# Decay of Strong Solutions for 4D Navier-Stokes Equations Posed on Lipschitz Domains

Authors: N. A. Larkin
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

Initial-boundary value problems for 4D Navier-Stokes equations posed on bounded and unbounded 4D parallelepipeds were considered. The existence and uniqueness of regular global solutions on bounded parallelepipeds and their exponential decay as well as the existence, uniqueness, and exponential decay of strong solutions on an unbounded parallelepiped have been established provided that initial data and domains satisfy some special conditions.


---

## Body

## 1. Introduction

This work concerns the existence and uniqueness of global strong solutions and sharp decay estimates of solutions to initial-boundary value problems for the 4D Navier-Stokes equations:(1)
u
t
+
u
·
∇
u
=
ν
Δ
u
-
∇
p
,
i
n
Ω
×
0
,
t
,
(2)
∇
·
u
=
0
i
n
Ω
,
u
∂
Ω
=
0
,
(3)
u
x
,
0
=
u
0
x
,where Ω is either a bounded or an unbounded parallelepiped in R
4 with the homogeneous Dirichlet condition on the boundary of Ω.The question of decay of the energy for weak solutions had been stated by J. Leray in [1] and attracts till now attention of many pure and applied mathematicians [2–9]. In all of these papers, the decay rate of u
(
t
)
L
2
(
Ω
) was controlled by the first eigenvalue of the operator A
=
-
P
Δ, where P is the projection operator on the solenoidal subspace of L
2
(
Ω
). Obviously, this approach does not work in unbounded domains; see [6, 7, 9].It is well known that solutions of the 2D Navier-Stokes equations posed on smooth bounded domains with the Dirichlet boundary conditions are globally regular [4, 6–9]. On the other hand, the question of regularity for 3D and 4D NSE with arbitrary initial data is till now an open problem even for smooth domains; see [6, 7, 9]. Small initial data help to solve this problem [6, 7, 9] as well as the so-called “thin” domains when some size of a domain is small [10, 11]. The question of regularity becomes more difficult while a domain is Lipschitzian [10, 12, 13].In [6, 7, 9, 14], it has been proved that for 3D Lipschitz domains, bounded and unbounded and small initial data there exists a unique global strong solution (4)
u
,
u
t
∈
L
∞
0
,
∞
;
L
2
Ω
∩
L
2
0
,
∞
;
H
1
Ω
, but it was not clear whether (5)
u
∈
L
∞
0
,
∞
;
H
2
Ω at least for bounded Lipschitz domains.Our goal here is making some geometrical restrictions, to prove the existence and uniqueness of strong global solutions in 4D Lipschitz domains for arbitrary regular initial data as well as exponential decay of solutions.In this work, making use of ideas of [15], we have established that u
∈
W
2,4
/
3
(
Ω
) for a 4D bounded parallelepiped. The following inequality holds: (6)
u
H
0
1
Ω
2
t
+
u
t
L
2
Ω
2
t
≤
C
u
0
H
2
Ω
2
exp
⁡
-
ν
χ
t
, where χ
>
0
.Our paper has the following structure: Section1 is Introduction. Section 2 contains notations and auxiliary facts. In Section 3, existence, uniqueness, and decay of global strong solutions on a bounded 4D parallelepiped have been established. In Section 4, the existence, uniqueness, and decay of regular solutions on bounded 4D parallelepipeds and strong solutions on 4D unbounded parallelepipeds have been demonstrated.


## 2. Notations and Auxiliary Facts

Letx
=
(
x
1
,
x
2
,
x
3
,
x
4
) and Ω be a domain in R
4
. Define as in [9], p.2-4 (7)
D
i
=
∂
∂
x
i
,
i
=
1,2
,
3,4
;
D
α
=
∏
i
=
1
4
D
i
α
i
,
α
=
∑
i
=
1
4
α
i
,
u
=
u
1
,
u
2
,
u
3
,
u
4
;
u
t
=
u
1
t
,
u
2
t
,
u
3
t
,
u
4
t
;
D
i
u
=
D
i
u
1
,
D
i
u
2
,
D
i
u
3
,
D
i
u
4
. We denote for scalar functions f
(
x
) the Banach space L
p
(
Ω
)
,
1
<
p
<
+
∞ with the norm(8)
f
L
p
Ω
p
=
∫
Ω
f
p
d
x
,
p
∈
1
,
+
∞
,
f
L
∞
Ω
=
e
s
s
s
u
p
Ω
f
x
. For p
=
2
,
L
2
(
Ω
) is a Hilbert space with the scalar product (9)
u
,
v
=
∫
Ω
u
x
v
x
d
x
a
n
d
t
h
e
n
o
r
m
u
2
=
∫
Ω
u
x
2
d
x
.The Sobolev spaceW
m
,
p
(
Ω
) is a Banach space with the norm (10)
u
W
k
,
p
Ω
=
∑
0
≤
∣
α
≤
k
D
α
u
L
p
Ω
. When p
=
2
,
W
m
,
2
(
Ω
)
=
H
m
(
Ω
) is a Hilbert space with the following scalar product and the norm: (11)
u
,
v
H
m
Ω
=
∑
j
≤
m
D
j
u
,
D
j
v
,
u
H
m
Ω
2
=
∑
j
≤
m
D
j
u
2
.LetD
(
Ω
) or D
(
Ω
¯
) be the space of C
∞ functions with compact support in Ω or Ω
¯. The closure of C
∞ functions in W
m
,
p
(
Ω
) is denoted by W
0
m
,
p
(
Ω
) and (
H
0
m
(
Ω
) when p
=
2
)
.Define the auxiliary spaces which are projections for the solenoidal vector functions,(12)
V
=
u
∈
D
Ω
,
∇
·
u
=
0
,
V
=
t
h
e
c
l
o
s
u
r
e
o
f
V
i
n
H
0
1
Ω
,
H
=
t
h
e
c
l
o
s
u
r
e
o
f
V
i
n
L
2
Ω
.The space H is equipped with the natural L
2 inner product. The space V will be equipped with the scalar product (13)
u
,
v
=
∑
i
=
1
4
D
i
u
,
D
i
vwhen Ω is bounded. If Ω is unbounded, we define the inner product as the sum of the inner products as follows: (14)
u
,
v
=
u
,
v
+
u
,
v
.We use the usual notations of Sobolev spacesW
k
,
p, L
p, and H
k for vector functions and the following notations for the norms:(i) For vector functionsu
(
x
)
=
(
u
1
(
x
)
,
u
2
(
x
)
,
u
3
(
x
)
,
u
4
(
x
)
), (15)
u
L
p
Ω
p
=
∫
Ω
∑
i
=
1
4
u
i
p
d
x
,
u
W
k
,
p
Ω
=
∑
0
≤
α
≤
k
∑
i
=
1
4
D
α
u
i
L
p
Ω
,
p
∈
1
,
+
∞
.The closures ofV in L
2
(
Ω
) and in H
0
1
(
Ω
) are the basic spaces in our study. We denote them by H and V, respectively.Remark 1.
By definition,V is a proper subspace of H
0
1
(
Ω
)
.Define the operator(16)
u
·
∇
u
=
∑
i
=
1
4
u
i
D
i
u
.Lemma 2 (the Steklov inequality [16]).
Letv
∈
H
0
1
(
0
,
L
)
. Then(17)
π
2
L
2
v
2
t
≤
v
x
2
t
.Proof.
Letv
(
t
)
∈
H
0
1
(
0
,
π
), then by the Fourier series, (18)
∫
0
π
v
t
2
t
d
t
≥
∫
0
π
v
2
t
d
t
.Inequality (17) follows by a simple scaling.The next lemmas will be used in estimates.Lemma 3 (see: [17] Theorem 7.1, p.14).
Letv
∈
H
0
1
(
Ω
), then(19)
v
L
4
Ω
≤
3
∇
v
L
2
Ω
.Lemma 4.
Letb
(
u
,
v
,
w
)
=
(
(
u
·
∇
)
v
,
w
), then (20)
b
u
,
v
,
w
≤
u
L
4
Ω
v
V
w
L
4
Ω
≤
3
2
u
V
v
V
w
V
∀
u
,
v
,
w
∈
H
0
1
(
Ω
). If u
∈
L
∞
(
0
,
∞
;
V
), then we can define the operator B
u such that B
u belongs to L
∞
(
0
,
∞
;
V
′
) and (21)
B
u
,
v
=
b
u
,
u
,
v
,
B
u
L
∞
0
,
∞
;
V
′
≤
3
2
u
L
∞
0
,
∞
;
V
2
.


## 3. Existence Theorems

LetΩ be a bounded 4D parallelepiped: Ω
=
[
x
∈
R
4
;
x
i
∈
(
0
,
L
i
)
;
L
i
>
0
,
i
=
1,2
,
3,4
] which is a Lipschitz domain. Denote L
=
m
i
n
(
L
1
,
L
2
,
L
3
,
L
4
). Given u
0
∈
H, consider the following problem:(22)
u
t
-
ν
Δ
u
+
∇
p
+
u
·
∇
u
=
0
i
n
Ω
×
0
,
t
,
∇
·
u
=
0
i
n
Ω
×
0
,
t
,
u
=
0
o
n
∂
Ω
×
0
,
t
,
t
>
0
,
u
x
,
0
=
u
0
x
,
i
n
Ωequivalent to the variational problem given by (see [7], [9] Problem 3.2, p. 191.)(23)
u
′
+
A
u
+
B
u
=
0
i
n
0
,
t
,
t
>
0
u
0
=
u
0
,where A
u
∈
V
′ such that 〈
A
u
,
v
〉
=
ν
(
(
u
,
v
)
) for all v
∈
V and B
u
∈
V
′ such that(24)
B
u
,
v
=
b
u
,
u
,
v
.Theorem 5.
Givenu
0
∈
H
2
(
Ω
)
∩
V and Ω such that(25)
π
L
1
/
2
ν
7
/
4
-
18
u
0
1
/
4
u
t
0
3
/
4
>
0
,there exists a unique strong solution u
(
x
,
t
) to (22): (26)
u
∈
L
∞
0
,
∞
;
V
,
u
t
∈
L
∞
0
,
∞
;
H
∩
L
2
0
,
∞
;
Vsuch that for all Φ
:
R
4
→
R
4
∈
V
,
Φ
∂
Ω
=
0 it satisfies the following identity:(27)
u
t
,
Φ
t
+
ν
∑
i
=
1
4
D
i
u
,
D
i
Φ
t
+
b
u
,
Φ
,
u
t
=
0
,
t
>
0
.Moreover, the following inequalities hold:(28)
u
t
≤
u
0
e
-
χ
t
,
u
t
t
≤
u
t
0
e
-
χ
/
2
t
,
u
t
V
≤
C
u
0
u
t
0
e
-
3
/
4
χ
t
,
whereχ
=
ν
π
2
∑
i
=
1
4
1
/
L
i
2
.Proof.
The estimates that follow may be established on Galerkin’s approximations (see [14], [6] p. 136-140, [7], [9], p. 192-197.).
Estimate I (u
∈
L
∞
(
0
,
∞
;
H
)
∩
L
2
(
0
,
∞
;
V
)). Multiply (23) by u
∈
V to obtain(29)
u
t
,
u
t
+
A
u
,
u
t
=
0
.It follows from here that(30)
d
d
t
u
2
t
+
2
ν
u
V
2
t
=
0
.Making use of the Steklov inequalities, we get (31)
∇
u
2
t
≥
π
2
∑
i
=
1
4
1
L
i
2
u
2
t
.Returning to (30), we obtain (32)
d
d
t
u
2
t
+
2
χ
u
2
t
≤
0 which implies(33)
u
2
t
≤
u
0
2
e
-
2
χ
t
,
χ
=
ν
π
2
∑
i
=
1
4
1
L
i
2
. This and (30) give (34)
u
∈
L
∞
0
,
∞
;
H
∩
L
2
0
,
∞
;
V
.
Estimate II (u
t
∈
L
∞
(
0
,
∞
;
H
)
∩
L
2
(
0
,
∞
;
V
)). Differentiating (23) and multiplying by u
t, we get(35)
d
d
t
u
t
2
t
+
2
ν
u
t
V
2
t
+
2
b
u
t
,
u
,
u
t
t
=
0
.
We estimate(36)
b
u
t
,
u
,
u
t
t
≤
u
t
t
L
4
Ω
u
V
t
u
t
t
L
4
Ω
.
By Lemma3, (37)
2
b
u
t
,
u
,
u
t
t
≤
2
·
3
2
u
t
V
2
t
u
V
t
and (35) becomes(38)
d
d
t
u
t
2
t
+
ν
u
t
V
2
t
+
ν
-
18
∇
u
t
∇
u
t
2
t
≤
0
.Making use of (30), we find(39)
ν
∇
u
2
t
≤
u
t
u
t
t
.Substituting this into (38), we get (40)
d
d
t
u
t
2
t
+
ν
u
t
V
2
t
+
ν
-
18
u
1
/
2
t
u
t
1
/
2
t
ν
1
/
2
∇
u
t
2
t
≤
0
.
By Lemma2, (41)
u
t
≤
L
π
∇
u
t
.
Substituting this into (40) and using (39), we find(42)
d
d
t
u
t
2
t
+
ν
u
t
V
2
t
+
ν
-
18
L
1
/
2
∇
u
1
/
2
t
u
t
1
/
2
t
ν
1
/
2
π
1
/
2
∇
u
t
2
t
≤
0
.
Again by (39),(43)
d
d
t
u
t
2
t
+
ν
u
t
V
2
t
+
ν
-
18
L
1
/
2
u
1
/
4
t
u
t
3
/
4
t
ν
3
/
4
π
1
/
2
∇
u
t
2
t
≤
0
.Rewrite this in the form(44)
d
d
t
u
t
2
t
+
ν
u
t
V
2
t
+
L
1
/
2
π
1
/
2
/
L
1
/
2
ν
7
/
4
-
18
u
0
1
/
4
u
t
3
/
4
t
π
1
/
2
ν
3
/
4
∇
u
t
2
t
≤
0
.Taking into account conditions (25) of Theorem 5, we get that(45)
π
1
/
2
L
1
/
2
ν
7
/
4
-
18
u
0
1
/
4
u
t
3
/
4
t
>
0
,
t
>
0
;hence (44) reduces to the form(46)
d
d
t
u
t
2
t
+
ν
u
t
V
2
t
≤
0
.By the Steklov inequality,(47)
d
d
t
u
t
2
t
+
χ
u
t
2
t
≤
0
.
We estimate(48)
u
t
0
≤
ν
Δ
u
0
+
u
0
L
4
Ω
∇
u
0
L
4
Ω
≤
ν
Δ
u
0
+
C
u
0
H
2
Ω
∩
V
2
≡
C
u
0
H
2
Ω
∩
V
2
.
This and (46), (47) imply that u
t
∈
L
∞
(
0
,
∞
;
H
)
∩
L
2
(
0
,
∞
;
V
), and(49)
u
t
2
t
≤
C
u
0
H
2
Ω
∩
V
2
e
-
χ
t
.
Returning to (39), we get u
∈
L
∞
(
0
,
∞
;
V
). This and (33), (49) prove validity of (27), (28), and consequently the existence part of Theorem 5.
Uniqueness of the Strong Solution. Let u and v be two strong solutions to (22) satisfying (23) and (27). Define z
=
u
-
v. Then for Φ
∈
V we have (50)
z
t
,
Φ
+
A
z
,
Φ
+
b
u
,
z
,
Φ
-
b
z
,
z
,
Φ
=
0
.
TakingΦ
=
z, we come to the inequality(51)
d
d
t
z
2
t
+
2
ν
z
V
2
t
=
b
z
,
z
,
u
t
≤
18
u
V
t
z
V
2
tthat can be rewritten as (52)
d
d
t
z
2
t
+
ν
z
V
2
t
+
ν
-
18
u
V
t
z
2
t
V
≤
0
.Acting in the same manner as by the proof of Estimate II, we come to the inequality(53)
d
d
t
z
2
t
+
ν
z
V
2
t
+
L
1
/
2
π
1
/
2
/
L
1
/
2
ν
7
/
4
-
18
u
1
/
4
t
u
t
3
/
4
t
π
1
/
2
ν
3
/
4
z
V
2
t
≤
0
.By conditions of Theorem 5, (54)
π
1
/
2
/
L
1
/
2
ν
7
/
4
-
18
u
1
/
4
t
u
t
3
/
4
t
π
1
/
2
ν
3
/
4
>
0for t
=
0
. Taking into account Estimates (28) and using standard arguments, we get for all t
>
0
(55)
π
1
/
2
/
L
1
/
2
ν
7
/
4
-
18
u
1
/
4
t
u
t
3
/
4
t
π
1
/
2
ν
3
/
4
>
0
.Hence, (53) becomes (56)
d
d
t
z
2
t
≤
0
.This implies z
≡
0 that proves uniqueness of the strong solution and completes the proof of Theorem 5.


## 4. More Regularity

Consider the Poisson problem in a bounded domainΩ
∈
R
n:(57)
Δ
u
=
f
x
,
x
∈
Ω
,
u
∂
Ω
=
0
.In [15] Theorem 11, p. 120-123, the following has been proved.Lemma 6.
Let(58)
Ω
π
=
x
=
x
1
,
…
,
x
n
,
0
<
x
i
<
π
;
i
=
1
,
…
,
nthen the unique weak solution u
∈
H
0
1
(
Ω
) of (57) satisfies the following inequality: (59)
u
W
2
,
p
Ω
π
≤
C
Ω
f
L
p
Ω
π
.It is possible to generalize this result for a bounded parallelepiped inR
4
.Theorem 7.
The problem (57) posed in a parallelepiped Ω
=
{
(
x
)
∈
R
4
,
0
<
x
i
<
L
i
;
i
=
1,2
,
3,4
}, where f
∈
L
p
(
Ω
), 1
<
p
≤
2, has a solution u
∈
W
2
,
p
(
Ω
). Moreover,(60)
u
W
2
,
p
Ω
≤
C
Ω
f
L
p
Ω
.Returning to the original problem for the Navier-Stokes equations,(61)
u
t
-
ν
Δ
u
+
∇
p
+
u
·
∇
u
=
0
i
n
Ω
×
0
,
t
,
∇
·
u
=
0
i
n
Ω
×
0
,
t
,
u
=
0
i
n
∂
Ω
×
0
,
t
,
t
>
0
,
u
x
,
0
=
u
0
x
i
n
Ω
,where u
(
x
) is a vector function from R
4 into R
4 and p is a real function from R
4 into R, and making use of Galerkin approximations, we establish the following result.Theorem 8.
Givenu
0
∈
H
2
(
Ω
)
∩
V and a domain Ω satisfying (25), then problem (61) has a unique regular solution (
u
,
p
) such that(62)
u
∈
L
∞
0
,
∞
;
V
∩
W
2,4
/
3
Ω
,
u
t
∈
L
∞
0
,
∞
;
H
∩
L
2
0
,
∞
;
V
,
∇
p
∈
L
∞
0
,
∞
;
L
4
/
3
Ωwhich for all Φ
(
x
)
∈
V satisfies the following integral identity: (63)
∫
Ω
u
t
+
ν
Δ
u
+
u
·
∇
u
Φ
d
x
=
0
.
Moreover,(64)
u
t
t
+
u
t
W
2,4
/
3
Ω
+
∇
p
L
4
/
3
Ω
t
≤
C
e
-
1
/
2
χ
t
,where χ
=
ν
π
2
∑
i
=
1
4
1
/
L
i
2 and C depends on u
0
H
2
(
Ω
).Proof (decay ofW
2,4
/
3
(
Ω
)-norm).
Taking into account that conditions of Theorem8 and of Theorem 5 are the same, by Theorem 5, we have a unique strong solution of (61). Hence, to prove Theorem 8, it is sufficient to establish that u
W
2,4
/
3
(
Ω
)
(
t
)
. First write (22) as(65)
Δ
u
=
f
=
1
ν
u
t
+
∇
p
+
u
·
∇
u
.We estimate(66)
b
u
,
u
,
v
t
=
u
·
∇
u
,
v
t
≤
C
u
t
L
4
Ω
u
t
V
v
t
L
4
Ω
≤
C
u
2
t
V
v
t
L
4
Ω
.Hence by (33), (49), (51),(67)
B
u
t
L
4
/
3
Ω
=
u
·
∇
u
t
L
4
/
3
Ω
≤
C
e
-
3
/
2
χ
t
.
Returning to (23) and making use of (67), (49), we obtain(68)
A
u
L
4
/
3
Ω
t
≤
B
u
L
4
/
3
Ω
t
+
u
t
L
2
Ω
t
≤
C
e
-
χ
/
2
t
.By the Theorem of de Rham (see [7, 18]), [9] Propositions 1.1, 1.2, p. 10, one can check that there exists ∇
p such that (see [9])(69)
-
∇
p
=
u
t
+
A
u
+
B
u
and(70)
∇
p
L
4
/
3
Ω
t
≤
u
t
L
2
Ω
t
+
A
u
L
4
/
3
Ω
t
+
B
u
L
4
/
3
Ω
t
≤
C
e
-
χ
/
2
t
.Returning to (65) and having f
∈
L
4
/
3
(
Ω
), we obtain, due to Theorem 7,(71)
u
t
W
2,4
/
3
Ω
≤
u
t
L
2
Ω
t
+
∇
p
L
4
/
3
Ω
t
+
u
·
∇
u
L
4
/
3
Ω
t
≤
C
e
-
χ
/
2
t
.By the Sobolev theorems,(72)
u
L
4
Ω
t
≤
C
u
W
2,4
/
3
Ω
t
≤
C
e
-
χ
/
2
t
.The proof of Theorem 8 is complete.Remark 9.
It follows from (72) that B
u
∈
L
4
/
3 and consequently, f
∈
L
4
/
3
(
Ω
) and u
∈
W
2,4
/
3
(
Ω
) in (60). This means that we cannot achieve better regularity then u
∈
W
2,4
/
3
(
Ω
)
. In some sense, this is the superior regularity for the problem (61). It looks like n
=
4 is the critical case of the Navier-Stokes system.Existence and Decay of a Strong Solution on an Unbounded 4D Domain. Define an unbounded four-dimensional parallelepiped D
=
{
x
∈
R
4
;
0
<
x
i
<
L
i
,
i
=
1,2
,
3
;
0
<
x
4
<
+
∞
} and let L
m
=
m
i
n
(
L
1
,
L
2
,
L
3
)
.Givenu
0
∈
H
2
(
D
)
∩
V, consider in D
×
(
0
,
t
) the following problem:(73)
u
t
-
ν
Δ
u
+
∇
p
+
u
·
∇
u
=
0
i
n
D
×
0
,
t
,
∇
·
u
=
0
i
n
D
×
0
,
t
,
u
=
0
o
n
∂
D
×
0
,
t
,
t
>
0
,
l
i
m
x
4
→
∞
u
x
,
t
=
0
,
t
>
0
,
u
x
,
0
=
u
0
x
i
n
D
.Theorem 10.
Givenu
0
∈
H
2
(
D
)
∩
V and D such that(74)
π
L
m
1
/
2
ν
7
/
4
-
18
u
0
1
/
4
u
t
3
/
4
0
>
0
,there exists a unique strong solution u
(
x
,
t
) to (73): (75)
u
∈
L
∞
0
,
∞
;
V
,
u
t
∈
L
∞
0
,
∞
;
H
∩
L
2
0
,
∞
;
Vsuch that for all Φ
:
R
4
→
R
4
∈
V,  Φ
|
∂
D
=
0, tending sufficiently rapidly to 0 as x
4
→
∞, it satisfies the following identity:(76)
u
t
,
Φ
t
+
ν
∑
i
=
1
4
D
i
u
,
D
i
Φ
t
+
b
u
,
Φ
,
u
t
=
0
,
t
>
0
.Moreover, the following inequalities hold:(77)
u
t
≤
u
0
e
-
θ
t
,
u
t
t
≤
u
t
0
e
-
θ
/
2
t
,
u
t
V
≤
C
u
0
u
t
0
e
-
3
/
4
θ
t
,
whereθ
=
ν
π
2
∑
i
=
1
3
1
/
L
i
2
.Proof.
Obviously, the variational formulation of (73) is also (23). Repeating the proof of Theorem 5, we can prove the existence and uniqueness of the strong solution to problem (73). Using the Steklov inequalities with respect to variables x
1
,
x
2
,
x
3, we obtain(78)
∇
u
2
≥
∑
i
=
1
3
π
2
L
i
2
u
2
+
D
4
u
2
.hence, (23) becomes(79)
d
d
t
u
2
t
+
2
ν
∑
i
=
1
3
π
2
L
i
2
u
2
t
+
2
ν
D
4
u
2
t
≤
0
.This implies(80)
u
2
t
≤
u
0
2
e
-
2
θ
t
.
Repeating the proof of Estimate II of Theorem5, we find(81)
u
t
2
t
≤
u
t
2
0
e
-
θ
t
and(82)
u
V
2
t
≤
1
ν
u
t
t
u
t
≤
1
ν
u
0
u
t
0
e
-
3
/
2
θ
t
.
Decay for Pressure. In order to obtain decay for ∇
p
L
4
/
3
(
D
)
(
t
), we start with(83)
u
·
∇
u
L
4
/
3
D
t
=
B
u
L
4
D
′
t
,where L
4
D
′ is the dual to the space L
4
(
D
). Since (84)
A
u
=
-
u
t
-
B
u
,then by (81) and (83), (85)
A
u
t
L
4
/
3
Ω
≤
c
1
e
-
1
/
2
θ
t
.Moreover, by (69),(86)
∇
p
L
4
/
3
D
t
≤
u
t
L
2
D
t
+
A
u
L
4
/
3
D
t
+
B
u
L
4
/
3
D
t
≤
c
2
e
-
1
/
2
θ
t
.
Jointly (80), (81), and (86) prove Theorem 10.Conclusions. In our work, we tried to respond to some questions posed by J. Leray [1], namely, regularity of global solutions of the Navier-Stokes equations and their decay. Therefore, our results can be divided into two parts: the first one concerns decay of global regular solutions of the 4D Navier-Stokes equations posed on bounded 4D parallelepipeds. It is known that there exist global regular solutions for the 2D Navier-Stokes equations posed on smooth bounded domains [4, 6, 8, 9], but regularity in nonsmooth (Lipschitz) domains is not obvious. For bounded 4D parallelepipeds, we have established the existence of a unique global regular solution which decays exponentially as t
→
+
∞ provided that initial data satisfies (25). We demonstrated that the decay rate is different for different norms; see (77), where θ is defined by the geometrical characteristics of a domain Ω.The second part of our work concerns decay of solutions for the 4D Navier-Stokes equations posed on an unbounded parallelepiped. In existing publications [3, 4, 6, 9], the decay rate of u
L
2
(
Ω
)
(
t
) is controlled by the first eigenvalue of the operator A
=
-
P
Δ, where P is the projection operator on a solenoidal subspace of L
2
(
Ω
)
. It is clear that this approach does not work in unbounded domains.On the other hand, our approach based on the Steklov inequalities allowed us to estimate the decay rate of a strong solution for the 4D Navier-Stokes equations posed on an unbounded 4D parallelepiped.We must emphasize that this estimate is the first one which gives an explicit value of the decay rate for unbounded 4D domains. Results established in our work can be used in constructing of numerical schemes for solving initial-boundary value problems for the Navier-Stokes equations appearing in Mechanics of viscous liquid. From the physical point of view, decay estimates show that the decay rate of perturbations of solutions caused by the initial data is bigger for bigger values of viscosityν and smaller sizes of 4D parallelepipeds.My interest for the 4D Navier-Stokes equations is purely mathematical and, on my opinion, can not be extended to higher dimensions beyond 4. I must also note that there are publications on the existence of weak solutions for 4D Navier-Stokes equations [7], [9] p.189-197.



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