rotation.
354
PAUL DEURING, STANISLAV KRAČMAR AND ŠÁRKA NEČASOVÁ
Concerning an approach to (2) in weighted spaces, Kračmar, Nečasová, Penel [27]
– [29] work in a L2 -framework with anisotropic weights, extending to (2) the theory
established in [31], [32] for a simplified Oseen-type equation. In particular, a positive
answer could be given to the question of existence of a wake, independently of [19],
where the wake phenomenon is captured by inequality (5). Another possibility to
deal with (2) consists in working in an Lq -framework; then weighted multiplier and
Littlewood-Paley theory are used, as well as the theory of one-sided Muckenhoupt
weights corresponding to one-sided maximal functions. This approach was first
introduced by Farwig, Hishida, Müller [10] (zero velocity at infinity) and Farwig
[4], [5] (nonzero velocity at infinity) for the case that no weight is present, and then
extended to the weighted case by Farwig, Krbec, Nečasová [11], [12] and Nečasová,
Schumacher [37]. The case of singular data was studied in this framework in [26].
Pointwise estimates, even for solutions of the nonlinear Navier-Stokes equations,
can be found in [15] (τ = 0) and [18] (τ 6= 0). Indeed, according to that latter
reference, there exists a stationary strong solution of the nonlinear problem (8)
with the velocity part u of this solution satisfying the inequality |u(x)| ≤ c/|x|.
This pointwise estimate suggests to discuss (1.2) in weak Lq -spaces (L3/2,∞ and
L3,∞ ) as done in [7], [24]. Stability estimates in the L2 -setting are proved in [18],
and in the L3,∞ -setting in [25].
2. Notation, definitions and auxiliary results. If A ⊂ R3 , we write Ac for
the complement R3 \A of A. The symbol | | denotes the Euclidean norm of R3 and
also the length of a multiindex in N30 , that is, |α| := α1 + α2 + α3 for α ∈ N30 .
The open ball centered at the origin and with radius r > 0 is denoted by Br . Put
e1 := (1, 0, 0). Let x × y denote the usual vector product of x, y ∈ R3 .
The parameters τ ∈ (0, ∞), % ∈ R\{0} and ω ∈ R3 \{0} introduced in Section 1
will be kept fixed throughout. Recall that ω = % · e1 ; further recall the definition of
the differential operator L in (3). We put
sτ (x) := 1 + τ · (|x| − x1 )
for x ∈ R3 .
By the symbol C, we denote constants only depending on τ or ω. We write
C(γ1 , ..., γn ) for constants that additionally depend on parameters γ1 , ..., γn ∈ R,
for some n ∈ N. Using this convention, we recall a result from [2].
Lemma 1 ([2, Lemma 2.4]). Let S ∈ (0, ∞). Then |x| ≥ C(S) · sτ (x) for x ∈ BSc .
The open bounded set D ⊂ R3 introduced in Section 1 will be kept fixed, too.
We suppose that D is C 2 -bounded, and write n(D) for its outward unit normal. For
T ∈ (0, ∞), we put DT := BT \D.
For p ∈ [1, ∞), k ∈ N, and for open sets A ⊂ R3 , we write W k,p (A) for the usual
Sobolev space of order k and exponent p. Its standard norm will be denoted by
k,p
k kk,p . If B ⊂ R3 is open, define Wloc
(B) as the set of all functions g : B 7→ R
k,p
such that g|A ∈ W (A) for any open set A ⊂ R3 with A compact, A ⊂ B.
Also we will need the fractional order Sobolev space W 2−1/p,p (∂D)
equipped with
its intrinsic norm, which we denote by k k2−1/p, p p ∈ (1, ∞) ; see [34] for the
corresponding definitions. If H is a normed space whose norm is denoted by k kH ,
(n)
(n)
and if n ∈ N, we equip the product space Hn with a norm k k defined by kvk :=
H
H
P
1/2
(n)
n
n
2
for v ∈ H . But for simplicity, we will write k kH instead of k k .
j=1 kvj kH
H
DECAY ESTIMATES
355
Next we introduce the Guenther-Thomann fundamental solution of (2). We begin
by setting
2
K(x, t) := (4 · π · t)−3/2 · e−|x|
/(4·t)
for x ∈ R3 , t ∈ (0, ∞)
(fundamental solution of the heat equation). The Kummer function 1 F1 (1, 5/2, · )
used below is defined by
1 F1 (1, 5/2, u) :=
∞
X
Γ(5/2) · Γ(n + 5/2)−1 · un
for u ∈ R.
n=0
Here the letter Γ denotes the usual Gamma function. Further put
Hjk (x) := xj · xk · |x|−2
for x ∈ R3 \{0},
Λjk (x, t)
:= K(x, t) · δjk − Hjk (x) − 1 F1 1, 5/2, |x|2 /(4 · t) · δjk /3 − Hjk (x)
for x ∈ R3 \{0}, t ∈ (0, ∞), j, k ∈ {1, 2, 3},




0 −ω3
ω2
0 0
0
0 −ω1  = % ·  0 0 −1  ,
Ω :=  ω3
0 1
0
−ω2
ω1
0
−t·Ω
Γjk (y, z, t) 1≤j,k≤3 := Λrs (y − τ · t · e1 − e
· z, t) 1≤r,s≤3 · e−t·Ω
for y, z ∈ R3 , t ∈ (0, ∞) with y − τ · t · e1 − e−t·Ω · z 6= 0. The function (Γjk )1≤j,k≤3
is the velocity part of a fundamental solution of the time-dependent variant of (2);
see [21]. According to [1, Lemma 3.3] or [3, Lemma 2.6], we have
Lemma 2. Let y, z ∈ R3 with y 6= z, j, k ∈ {1, 2, 3}. Then
|Γjk (y, z, t)| ≤ C(y, z) · (χ(0,1] (t) + χ(1,∞) (t) · t−3/2 )
so that
R∞
0
for t ∈ (0, ∞),
|Γjk (y, z, t)| dt < ∞.
Due to Lemma 2, we may define
Z ∞
Zjk (y, z) :=
Γjk (y, z, t) dt,
for y, z ∈ R3 with y 6= z, j, k ∈ {1, 2, 3}.
0
We further set
E4j (x) := (4 · π)−1 · xj · |x|−3
(1 ≤ j ≤ 3, x ∈ R3 \{0}).
The function (Zjk )1≤j,k≤3 is the velocity part of the Guenther-Thomann fundamental solution to (2), and the function (E4j )1≤j≤3 constitutes its pressure part. Note
that the latter function arises as pressure part not only of the Guenther-Thomann
fundamental solution of the rotational problem (2), but also of the usual fundamental solution of the Stokes and Oseen system, respectively, in the stationary and in
the nonstationary case. The following estimate of Zjk , established in [2] and [3],
will play a key role in the proof of our decay estimates.
Theorem 1. Let j, k ∈ {1, 2, 3}. Then Zjk ∈ C 2 (R3 × R3 )\{(x, x) : x ∈ R3 } .
Let S1 , S ∈ (0, ∞) with S1 < S. Then
−1−|α+β|/2
|∂yα ∂zβ Zjk (y, z)| ≤ C(S1 , S) · |y| · sτ (y)
(9)
for y ∈ BSc , z ∈ BS1 , α, β ∈ N30 with |α + β| ≤ 2.
356
PAUL DEURING, STANISLAV KRAČMAR AND ŠÁRKA NEČASOVÁ
Proof. We refer to [3, Lemma 3.2] as concerns the first part of the theorem. Inequality (9) in the case |α + β| ≤ 1 holds according to [2, Theorem 2.19]. If |α + β| = 2,
we refer to [3, Theorem 3.1].
3. Decay estimates. For the convenience of the reader, we state two lemmas
proved in [3]. They involve estimates of certain boundary and volume potentials.
Lemma 3. Let j, k, l ∈ {1, 2, 3}, g ∈ L1 (∂D), and put
Z
Z
F (y) :=
∂zl Zjk (y, z) · g(z) doz , G(y) :=
Zjk (y, z) · g(z) doz ,
∂D
∂D
Z
c
H(y) :=
E4j (y − z) · g(z) doz for y ∈ D .
∂D
c
Then F, G, H ∈ C 1 (D ), and
Z
∂ α F (y) =
∂D
Z
∂ α G(y) =
∂D
Z
α
∂ H(y) =
∂D
∂yα ∂zl Zjk (y, z) · g(z) doz ,
(10)
∂yα Zjk (y, z) · g(z) doz ,
∂yα E4j (y − z) · g(z) doz
c
for α ∈ N30 with |α| ≤ 1, y ∈ D .
Let S1 , S ∈ (0, ∞) with D ⊂ BS1 , S1 < S. Then
|∂ α F (y)| ≤ C(S1 , S) · kgk1 · |y| · sτ (y)
−3/2−|α|/2
|∂ α G(y)| ≤ C(S1 , S) · kgk1 · |y| · sτ (y)
−1−|α|/2
,
(11)
,
−2−|α|
α
|∂ H(y)| ≤ C(S1 , S) · kgk1 · |y|
for y ∈ BSc , α ∈ N30 with |α| ≤ 1.
Lemma 4. Let j, k ∈ {1, 2, 3}, R > 0, g ∈ L1 (BR ), and put
Z
Z
F (y) :=
Zjk (y, z) · g(z) dz, G(y) :=
E4j (y − z) · g(z) dz
BR
c
BR
c
1
for y ∈ BR . Then F, G ∈ C (BR ) and
Z
∂ α F (y) =
∂yα Zjk (y, z) · g(z) dz,
BR
Z
α
∂ G(y) =
∂yα E4j (y − z) · g(z) dz
BR
c
for α ∈ N30 with |α| ≤ 1, y ∈ BR . Let T ∈ (R, ∞). Then
|∂ α F (y)| ≤ C(R, T ) · kgk1 · |y| · sτ (y)
α
|∂ G(y)| ≤ C(R, T ) · kgk1 · |y|
−1−|α|/2
,
−2−|α|
for y ∈ BTc , α ∈ N30 with |α| ≤ 1.
In view of stating the representation formula mentioned in Section 1 and established in [2], we introduce some further notation. First we define two volume
DECAY ESTIMATES
357
potentials. For p ∈ (1, ∞), q ∈ (1, 2), f : R3 7→ R3 with f |BT ∈ Lp (BT )3 for any
T ∈ (0, ∞), f |BSc ∈ Lq (BSc )3 for some S ∈ (0, ∞), we put
Z X
3
Rj (f )(y) :=
Zjk (y, z) · fk (z) dz (1 ≤ j ≤ 3, y ∈ R3 ).
R3 k=1
Moreover, for p ∈ (1, ∞), q ∈ (1, 3), g : R3 7→ R with g|BT ∈ Lp (BT ) for any
T ∈ (0, ∞), g|BSc ∈ Lq (BSc ) for some S ∈ (0, ∞), define
Z
Sj (g)(y) :=
E4j (y − z) · g(z) dz (1 ≤ j ≤ 3, y ∈ R3 ).
R3
According to [2, Lemma 3.1, 3.4], the integral appearing in the definition of Rj (f )
and Sj (f ), respectively, is well defined at least for almost every y ∈ R3 . Next, for
p ∈ (1, ∞), we introduce the function space Mp as the set of all pairs (u, π) such
that
c
c
1,p
2,p
(D ), u|DT ∈ W 1,p (DT )3 , π|DT ∈ Lp (DT ),
u ∈ Wloc
(D )3 , π ∈ Wloc
u|∂D ∈ W 2−1/p, p (∂D)3 , div u|DT ∈ W 1,p (DT ), L(u) + ∇π|DT ∈ Lp (DT )3
for some T ∈ (0, ∞) with D ⊂ BT .
Note that if (u, π) ∈ Mp , then u belongs to W 2,p near ∂D, and π to W 1,p . This
is stated in the ensuing theorem, which is proved in [2] via standard Lp -regularity
theory for the Stokes system.
Theorem 2 ([2, Theorem 4.4]). Let p ∈ (1, ∞), (u, π) ∈ Mp , T ∈ (0, ∞) with
D ⊂ BT . Then u|DT ∈ W 2,p (DT )3 and π|DT ∈ W 1,p (DT ).
Theorem 2 means in particular that if (u, π) ∈ Mp , then the traces of u, ∇u and
π on ∂D are well defined. Hence, for p ∈ (1, ∞), (u, π) ∈ Mp , j ∈ {1, 2, 3}, we
c
may introduce a boundary potential Bj = Bj (u, π) : D 7→ R by setting
Bj (y)
Z
3 hX
3 X
:=
Zjk (y, z) · −∂l uk (z) + δkl · π(z) + uk (z) · (τ · e1 − ω × z)l
∂ D k=1 l=1
i
(D)
(D)
+∂zl Zjk (y, z) · uk (z) · nl (z) + E4j (y − z) · uk (z) · nk (z) doz
c
for y ∈ D . The representation formula mentioned above may now be stated as
follows.
Theorem 3 ([2, Theorem 4.6]). Let p ∈ (1, ∞), (u, π) ∈ Mp . Put f := L(u) + ∇π,
and suppose there are numbers q ∈ (1, 2), S ∈ (0, ∞) such that D ⊂ BS ,
u|BSc ∈ L6 (BSc )3 , ∇u|BSc ∈ L2 (BSc )9 , π|BSc ∈ L2 (Bsc ), f |BSc ∈ Lq (BSc )3 .
Let j ∈ {1, 2, 3}. Then
uj (y) = Rj (f )(y) + Sj (div u)(y) + Bj (y)
c
(12)
c
for a. e. y ∈ D . If p > 3/2, equation (12) holds for any y ∈ D , without the
restriction “a. e.”.
The idea now is to estimate the three terms on the right-hand side of (12) seperR
(D)
ately. But due to the integral ∂ D ∂zl Zjk (y, z) · uk (z) · nl (z) dz involved in the
358
PAUL DEURING, STANISLAV KRAČMAR AND ŠÁRKA NEČASOVÁ
definition of Bj (y), a second-order derivative of Zjk arises according to (10) whenever a first-order derivative is applied to Bj (y). This is the reason why the gradient
of Bj (y) cannot be estimated directly if no estimate of the second derivatives of
Zjk is available. In [2], we circumvented this difficulty by moving the derivative ∂zl
away from Zjk , using a partial integration. As part of this approach, we needed
an extension operator Ep : W 2−1/p, p (∂D) 7→ W 2,p (D), which gave rise to the term
ku|∂Dk2−1/p, p in our decay estimate of ∇Bj , and hence of ∇u. In the present context, however, we may refer to [2] for an estimate of the second derivatives of Zjk ,
obtaining inequality (11) in Lemma 3. This inequality and the other two estimates
in the same lemma yield the ensuing decay bounds for Bj and its gradient.
c
Corollary 1. Let p ∈ (1, ∞), (u, π) ∈ Mp , j ∈ {1, 2, 3}. Then Bj ∈ C 2 (D ).
Let S1 , S ∈ (0, ∞) with D ⊂ BS1 and S1 < S. Let α ∈ N30 with |α| ≤ 1, y ∈ BSc .
Then
|∂ α Bj (y)|
−1−|α|/2
≤ C(S1 , S) · ku|∂Dk1 + k∇u | ∂Dk1 + kπ|∂Dk1 · |y| · sτ (y)
.
Proof. Use Lemma 3 and observe that |y|−1 ≤ C(S) · sτ (y)−1 for y ∈ BSc by Lemma
1.
In view of (12), we still have to estimate the volume potentials Rj (f ) and
Sj (div u) in order to obtain decay estimates of u. If f and div u have compact
support, we may use Lemma 4. But concerning f , we consider the more difficult
case studied in [2], where the function f is supposed only to decay sufficiently fast,
without necessarily having compact support. As for div u, we distiguish two cases.
In one of them, we suppose that supp(div u) is compact and use Lemma 4. In the
other one, we refer to the theory in [2], which yields a bound for Sj (div u), but
not for ∇Sj (div u), under a decay condition on div u. For the convenience of the
reader, we restate the relevant results from [2] in the form of two theorems, the first
pertaining to Rj (f ), and the second to Sj (div u).
Theorem 4 ([2, Theorem 3.3]). Let S, S1 , γ ∈ (0, ∞) with S1 < S, p ∈ (1, ∞),
A ∈ [2, ∞), B ∈ R, f : R3 7→ R3 measurable with A + min{1, B} ≥ 3,
f |BS1 ∈ Lp (BS1 )3 ,
|f (z)| ≤ γ · |z|−A · sτ (z)−B
for z ∈ BSc 1 .
Let i, j ∈ {1, 2, 3}, y ∈ BSc . Then
|Rj (f )(y)| ≤ C(S, S1 , A, B) · (kf |BS1 k1 + γ) · |y| · sτ (y)
−1
· lA,B (y),
|∂yi Rj (f )(y)| ≤ C(S, S1 , A, B) · (kf |BS1 k1 + γ)
−3/2
· |y| · sτ (y)
· sτ (y)max(0, 7/2−A−B) · lA,B (y),
1
if A + min{1, B} > 3
where lA,B (y) :=
.
max(1, ln |y|) if A + min{1, B} = 3
Theorem 5 ([2, Theorem 3.5]). Let S, S1 , γ
e ∈ (0, ∞) with S1 < S, p ∈ (1, ∞), C ∈
(5/2, ∞), D ∈ R, g : R3 7→ R measurable with C + min{1, D} > 3,
g|BS1 ∈ Lp (BS1 ),
|g(z)| ≤ γ
e · |z|−C · sτ (z)−D
for z ∈ BSc 1 .
Let j ∈ {1, 2, 3}, y ∈ BSc . Then
|Sj (g)(y)| ≤ C(S, S1 , C, D) · (kg|BS1 k1 + γ
e) · |y|−2 .
DECAY ESTIMATES
359
Now we may prove the ensuing decay estimates for u and ∇u.
Theorem 6. Let p ∈ (1, ∞), (u, π) ∈ Mp . Put f := L(u) + ∇π. Suppose there are
numbers S1 , S, γ ∈ (0, ∞), A ∈ [2, ∞), B ∈ R such that S1 < S, D ⊂ BS1 ,
u|BSc ∈ L6 (BSc )3 , ∇u|BSc ∈ L2 (BSc )9 , π|BSc ∈ L2 (BSc ), supp(div u) ⊂ BS1 ,
A + min{1, B} ≥ 3, |f (z)| ≤ γ · |z|−A · sτ (z)−B for z ∈ BSc 1 .
Let j, l ∈ {1, 2, 3}, y ∈ BSc . Then
|uj (y)| ≤ C(S, S1 , A, B) · (K + kdiv uk1 ) · |y| · sτ (y)
−1
|∂yl uj (y)| ≤ C(S, S1 , A, B) · (K + kdiv uk1 ) · |y| · sτ (y)
· lA,B (y),
−3/2
(13)
·sτ (y)max(0, 7/2−A−B) · lA,B (y),
with the abbreviation
K := γ + kf |BS1 k1 + ku|∂Dk1 + k∇u | ∂Dk1 + kπ|∂Dk1 ,
and with the function lA,B (y) introduced in Theorem 4.
If the assumption supp(div u) ⊂ BS1 is replaced by the condition
|div u(z)| ≤ γ
e · |z|−C · sτ (z)−D
for z ∈ BSc 1 ,
for some γ
e ∈ (0, ∞), C ∈ (5/2, ∞), D ∈ R with C + min{1, D} > 3, then inequality
(13) remains valid if the term kdiv uk1 on the right-hand side is replaced by γ
e+
kdiv u|BS1 k1 . Of course, in that case the constant in (13) additionally depends on
C and D.
Note that if A + min{1, B} > 3, A + B ≥ 7/2 in Theorem 6, we have
sτ (y)max(0, 7/2−A−B) · lA,B (y) = 1
on the right-hand side of (13). The preceding conditions on A and B are verified if
for example A = 5/2, B = 1, or B = 3/2 and A = 2 + for some ∈ (0, 1/2).
Proof of Theorem 6. Starting from (12), we estimate Bj (y) by applying Corollary
1, Rj (f )(y) by using Theorem 4, and Sj (div u)(y) by referring to Lemma 4 (with
R, T replaced by S1 , S, respectively). In the estimate of Sj (div u)(y), we further
take into account that |y|−1 ≤ C(S) · sτ (y)−1 for y ∈ BSc (Lemma 1). Concerning
the last part of the theorem, pertaining to the case that supp(div u) is not compact,
we make use of Theorem 5 instead of Lemma 4.
Acknowledgments. The research of Š. N. was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503 and
by the Grant Agency of the Academy of Sciences No. IAA100190804. The research
of S. K. was supported by the Research Plan of the Ministry of Education of the
Czech Republic No. 6840770010 and by the Grant Agency of the Academy of Sciences No. IAA100190804. Final version of paper was supported by Grant Agency
of Czech Republic P 201/11/1304 for second and third authors.
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Received July 2010; revised March 2011.
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E-mail address: [email protected]