Acta Polytechnica


https://doi.org/10.14311/AP.2021.61.0005
Acta Polytechnica 61(SI):5–13, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID
AROUND A ROTATING AND TRANSLATING RIGID BODY

Paul Deuringa, Stanislav Kračmarb, c, Šárka Nečasovác, ∗

a Université du Littoral Côte d’Opale, Centre Universitaire de la Mi-Voix 50, rue F.Buisson CS 80699, 62228
Calais Cedex, France

b Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Technical
Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic

c Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 11567 Praha 1, Czech Republic
∗ corresponding author: matus@math.cas.cz

Abstract. We consider the linearized and nonlinear systems describing the motion of incompressible
flow around a rotating and translating rigid body D in the exterior domain Ω = R3 \D, where D ⊂ R3
is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and
investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity
essentially behaves near the infinity as a constant times the first column of the fundamental solution
of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω
under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare
this solution with the solution in the exterior domain Ω to get the truncation error estimate.

Keywords: Incompressible fluid, rigid body, exterior domain, estimates of pressure, leading terms,
artificial boundary conditions.

1. Introduction
The boundary problem of Navier–Stokes equations describing flows past a rigid body translating with a constant
velocity (with or without rotation) is one of the challenging problems in fluid mechanics. In recent decades,
much effort has been made to analyze the properties of solutions of both stationary and non-stationary solutions,
both linear and nonlinear mathematical models, both in the whole space and in exterior domains. The difficulty
which arises in this type of problem is the variability of the spatial domain in time. To solve it there are two
possibilities: (i) to study a problem in the time dependent domain, see Conca, Starovoitov and Tucsnak [1],
Desjardins and Esteban [2], Gunzburger, Lee and Seregin [3], Hoffman and Starovoitov [4], etc. (ii) to use a
transformation in order to transform the spatial domain varying in time in to a fixed domain. For this approach
the global or local transformation can be applied. The global linear transformation implies that the whole space
is rigidly rotated and shifted back to its original position at each time t > 0 (cf. [5]). The equations of motion
of the fluid-rigid body system is in a frame attached to the rigid body, with its origin in the center of mass of
the latter and coinciding with an inertial frame at time t = 0. (Works related to this type of transformation see
[6–12]). The local transformation implies that the change of variables only acts in a bounded neighbourhood of
the body, the solenoidal condition of the fluid velocity are preserved and the regularity of the solution are not
changed. See e.g. works of Tucsnak, Cumsille and Takahashi (cf. [13–15]).

1.1. Formulation of the problem
Let us formulate our problem in the fixed domain, which is a result of applying the global linear transformation,
for more details, see [5]. The systems of equations are as follows

−∆u(z) + τ∂1u(z) − (ω ×z) ·∇u(z) + ω ×u(z)
+τ(u(z) ·∇)u(z) + ∇π(z) = F(z)

div u(z) = 0 for z ∈ Ω
(1.1)

−∆u(z) + τ∂1u(z) − (ω ×z) ·∇u(z) + ω ×u(z) + ∇π(z) = F(z)
div u(z) = 0 for z ∈ Ω (1.2)

where D ⊂ R3 is open and bounded, with Lipschitz boundary. The systems (1.1) and (1.2) together with some
boundary conditions on ∂Ω = ∂D constitute the mathematical models (linear and non-linear, respectively)
describing the stationary flow of a viscous incompressible fluid around a rigid body which moves at a constant
velocity and rotates at a constant angular velocity. In this study we consider that the rotation is parallel to the
velocity at infinity. (For more details concerning the derivation of the model, see [5, 7]. The description and the

5

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P. Deuring, S. Kračmar, Š. Nečasová Acta Polytechnica

analysis in the case where the rotation is not parallel to the velocity at infinity can be found in the following
works, see [16, 17]).

The aim is to obtain the L∞ estimates for the pressure in the linear and nonlinear cases, since such estimates
are missing in the literature. Only the estimates of the velocity field and the gradient of the velocity field in L∞
are available. This implies that complete information about the decay of the solution (u,π) of the systems (1.1),
(1.2) for |x|→∞. (For other works see [18], [19].)

Second, we are interested in the “Leray solutions” of (1.1), supplemented by a decay condition at infinity,

u(x) → 0 for |x|→∞, (1.3)

and the suitable boundary conditions on ∂Ω. Weak solutions are characterized by the conditions u ∈ L6(Ω)3 ∩
W

1,1
loc (Ω)

3, ∇u ∈ L2(Ω)9 and π ∈ L2loc(Ω).
From [18] and [20], it follows that the velocity part of the Leray solution (u,π) in (1.1) and (1.3) decays for

|x|→∞ as the estimates express below

|u(x)| ≤ C
(
|x|s(x)

)−1
, |∇u(x)| ≤ C

(
|x|s(x)

)−3/2 (1.4)
for x ∈ R3 with |x| sufficiently large, where s(x) := 1 + |x|−x1 (x ∈ R3) and C > 0 a constant independent
of x. The factor s(x) may be considered as a mathematical manifestation of the wake extending downstream
behind a body moving in a viscous fluid. In the work by M. Kyed, (see [21]) it was shown that

uj(x) = γ Ej1(x) + Rj(x), ∂luj(x) = γ ∂lEj1(x) + Sjl(x) (x ∈ D
c
, 1 ≤ j, l ≤ 3), (1.5)

where E : R3\{0} 7→ R4 ×R3 denotes a fundamental solution to the Oseen system

−∆v + τ ∂1v + ∇% = f, divv = 0 in R3. (1.6)

The term Ej1(x) can be expressed explicitly in terms of elementary functions. The coefficient γ is also given
explicitly, its definition involving the Cauchy stress tensor. The remainding terms R and S are characterized
by the relations R ∈ Lq(Ω)3 for q ∈ (4/3, ∞), S ∈ Lq(Ω)3 for q ∈ (1,∞). From [22, Section VII.3] it is known
that Ej1|Bcr /∈ Lq(Bcr) for r > 0, q ∈ [1, 2], and ∂lEj1|Bcr /∈ Lq(Bcr) for r > 0, q ∈ [1, 4/3], j, l ∈{1, 2, 3}. The
function R decays faster than Ej1, and Sjl decays faster than ∂lEj1, in the sense of Lq-integrability. Thus, the
equations in (1.5) can be viewed in fact as asymptotic expansions of u and ∇u, respectively. Let us mention
that the result in [21] are valid under the assumption that u verifies the boundary conditions

u(x) = e1 + (ω ×x) for x ∈ ∂Ω, (1.7)

which is not our case.
Reference [21] does not deal with L∞-decay of R and S, nor does it indicate whether S = ∇R.
Below, in Theorem 4.1 we derive an L∞-decay of u and ∇u respectively, which is independent on the boundary

conditions. However, in comparison with [21] and indicated in (1.5), our leading term is less explicit than the
term γ Ej1(x) in (1.5) and instead of the fundamental solution Ej1(x) of the stationary Oseen system, we use
the time integral of the fundamental solution of the evolutionary Oseen system.

In [23] it was proved that Zj1(x, 0) = Ej1(x) for x ∈ R3\{0}, 1 ≤ j ≤ 3, and lim|x|→∞ |∂αxZjk(x, 0)| =
O
(

(|x|s(x))−3/2−|α|/2
)
for 1 ≤ j ≤ 3, k ∈{2, 3} ([23, Corollary 4.5, Theorem 5.1]). Thus, setting

Gj(x) :=
3∑
k=2

βk Zjk(x, 0) + Fj(x) (x ∈ BS1
c
, 1 ≤ j ≤ 3), (1.8)

we may obtain from (4.3) that

uj(x) = β1 Ej1(x) +
(∫

∂Ω
u ·ndox

)
xj (4 π |x|3)−1 + Gj(x) (x ∈ BS1

c
, 1 ≤ j ≤ 3) (1.9)

and

lim
|x|→∞

|∂αG(x)| = O
(

(|x|s(x))−3/2−|α|/2 ln(2 + |x|)
)

for α ∈ N30 with |α| ≤ 1 (1.10)

(Theorem 4.2, Corollary 4.3).
Comparing the coefficient γ from (1.5) in the work [21] with the coefficient β1 from (1.9) in [24], see Theorem

4.1 below, and taking into account the boundary condition (1.7) in [21], it follows that γ and β1 are equal.
Third, we are solving the linear system (1.2) in a truncation ΩR := BR ∩ Ω of the exterior domain R3 \D

under certain artificial boundary conditions on the truncating boundary ∂BR. Then we compare this solution
with the solution of (1.2) in the exterior domain, i.e. to find the error estimates of the method of an artificial
boundary condition. For this aim we use L∞-estimates of the velocity and of the pressure.

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vol. 61 Special Issue/2021 Note on the problem of motion of viscous fluid. . .

2. Definitions and notation
Let us define

s(y) := 1 + |y|−y1 for y ∈ R3, Ω = R3 \D , ΩR := BR ∩ Ω, BcR := R
3 \BR,

where BR := {x ∈ R3; |x| < R}, for R > 0 such that BR ⊃D.
So, ΩR is the truncation of the exterior domain Ω = R3 \D by the ball BR. The boundary ΩR consists of parts
∂Ω and ∂BR, the later we call the truncating boundary.

Fix τ ∈ (0,∞), e1 := (1, 0, 0), ω = |ω|e1, |ω| 6= 0, and Ω := |ω|


0 0 00 0 −1

0 1 0


.

So, Ω ·z = ω ×z for z ∈ R3. For U ⊂ R3 open, u ∈ W 2,1loc (U)
3, z ∈ U, put

(Lu)(z) := − ∆u(z) + τ∂1u(z) − (ω ×z) ·∇u(z) + ω ×u(z),
(L∗u)(z) := − ∆u(z) − τ∂1u(z) + (ω ×z) ·∇u(z) −ω ×u(z).

Put N(x) := (4 π |x|)−1 for x ∈ R3\{0} (”Newton potential”, fundamental solution of the Poisson equation
in R3), O(x) := (4 π |x|)−1 e−τ (|x|−x1)/2 for x ∈ R3\{0} (fundamental solution of the scalar Oseen equation
−∆v + τ ∂1v = g in R3),

Put K(z,t) := (4πt)−3/2e−|z|
2/(4t) (z ∈ R3, t ∈ (0,∞)),

Λ(z,t) :=
(
K(z,t)δjk + ∂zj∂zk

(∫
R3

(4π|z −y|)−1K(y,t)dy
))

1≤j,k≤3
(z ∈ R3, t > 0),

Γ(x,y,t) := Λ(x− τte1 −e−tΩy,t) ·e−tΩ,
Γ̃(x,y,t) := Λ(x + τte1 −etΩy,t) ·etΩ (x,y ∈ R3, t > 0),

Z(x,y) :=
∫ ∞

0
Γ(x,y,t)dt, Z̃(x,y) :=

∫ ∞
0

Γ̃(x,y,t)dt, (x,y ∈ R3, x 6= y).

ψ(r) :=
∫ r

0 (1 −e
−t) t−1 dt (r ∈ R), Φ(x) := (4 π τ)−1 ψ

(
τ (|x|−x1)/2

)
(x ∈ R3),

Ejk(x) := (δjk ∆ −∂j∂k)Φ(x), E4k(x) := xk (4 π |x|3)−1 (x ∈ R3\{0}, 1 ≤ j,k ≤ 3) (fundamental solution
of the Oseen system (1.6), with (Ejk)1≤j,k≤3 the velocity part and (E4k)1≤k≤3 the pressure part).

For q ∈ (1, 2), f ∈ Lq(R3)3, put

R(f)(x) :=
∫
R3
Z(x,y)f(y)dy (x ∈ R3);

see [25, Lemma 3.1].
We will use the space D1,20 (Ω)

3 := {v ∈ L6(Ω)3 ∩H1loc(Ω)
3 : ∇v ∈ L2(Ω)9, v|∂Ω = 0}

equipped with the norm ‖∇u‖2, where v|∂Ω means the trace of v on ∂Ω. For p ∈ (1,∞), define Mp as the space
of all pairs of functions (u,π) such that u ∈ W 2,ploc (Ω)

3, π ∈ W 1,ploc (Ω),

u|ΩR ∈ W 1,p(ΩR)3, π|ΩR ∈ Lp(DR), u|∂Ω ∈ W 2−1/p,p(∂Ω)3,
divu|ΩR ∈ W 1,p(ΩR), L(u) + ∇π|ΩR ∈ Lp(ΩR)3

for some R ∈ (0,∞) with Ω
c
⊂ BR.

We write C for generic constants. In order to romove possible ambiguities, we sometimes use the notation
C(γ1, ..., γn) in order to indicate that the constant in question depends particularly on γ1, ..., γn ∈ (0,∞), for
some n ∈ N. But the relevant constant may depend on other parameters as well.

3. Decay estimates
In the first part of this section, we recall some known results from [25] and [26] about the decay of the velocity
part of the solution of the system (1.2). In order to get the full decay characterization of the solution, we derive
the decay of the pressure part of the solution of (1.2). In the second part of this section, we extend the result
for the pressure to the non-linear case of (1.1).

7



P. Deuring, S. Kračmar, Š. Nečasová Acta Polytechnica

3.1. Decay estimates in the linear case
Our starting point is a decay result from [26] for the velocity part u of a solution to (1.2).

Theorem 3.1. ([26, Theorem 3.12]) Suppose that Ωc is C2-bounded. 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,

Ωc ∪ supp(div u) ⊂ BS1, u|B
c
S ∈ L

6(BcS)
3, ∇u|BcS ∈ L

2(BcS)
9,

A + min{1,B}≥ 3, |F(z)| ≤ γ|z|−As(z)−B for z ∈ BcS1.

Then
|u(y)| ≤ C (|y|s(y))−1 lA,B(y), (3.1)

|∇u(y)| ≤ C (|y|s(y))−3/2 s(y)max (0,7/2−A−B) lA,B(y) (3.2)

for y ∈ BcS, where function lA,B is given by{
1 if A + min{1,B} > 3

max(1, ln(y)) if A + min{1,B} = 3.

Corollary 3.2. Let p ∈ (1,∞), γ, S1, S ∈ (0,∞) with Ωc ⊂ BS1, S1 < S, A ∈ [2,∞), B ∈ R with
A + min{1,B}≥ 3. Let F : Ω 7→ R3 be measurable with F |ΩS1 ∈ Lp(ΩS1 )3 and |F (z)| ≤ γ|z|−As(z)−B for z ∈
BcS1.

Let u ∈ W 1,ploc (Ω)
3 with u|BcS ∈ L

6(BcS)
3, ∇u|BcS ∈ L

2(BcS)
9, supp(div u) ⊂ BS1 ,∫

D
c

[
∇u ·∇ϕ +

(
τ ∂1u− (ω ×z) ·∇u + (ω ×u) −F

)
·ϕ
]
dz (3.3)

= 0 for ϕ ∈ C∞0 (Ω)
3 with div ϕ = 0.

Then inequalities (3.1) and (3.2) hold for y ∈ BcS.
Moreover F ∈ Lq(Ω)3 for q ∈ (1,p]. If p ≥ 6/5, the function F may be considered as a bounded linear

functional on D1,20 (Ω)
3, in the usual sense.

Let π ∈ Lploc(Ω) with ∫
D

c

[
∇u ·∇ϕ +

(
τ ∂1u− (ω ×z) ·∇u + (ω ×u) −F

)
·ϕ (3.4)

−π div ϕ
]
dz = 0 for ϕ ∈ C∞0 (Ω)

3.

Fix some number S0 ∈ (0,S1) with D∪ supp(div u) ⊂ BS0. Then the relations u|BS0
c
∈ W 2,ploc (BS0

c
)3, π ∈

W
1,p
loc (BS0

c
) and L(u|BS0

c
) + ∇π = F|BS0

c
hold.

The main result of this section, dealing with the L∞-estimates of the pressure, is stated in

Theorem 3.3. Let p, γ, S1, S, A, B, F, u be given as in Corollary 3.2, but with the stronger assumptions
A = 5/2, B ∈ (1/2, ∞) on A and B. Let π ∈ Lploc(Ω) such that (3.4) holds Then there is c0 ∈ R such that

|π(x) + c0| ≤ C |x|−2 for x ∈ BcS. (3.5)

Corollary 3.4. Let p, γ, S1, S, A, B, F, u be given as in Corollary 3.2, but with the stronger assumptions
A ≥ 5/2, A + min{1,B} > 3 on A and B. Let π ∈ Lploc(Ω) such that (3.4) holds. Then there is c0 ∈ R such
that inequality (3.5) is valid.

Proof: Put B′ := A− 5/2 + min{1,B}. Since A + min{1,B} > 3, we have B′ ∈ (1/2, ∞). Moreover, since
A ≥ 5/2, we find for z ∈ BcS1 that

|F(z)| ≤ γ C(S1,A) |z|−5/2 s(z)−A+5/2−B ≤ γ C(S1,A) |z|−5/2 s(z)−B
′
.

Thus the assumptions of Theorem 3.3 are satisfied with B replaced by B′ and with a modified parameter γ.
This implies the conclusion of Theorem 3.3. �

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vol. 61 Special Issue/2021 Note on the problem of motion of viscous fluid. . .

3.2. Decay estimates in the non-linear case
Let us assume now the non-linear case, i.e. the system (1.1). First, recall the result about the decay properties
of the velocity in this non-linear case:

Theorem 3.5. [20, Theorem 1.1] Let γ, S1 ∈ (0,∞), p0 ∈ (1,∞), A ∈ (2,∞), B ∈ [0, 3/2] with Ωc ⊂ BS1,
A + min{B, 1} > 3, A + B ≥ 7/2. Take F : R3 7→ R3 measurable with F |BS1 ∈ Lp0 (BS1 )3,

|F(y)| ≤ γ · |y|−A ·s(y)−B for y ∈ BcS1.

Let u ∈ L6(Ω)3 ∩W 1,1loc (Ω)
3,π ∈ L2loc(Ω) with ∇u ∈ L

2(Ω)9,divu = 0 and∫
D

c
[∇u ·∇ϕ + τ∂1u− (ω ×z) ·∇u + ω ×u

+τ(u ·∇)u−F) ·ϕ−π divϕ] dx = 0

for ϕ ∈ C∞0 (Ω)3. Let S ∈ (S1,∞). Then

|∂αu(x)| ≤ C (|x|s(x))−1−|α|/2 for x ∈ BcS, α ∈ N
3
0 with |α| ≤ 1. (3.6)

Now, using Theorems 3.3 and 3.5, we are in the position to prove the result on the decay of the pressure in
the non-linear case:

Theorem 3.6. Consider the situation in Theorem 3.5. Suppose in addition that A ≥ 5/2. Then there is c0 ∈ R
such that inequality (3.5) holds.

4. Leading term
In this section we study the asymptotic behavior of the velocity profile of the system (1.2). Let us recall known
results from [26] and [24].

Theorem 4.1. Let D ⊂ R3 be open, p ∈ (1,∞), f ∈ Lp(R3)3 with supp(f)compact. Let S1 ∈ (0,∞) with
D∪supp(f) ⊂ BS1, Ω = D

c
.

Let u ∈ L6(Ω)3 ∩W 1,1loc (Ω)
3, π ∈ L2loc(Ω) with ∇u ∈ L

2(Ω)9, divu = 0 and∫
Ω

[
∇u ·∇ϕ +

(
τ ∂1u + τ (u ·∇)u− (ω ×z) ·∇u + ω ×u

)
·ϕ−π div ϕ

]
dz (4.1)

=
∫

Ω
f ·ϕdz for ϕ ∈ C∞0 (Ω)

3.

(This means the pair (u,π) is a Leray solution to (1.2), (1.3).) Suppose in addition that

Ωc is C2-bounded, u|∂Ω ∈ W 2−1/p,p(∂Ω)3, π|BS1\D ∈ L
p(BS1\D). (4.2)

Let n denote the outward unit normal to Ω, and define

βk :=
∫

Ω
fk(y) dy

+
∫
∂Ω

3∑
l=1

(
−∂luk(y) + δkl π(y) + (τ e1 −ω ×y)l uk(y) − τ (ul uk)(y)

)
nl(y) doy

for 1 ≤ k ≤ 3,

Fj(x) :=
∫

Ω

[ 3∑
k=1

(
Zjk(x,y) −Zjk(x, 0)

)
fk(y) − τ ·

3∑
k,l=1

Zjk(x,y) (ul ∂luk)(y)
]
dy

+
∫
∂Ω

3∑
k=1

[(
Zjk(x,y) −Zjk(x, 0)

) 3∑
l=1

(
−∂luk(y) + δkl π(y) + (τ e1 −ω ×y)l uk(y)

)
nl(y)

+
(
E4j(x−y) −E4j(x)

)
uk(y) nk(y)

+
3∑
l=1

(
∂ylZjk(x,y) (uk nl)(y) + τZjk(x, 0) (ul uk nl)(y)

)]
doy

9



P. Deuring, S. Kračmar, Š. Nečasová Acta Polytechnica

for x ∈ BS1
c
, 1 ≤ j ≤ 3. The preceding integrals are absolutely convergent. Moreover F ∈ C1(BS1

c
)3 and

equation

uj(x) =
3∑
k=1

βk Zjk(x, 0) +
(∫

∂Ω
u ·ndox

)
xj (4 π |x|3)−1 + Fj(x). (4.3)

holds. In addition, for any S ∈ (S1,∞), there is a constant C > 0 which depends on τ, ω, S1, S, f, u and π,
and which is such that

|∂αF(x)| ≤ C
(
|x|s(x)

)−3/2−|α|/2 ln(2 + |x|) for x ∈ BSc, α ∈ N30 with |α| ≤ 1. (4.4)
Theorem 4.2. Let D, p, f, S1, u, π satisfy the assumptions of Theorem 4.1, including (4.2). Let β1, β2, β3
and F be defined as in Theorem 4.1. Define the function G as

Gj(x) :=
3∑
k=2

βk Zjk(x, 0) + Fj(x) (x ∈ BS1
c
, 1 ≤ j ≤ 3). (4.5)

Then G ∈ C1(BS1
c
)3, equation

uj(x) = β1 Ej1(x) +
(∫

∂Ω
u ·ndox

)
xj (4 π |x|3)−1 + Gj(x) (x ∈ BS1

c
, 1 ≤ j ≤ 3) (4.6)

holds, and for any S ∈ (S1,∞), there is a constant C > 0 which depends on τ, ω, S1, S, f, u and π, and which
is such that

|∂αG(x)| ≤ C
(
|x|s(x)

)−3/2−|α|/2 ln(2 + |x|) for x ∈ BSc, α ∈ N30 with |α| ≤ 1.
Corollary 4.3. Take D, p, f, S1, u, π as in Theorem 4.1, but without requiring (4.2). (This means that
(u, π) is only assumed to be a Leray solution of (1.2), (1.3).) Put p̃ := min{3/2, p}. Then u ∈ W 2,p̃loc (Ω)

3 and
π ∈ W 1,p̃loc (Ω).

Fix some number S0 ∈ (0,S1) with D∪ supp(f) ⊂ BS0 , and define β1, β2, β3 and F as in Theorem 4.1,
but with D replaced by BS0 , and n(x) by S

−1
0 x, for x ∈ ∂BS0. Moreover, define G as in (4.5). Then all the

conclusions of Theorem 4.2 are valid.

5. Formulation of the problem with artificial boundary conditions
Recall that we defined ΩR = BR ∩ Ω. We introduce the subspace WR of H1(ΩR) denoting

WR := {v ∈ H1(ΩR)3 : v|∂Ω = 0},

where v|∂Ω means the trace of v on ∂Ω.

Lemma 5.1. ([27, Lemma 4.1]) The estimate

‖u‖2 ≤ C (R ‖∇u‖2 + R
1/2 ‖u|∂BR‖2)

holds for R ∈ (0,∞) with Ωc ⊂ BR and for u ∈ WR.

We introduce an inner product (·, ·)(R) in WR by defining

(v,w)(R) =
∫

ΩR
∇v ·∇wdx +

∫
∂BR

(τ/2)v ·wdox for v,w ∈ WR.

The space WR equipped with this inner product is a Hilbert space. The norm generated by this scalar product
(·, ·)(R) is denoted by | · |(R), that is

|v|(R) :=
(
‖∇v‖22 + (τ/2)‖v|∂BR‖

2
2

)1/2
for v ∈ WR.

10



vol. 61 Special Issue/2021 Note on the problem of motion of viscous fluid. . .

We define the bilinear forms

AR : H1(ΩR)3 ×H1(ΩR)3 → R,
BR : H1(ΩR)3 ×L2(ΩR) → R,

AR(u,w) :=
∫

ΩR
[∇u ·∇w + τ∂1u ·w]dx +

τ

2

∫
∂BR

(u(x) ·w(x))
(

1 −
x1
R

)
dox,∫

ΩR

[
−
(

(ω ×x) ·∇
)
u(x) +

(
ω ×u(x)

)]
·w(x) dx

BR(w,σ) := −
∫

ΩR
(div w) σdx, +

(
ω ×u(x)

)]
·w(x) dx

for u,w ∈ H1(ΩR)3, σ ∈ L2(ΩR), R ∈ (0,∞) with Ωc ⊂ BR.

Lemma 5.2. Let R ∈ (0,∞) with Ωc ⊂ BR. Then

|AR(u,w)| ≤C(R) |u|(R) |w|(R)

for u,w ∈ H1(ΩR)3.

The key observation in this section is stated in the following lemma, which is the basis of the theory presented
in this section.

Lemma 5.3. Let R ∈ (0,∞) with Ωc ⊂ BR, and let w ∈ WR. Then the equation (|w|(R))2 = AR(w,w) holds.

Proof: Using the definition AR(·, ·), we get

AR(w,w) =
∫
DR

[
|∇w|2 + τ∂1

(
|w|2

2

)
− (ω ×x) ·∇

(
|w|2

2

)]
dx

+
τ

2

∫
∂BR

|w(x)|2
(

1 −
x1
R

)
dox

=
∫
DR
|∇w|2 dx +

∫
∂BR

(
τ

2
|w(x)|2

x1
R
−

1
2

(ω ×x) ·
x

R
|w(x)|2

)
dox

+
τ

2

∫
∂BR

|w(x)|2
(

1 −
x1
R

)
dox

=
∫
DR
|∇w|2 dx +

τ

2

∫
∂BR

|w(x)|2 = (|w|(R))2.

We applied that (ω ×x) ·x = 0 for x, ω ∈ R3. �
As in [28], we obtain that the bilinear form βR is stable:

Theorem 5.4. ([28, Corollary 4.3]) Let R > 0 with Ωc ⊂ BR. Then

inf
ρ∈L2(ΩR),ρ 6=0

sup
v∈WR,v 6=0

BR(v,ρ)
|v|(R)‖ρ‖2

≥ C(R).

We note that functions from W 1,1loc (Ω) with L
2-integrable gradient are L2-integrable on truncated exterior

domains:

Lemma 5.5. [29, Lemma II.6.1] Let w ∈ W 1,1loc (Ω) with ∇w ∈ L
2(Ω)3, and let R ∈ (0,∞) with Ωc ⊂ BR. Then

w|ΩR ∈ L2(ΩR). In particular the trace of w on ∂Ω is well defined.

The preceding lemma is implicitly used in the ensuing theorem, where we introduce an extension operator
E : H1/2(∂Ω)3 7→ W 1,1loc (Ω)

3 such that div E(b) = 0.

Theorem 5.6. [29, Exercise III.3.8] There is an operator E from H1/2(∂Ω)3 into W 1,1loc (Ω)
3 satisfying the

relations ∇E(b) ∈ L2(Ω)9, E(b)|∂Ω = b and div E(b) = 0 for b ∈ H1/2(∂Ω)3.

In view of Lemma 5.2 and 5.3 and Theorem 5.6 and 5.4, the theory of mixed variational problems yields

11



P. Deuring, S. Kračmar, Š. Nečasová Acta Polytechnica

Theorem 5.7. Let S > 0 with Ωc ⊂ BS, R ∈ [2S,∞), F ∈ L6/5(ΩR)3, b ∈ H1/2(∂Ω)3. Then there is a
uniquely determined pair of functions (Ṽ ,P) =

(
Ṽ (R,F,b), P(R,F,b)

)
∈ WR ×L2(ΩR) such that

AR(Ṽ ,g) + BR(g,P ) =
∫
DR

F ·g dx−AR
(
E(b)|ΩR, g

)
for g ∈ WR, (5.1)

BR(Ṽ ,σ) = 0 for σ ∈ L2(ΩR), (5.2)

where the operator E was introduced in Theorem 5.6.

Let us interpret variational problem (5.1), (5.2) as a boundary value problem. Define the expression used in
the boundary condition on the artificial boundary ∂BR :

LR(u,π)(x) :=


 3∑
j=1

∂juk(x)
xj
R
−π(x)

xk
R

+
τ

2

(
1 −

x1
R

)
uk(x)


 1≤k≤3

for x ∈ ∂BR, R ∈ (0,∞) with D ⊂ BR, u ∈ W 2, 6/5(ΩR)3, π ∈ W 1, 6/5(ΩR).

Lemma 5.8. Assume that Ωc is C2-bounded. Let S ∈ (0,∞) with Ωc ⊂ BS, R ∈ [2S,∞), F ∈ L6/5(ΩR)3 and
b ∈ W 7/6, 6/5(∂Ω)3. Put V := Ṽ (R,F,b) + E(b)|ΩR, with V (R,F,b) from Theorem 5.7 and E(b) from Theorem
5.6. Suppose that V ∈ W 2,6/5(ΩR)3 and P = P(R,F,b) ∈ W 1, 6/5(ΩR), with P(R,F,b) also introduced in
Theorem 5.7. Then

−∆V (z) + τ∂1V (z) − (ω ×z) ·∇V (z) + ω ×V (z) + ∇P(z) = F(z),
div V (z) = 0 (5.3)

for z ∈ ΩR, and V |∂Ω = b, LR(V,P) = 0.

Theorem 5.9. Suppose that Ωc is C2-bounded. Let γ, S1 ∈ (0,∞) with Ωc ⊂ BS1, A ∈ [5/2, ∞), B ∈ R with
A+min{1,B} > 3. Let F : Ω 7→ R3 be measurable with F|ΩS1 ∈ L6/5(ΩS1 )3 and |F (z)| ≤ γ |z|−As(z)−B for z ∈
BcS1 .

Let b ∈ W 7/6, 6/5(∂Ω)3, u ∈ W 1,1loc (Ω)
3 ∩L6(Ω)3 such that ∇u ∈ L2(Ω)9, div u = 0, u|∂Ω = b and equation

(3.3) is satisfied.
For R ∈ [2S1, ∞), put VR := Ṽ (R,F,b) + E(b), with E(b) from Theorem 5.6, and Ṽ (R,F,b) from Theorem

5.7. Then

|u|ΩR −VR|
(R) ≤ C R−1 for R ∈ [2S,∞).

Acknowledgements
The works of S.K. and Š. N. were supported by Grant No. 19-04243S of GAČR in the framework of RVO 67985840, S.K.
is supported by RVO 12000.

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	Acta Polytechnica 61(SI):5–13, 2021
	1 Introduction
	1.1 Formulation of the problem

	2 Definitions and notation 
	3 Decay estimates 
	3.1 Decay estimates in the linear case
	3.2 Decay estimates in the non-linear case

	4 Leading term
	5 Formulation of the problem with artificial boundary conditions 
	Acknowledgements
	References