Journal of ****, 2025, *(*), *-*

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ISSN Online:****-****
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Classical Fundamental Unique Solution for the In-

compressible Navier-Stokes Equation in RN

Waleed S. Khedr

Independent Researcher, Cairo, Egypt

Email: waleedshawki@yahoo.com

How to cite this paper: Khedr,

W. S. (2025) Classical Fundamen-
Abstract
tal Unique Solution for the Incom- We present a class of non-convective classical solutions for the multi-
pressible Navier-Stokes Equation in dimensional incompressible Navier-Stokes equation. We validate such
RN , Journal of ***, *, *-*. class as a representative for solutions of the equation in bound-
http://dx.doi.org/10.4236/***.
2025.*****
ed and unbounded domains by investigating the compatibility condi-
Received: **** **, *** tion on the boundary, the smoothness of the solution inside the do-
Accepted: **** **, *** main and the boundedness of the energy. Eventually, we show that
Published: **** **, *** this solution is indeed the unique classical solution for the prob-
Copyright c 2025 by author(s) and
lem given some appropriate and convenient assumptions on the data.
Scientific Research Publishing Inc.
This work is licensed under the Cre-
ative Commons Attribution Inter-
national License (CC BY 4.0). Keywords
http://creativecommons.org/
Fluid Mechanics; Navier-Stokes Equation; Fundamental Solutions.
licenses/by/4.0/

Lorem ipsum dolor sit amet, con-

sectetuer adipiscing elit. Ut pu-
rus elit, vestibulum ut, placerat 1. Introduction
ac, adipiscing vitae, felis. Cur-
abitur dictum gravida mauris. Nam In this article a well known model is to be investigated that represents the
arcu libero, nonummy eget, con- flow of an incompressible fluid in both bounded and unbounded domains
sectetuer id, vulputate a, magna. N
Donec vehicula augue eu neque. of R . This model is commonly called the Navier-Stokes equation follow-
Pellentesque habitant morbi tris- ing the French engineer Navier who was the first to propose this model.
tique senectus et netus et malesua- This model was investigated later by Poisson and de Saint Venant. How-
da fames ac turpis egestas. Mauris ever, Stokes was the one who justified the model based on the principles
ut leo. Cras viverra metus rhon-
cus sem. Nulla et lectus vestibu-
of continuum mechanics. By advent of 1930 the interest in this model in-
lum urna fringilla ultrices. Phasel- creased significantly and outstanding results were obtained by Leray, Hopf,
lus eu tellus sit amet tortor gravi- Ladyzhenskaya and Finn.
da placerat. Integer sapien est, i- This equation describes the flow of what is so called the Newtonian
aculis in, pretium quis, viverra ac,
nunc. Praesent eget sem vel leo ul- fluid. These are the fluids that exhibit shearing stress due to the presence
trices bibendum. Aenean faucibus. of frictional forces. Frictional forces within fluids are consequences of its
Morbi dolor nulla, malesuada eu, viscosity. Also, the gradient of the velocity represents a measure for the
pulvinar at, mollis ac, nulla. Cur- relative motion of the fluid’s particles. Moreover, deformation of fluids is
abitur auctor semper nulla. Donec
varius orci eget risus. Duis nibh
mi, 10.4236/***.2025.*****
DOI: congue eu, accumsan eleifend,
**** **, 2025
sagittis quis, diam. Duis eget orci
sit amet orci dignissim rutrum.
Nam dui ligula, fringilla a, euismod
sodales, sollicitudin vel, wisi. Mor-
bi auctor lorem non justo. Nam la-
cus libero, pretium at, lobortis vi-
W. S. Khedr
commonly associated with internal and external body forces; the internal
force is what we refer to as the pressure of the fluid. The derivation of
the Navier-Stokes equation is a natural application of Newton’s second
law of motion, the balance of momentum and the mass conservation, which
eventually leads to the definition of the Cauchy stress tensor. In Newtonian
fluids this stress tensor is a function in the pressure, the viscosity and
the gradient of the velocity. For a convenient physical background about
the basics of continuum mechanics and how we derive the Navier-Stokes
equation we propose [1, 2]. Also a very interesting work from a physical
point of view can be found in [3, 4]. In particular, the work of Kambe in
[4] was the source of inspiration for the ideas in this article.
This model poses a serious challenge when it comes to proving the ex-
istence and the smoothness of its solution. This problem was perfectly
addressed by Ladyzhenskaya in two dimensional spaces among many other
issues in higher dimensional spaces [5]. However, a decisive answer in the
three dimensional space or higher remains unavailable. It is almost im-
possible to enlist all the results obtained for this equation. Therefore, we
suggest for the interested reader to review the monographs [5, 6, 7] and the
references within for much more details.
Recently, the interest in this equation is not fading at all. There are
persistent efforts to clarify the properties of the solution, especially its s-
moothness. Among many respectful results, we mention the outstanding
analysis by Tao in [8], the work of Constantin in [9, 10, 11]. A very inter-
esting result for partial regularity of suitable weak solutions was obtained
by Caffarelli in [12].
In this article the idea is simple. A class of possible solutions is pro-
posed and then it is proved that it indeed represents the unique classical
solution of the problem. Most of the results are obtained by considering
standard theories of partial differential equations. Some of the results in
the monograph [7] are also used repeatedly. In the next section a statemen-
t of the problem is introduced along with some definitions, notations and
employed functional spaces. Afterwards, the proofs of the main results are
established.

2. Statement of the problem

The spatial domain is Ω which is either a bounded region in RN or the whole
of RN and this point shall be specified explicitly. For the sake of conciseness,
the notation Ωt is used to denote Ωt = {(x, t) : x ∈ Ω, t ∈ (0, ∞)}. Clearly,
such notation should not be taken to imply a moving boundary. The main
model equation is in the form

[5pt2pt][c]lvt + (v · ∇)v = µ∆v − ∇p + f in Ωt , ω = ∇ × v in Ωt , v(x, 0) = v
(1)
where the last equation in the above model is what many authors com-
monly refer to as the incompressibility condition or the solenoidal
condition. The first term in the first equation is the acceleration of the
fluid’s flow in time, the second is the convective term that represents the
acceleration of the flow in space, the third represents the diffusion scaled by
the kinematic viscosity constant µ, the fourth is the pressure, and the last
one represents the total of the external body forces. The initial profile is

2
 W. S. Khedr
denoted by v0 and the boundary datum is denoted by v∗ . The solution v is
the vector field representing the velocity of the flow in each direction, and
its rotation ω is the vorticity. Note that ∇ · ω = 0 in Ωt by compatibility.
The well known Lebesgue spaces Lq (Ω) will be used repeatedly to repre-
sent the functions with bounded mean of order q. The Sobolev space H m (Ω)
is used to represent functions with bounded derivatives such that for a vec-
tor field v = {v1 , . . . , vN } one has ∂ |α| vi ∈ L2 (Ω) for every |α| = 1, . . . , m
and i = 1, . . . , N . This motivates the usage of the space V m (Ω), which is
a well known space of functions in the theory of incompressible fluids as
a representative for divergence free (solenoidal) bounded vector fields such
that V m (Ω) = {v ∈ H m (Ω) : ∇ · v = 0 in Ω}.
By laws of classical mechanics, the energy generated by a moving ob-
ject is proportional to the square of its velocity. Hence, the energy E(t)
generated by the flow v is defined as follows
Z
E(t) = |v(x, t)|2 dx. (2)
Ω

Recall that the above integral represents the norm of v in the Lebesgue
space L2 (Ω). The smoothness of v0 (x) is such that

v0 (x) ∈ C2 (Ω) ∩ V N +2 (Ω). (3)

The smoothness of the boundary datum v∗ (xN −1 , t) is such that

∗
v∗ (xN −1 , t) ∈ C∞ (∂Ωt ) and v∗ (·, t) ∼ t−K for any K ∗ > 1. (4)

Finally, the forcing term f is smooth in space and time such that

f (x, t) ∈ C1 ([0, ∞]; C1 (Ω)∩H 3 (Ω)) and f (·, t) ∼ t−K for any K > 0. (5)

Note that the intersections in the above conditions are not really re-
quired in the case of bounded domain since boundedness of the domain
and continuity of the functions are enough to imply boundedness in the
sense of the mean. However, these requirements are of significant impor-
tance in the case of unbounded domain as will be shown later.
The target is to define a class of possible solutions to Model Equation
(1) from which v, ω and p can be concluded. Once v is obtained, then p
can easily be recovered from the main model. The validity of this solution
as a meaningful physical solution will be investigated when inserted in the
main model. A meaningful solution is a unique and smooth solution that
vanishes as t → ∞, and in the case of unbounded domain it vanishes as
|x| → ∞ as well.

Remark. The curl operator or the rotation of a vector field has a physical
meaning only in three dimensional space. However, it will be used in RN for
the sake of generality. Most of the results depend on the curl operator in
the sense of a differential operator without direct exposure to its definition.
Note that the main interest is to find the velocity, which means that any
use of the rotation, in spite of its significance in this work, is nothing
more than a transient step. It can always be assumed that the space is
three dimensional when needed, and a generalization becomes possible by
reverting back to the results of the velocity. In particular, some of the
vorticity ideas introduced in [7] are adopted in this study.

3
W. S. Khedr
3. Main results
In this section a class of possible solutions is proposed and the insertion of
these solutions in the main model is investigated to check where they will
lead to. This is initiated by the statement of the following claim.

Claim 1. The unique solution of Model Equation (1) is in the form

v(x, t) = ψ(x, t)u(t) where ψ : RN × R → R is a scalar field and
u = (u1 (t), . . . , uN (t)) is a vector field such that, at least, ψ(x, t) ∈ C2 (Ωt )
and u ∈ C(R).

An important question in the theory of Navier-Stokes equation is the

ability to verify the compatibility condition on the boundary with mini-
mum restrictions on the flux passing through the boundary especially if
∂Ω is divided into several parts [6, pp. 4-8]. This condition is a natural
consequence of the incompressibility of the flow. Hence, it takes the form
Z
(v · ~n)dxN −1 = 0, (6)
∂Ω

where ~n is the outward unit vector normal to ∂Ω. This motivates the
introduction of the following lemma.

Lemma 1 (Tangential flow). Let Ω be an arbitrary domain, ψ(x, t) :

RN × R → R be any scalar field such that ψ ∈ C1 (Ω) and let u(t) =
(u1 (t), . . . , uN (t)) be any vector field independent of x. The Compat-
ibility Condition (6) is satisfied for every divergence free vector field
v(x, t) : RN × R → RN in the form v = ψ(x, t)u(t). In particular, on
every part of ∂Ω, v and its rotation ω are tangents to ∂Ω such that
v · ~n = ω · ~n = 0.

Proof. Given that v = ψ(x, t)u(t) is divergence free such that ∇ · v = 0

leads to
N
X ∂ψ
ui = 0. (7)
∂xi
i=1

Identity (7) can be used to deduce that (v · ∇)v = (∇v)v = 0. Upon dot
product by v one obtains
1 1
0 = (∇v)v · v = (∇v)T v · v = v · ∇|v|2 = ∇ · (|v|2 v), (8)
2 2
and when integrated over Ω for any t > 0 provides
Z Z
2
0= ∇ · (|v| v)dx = (|v|2 v · ~n)dxN −1 , (9)
Ω ∂Ω

which is a true identity for every arbitrary Ω, ψ and u and for every t > 0.
On the other hand, given that (v · ∇)v = 0, one can use the identity

1
∇|v|2 = (v · ∇)v + v × ∇ × v = v × ω, (10)
2
where ω = ∇ × v, which upon dot product by ω provides

∇ · (|v|2 ω) = 0,

4
 W. S. Khedr
where we used also that ∇ · ω = 0. Integrate over Ω for any t > 0 to get
Z Z
0= ∇ · (|v|2 ω)dx = (|v|2 ω · ~n)dxN −1 , (11)
Ω ∂Ω

which is true for arbitrary Ω, ψ and u. Now, since ∇ · v = 0 one also has
Z Z
0 = (∇ · v)dx = (v · ~n)dxN −1 . (12)
Ω ∂Ω

Identity (10) implies that ∇|v|2 is orthogonal to the space spanned by v

and ω. Combine Identities (9), (11) and (12), and exclude the cases |v| = 1
and |v| = 0 by the arbitrariness of the choice to deduce that it is necessary
that v · ~n = 0; that is to say v is tangential to every part of ∂Ω.
It can also be deduced that either v = ω on every part of ∂Ω (this
actually means v = ω = 0 on ∂Ω), or ω · ~n = 0 on every part of ∂Ω. Both
cases imply that ω · ~n = 0 on every part of ∂Ω. Hence, both v and ω are
tangential to the boundary.

Remark. As pointed out, the question of verifying the Compatibility Con-

dition (1) on the boundary of Ω is an important open question in the math-
ematical theory of Navier-Stokes equation. Some results were obtained to
justify the validity of such compatibility under certain restrictions on the
flux of the flow in terms of the viscosity of the fluid, for details on this issue
refer to [6]. In the present case, the compatibility is naturally achieved
given, of course, that the solution of Model Equation (1) is indeed in the
form proposed in Claim 1. This shall be verified by the statements of the
following theorems.

Theorem 2 (Bounded Domain). Let Ω ⊂ RN be a bounded domain with

sufficiently smooth boundaries ∂Ω and let Ωt = Ω × (0, ∞). Suppose v0 (x),
v∗ (xN −1 , t) and f (x, t) satisfy Conditions (3), (4) and (5) respectively. If
v(x, t) is in the form proposed in Claim 1, then Model Equation (1) has
a classical solution (v, ω, p) with bounded energy E(t) such that v(x, t) ∈
C∞ (Ωt ), ω(x, t) ∈ C∞ (Ωt ) and p(x, t) ∈ C 1 ([0, ∞]; C 3 (Ω)). In particular,
the exact solution is given by solving the following system

[5pt2pt][c]l [5pt2pt][c]lωt = µ∆ω + ∇ × f ∆v = −∇ × ω∆p = ∇ · f } in Ωt , [5pt
(13)
where ∇p can be defined uniquely in terms of the values of v and f on the
boundary. Moreover, if f ∈ C∞ (Ωt ) then p ∈ C ∞ (Ωt ).

Proof. The proof is quite simple and it depends mostly on classical results
and the standard theory of linear parabolic and elliptic equations of second
order. If v = ψ(x, t)u(t), then by virtue of Identity (7) one has (v·∇)v = 0.
Hence, the main equation takes the form

vt = µ∆v − ∇p + f . (14)

Apply the divergence operator to get

∆p = ∇ · f , (15)

where the incompressibility condition ∇ · v = 0 is used. By the s-

tandard theory of elliptic equations, if f satisfies Condition (5), then

5
W. S. Khedr
p ∈ C 1 ([0, ∞]; C 3 (Ω)). However, if f ∈ C∞ (Ωt ), then p is actually C ∞ (Ωt ),
for details on such equation see [13, pp. 326-343]. This concludes one part
of System (13), however, a further discussion on a unique definition of p
will be introduced at the end of this proof.
Now, revert to Equation (14) and apply the curl operator to get

ωt = µ∆ω + ∇ × f , (16)

which is the first equation in System (13). Finally, given the incompress-
ibility of v and applying a simple vector identity lead to the third equation
in System (13) that is
−∆v = ∇ × ω. (17)
The fundamental solution ω to Equation (16) with initial profile ω0 and
N R |x−y|2
−
force ∇ × f is given formally by rCl ω(x, t) = (4πµt)− 2 Ω e 4µt ∇ ×
v0 (y)dy
|x−y|2
Rt N R −
+ 0 (4πµ(t−s))− 2 Ω e 4µ(t−s) ∇×f (y, s)dyds. As explained in [14, Chapter
7], because of the smoothing property of the Gaussian kernel, it is enough
to have a contentious data under the integral sign to guarantee that ω ∈
C∞ (Ωt ), which is what has been already assumed. Given the assumptions
on the growth of f in time and the form of Formula (3), the solution ω → 0
as t → ∞. Moreover, the continuity of the data, the clear decay in time
and the boundedness of Ω imply that |ω| < C in Ωt , which implies actually
that ω ∈ L2 (Ω) for every t > 0. Uniqueness of ω as per Expression
(3) is not clear unless we insert the boundary datum ∇ × v∗ explicitly in
Expression (3). This can be done by introducing the auxiliary variable
w = ω − ∇ × v∗ for which one obtains a homogeneous heat equation.
Anyhow, the presence of any form of boundary conditions guarantees the
uniqueness of ω. Moreover, since ω is actually a derivative of v, then it
suffices to show that v is unique, which is our main concern, to conclude
the uniqueness of ω.
Now, go back to Equation (17). By virtue of our assumptions on the
data, the results obtained above for ω and the standard theory of elliptic
equations one directly concludes that v ∈ C∞ (Ωt ), for more details see [13,
pp. 326-343]. Such regularity, the boundedness of Ω and the global decay
of ω in time imply that v is bounded in L2 (Ω) for every t > 0, which in
turn implies the boundedness of the energy of the flow E(t) as defined by
Expression (2). Since ω ∈ L2 (Ω), then [7, Proposition 2.16] can be used to
conclude that a formal solution for Equation (17) takes the form

x−y
Z
1
v(x, t) = × ω(y, t)dy, (18)
ωN Ω |x − y|N

where ωN is the area of the unit sphere in RN . The solution v can be

enforced to take the values v∗ on the boundary in a standard manner by
introducing Dirichlet Green’s function. We refrain from discussing these
details being highly dependent on the choice of the domain. The uniqueness
of v follows by the presence of the boundary condition v∗ , for details see
[13, 14].
Finally, go back to Equation (15) to solve for p. In this case one only
needs to calculate ∇p · ~n from the main model by knowing the values of vt ,
∆v and f on the boundary. This provides a form of boundary conditions

6
 W. S. Khedr
for p, which consequently guarantees its uniqueness up to a constant, for
details see [14]. The infinite differentiability of p follows also from the main
model and the fact that v ∈ C∞ (Ωt ) provided, of course, that f ∈ C∞ (Ωt )
as well. If f = 0, then p is certainly C ∞ (Ωt ). This completes the proof.

Remark. As explained in the proof of Theorem 2, the uniqueness of ω fol-

lows by the ability to define v uniquely. The order of solving the equations
in System (13) is not really important since none of the quantities (v, ω, p)
induces the other; they act simultaneously. Another way of solvability can
be introduced by which one can obtain the same results. This can be a
topic for a future study.

The problem of proving the existence of regular and smooth enough so-
lutions for the Navier-Stokes equation in bounded domain was exhaustively
investigated as pointed out in the introduction. The real problem was to
prove the boundedness of the solution in an unbounded domain, clearly
because of the unboundedness of the domain itself. This fact manifests the
need to show that the solution’s support is bounded in RN , or equivalently
to show that the solution v decays rapidly as |x| → ∞.
The solution obtained in Theorem 2 represents a perfect candidate as a
solution for Model Equation (1) in unbounded domains also except for one
issue. One needs to prove the boundedness of v in L2 (RN ) for every t > 0 so
that the boundedness of the energy E(t) can be claimed, and also to ensure
that the solution does indeed vanish as |x| → ∞. For v to be bounded in
L2 (RN ), it should attain a rate of decay, at least, |v| ≤ C(1 + |x|)−(N +δ)/2
for any δ > 0. One can argue that some of the results in the literature
require a rate of decay higher than that for the surface integrals to vanish;
these restrictions can be dropped because these integrals already vanish by
virtue of Lemma 1, see [7, Lemma 1.5]. However, this does not mean that
rapid rates of decay are not achievable, they are achievable as demonstrated
next.
In order to derive such an estimate one goes back to Formula (18) that
represents the fundamental solution for v. If |ω| ≤ C then it is expected
that the outcome of this integral will provide nothing less than a linear rate
of growth for |v|, which is a bad answer to the problem in hand. Therefore,
Formula (3) shall be used to help us estimate some rates of decay for ω and
consequently for v so that boundedness in L2 (Ω) can be proved for every
t > 0.

Theorem 3 (The Domain RN ). Suppose all the conditions of Theorem 2

are satisfied for Ω = RN . Then there exists a classical solution (v, ω, p) for
Model Equation (1) represented by the System (13) and defined as
Z |x−y|2
Z t Z |x−y|2
− − 4µ(t−s)
ω(x, t) = h(t) e 4µt ω0 (y)dy + h(t − s) e g(y, s)dyds,
RN 0 RN
(19)
and
x−y
Z
1
v(x, t) = × ω(y, t)dy, (20)
ωN RN |x − y|N
N
where h(t) = (4πµt)− 2 , g = ∇ × f and ωN is the surface area of a unit
sphere. The pressure p can be defined from Model Equation (1) up to a
constant where ∇p · ~n can be specified uniquely in terms of the values of v

7
W. S. Khedr
and f on the boundary. Moreover, the energy of the flow E(t) is bounded
for every t > 0 where v grows at most as |v| ∼ (1 + |x|)−(N +6)/2 .

Proof. The proof of smoothness and uniqueness is identical to the one in-
troduced in the proof of Theorem 2 and it follows by the standard the-
ory of linear second order elliptic and parabolic equations. The focus
here will be on proving the boundedness in L2 (RN ) for both ω and v,
which necessarily entails an estimation of appropriate decay rates as point-
ed out in the preceding discussion. Since f ∈ C1 ([0, ∞]; H 3 (RN )), then
g ∈ C([0, ∞]; H 2 (RN )). By assumption, v0 ∈ V N +2 (RN ) which implies
that ω0 ∈ V N +1 (RN ) and since N ≥ 2 for meaningful physical interpreta-
tion, then at least ω0 ∈ V 3 (RN ). Hence, ω ∈ L2 ((0, ∞]; V 4 (RN )) as per
the standard theory of linear second order parabolic equations, for details
refer to [13, Theorem 6, p. 386].
Now, it is needed to show that v is bounded in L2 (RN ) for every t > 0.
There are two ways to show this; in one of them a rough estimate will be
provided for the minimum rate of decay of v given the assumptions on the
data.
The first direction depends on the results in [7, Theorems 3.4 and 3.6].
In these theorems a regularization technique by mollifiers along with energy
estimates were used to prove global in time existence. In particular, [7,
Theorem 3.4] states that if v0 ∈ V m and m ≥ [N/2] + 2, then there exists
a unique continuous solution locally in time such that this is true up to
T ≤ C(kvkm )−1 , which coincides with the assumptions on v0 . The local
existence of a unique continuous solution was extended to a global in time
existence in [7, Theorem 3.6] given that
Z T
|ω(x, t)|L∞ (Ωt ) dt ≤ C
0

and such that v ∈ C1 ((0, T ]; C2 (Ω) ∩ V m (Ω)). Since ω0 and g are bounded
in L2 (RN ) for every t > 0, then it follows that there exists a ball B ⊂ RN
such that the supports of g and ω0 are entirely inside Bt where Bt =
B × [0, ∞]. Hence, the integrals in Formula (19) can be restricted to the
ball B, apply Hölder’s inequality, maximize the exponential term which is
bounded for every x, y ∈ B and for every t > 0, and with some estimation
procedures it becomes easy to find an estimate of the form |ω(x, t)| ≤ Ch(t)
in the whole of RN and for every t > 0. Since h(t) is a decreasing function
in time then by assuming a simple scale of time one obtains
Z ∞ Z ∞
|ω(x, t)|L∞ (Ωt ) dt ≤ C h(τ )dτ ≤ C.
0 1

It follows that all the conditions of [7, Theorem 3.4, Theorem 3.6] are
satisfied such that v exists globally in time and such that v ∈ V N +2 (RN )
for every t > 0, which implies the boundedness of the energy E(t) for every
t > 0 as well.
The second way is trying to get an estimate for v in terms of |x| to
confirm the boundedness in L2 (RN ). Consider the following argument:
fix t in Expression (19), calculate ∇ω, which is in L2 (RN ) for every t >
0 because ω ∈ V 4 (Ω). Take the absolute value of both sides, perform
some manipulation to the integrands, use x · y ≤ |x||y| and maximize the
time integral (the integrand is a decreasing function in time) so that one

8
 W. S. Khedr
finally gets the term with the highest power for |x| as follows rCl |∇ω| ≤
1
2 2 − |x−y|2

R |y|
H(t) B |x| 1 + |x| e 4µt (|ω0 (y)|L∞ (B) + |g(y, t)|L∞ (Bt ) )dy
≤ H(t)M(|x|)|x|C(|B|, |ω0 |L∞ (B) , |g|L∞ (Bt ) )
≤ CM(|x|)|x|, where M(|x|) is the collection of every possible appearance
of any power of |x| after the integration. Since ∇ω is in L2 (RN ) as pointed
out, then to obtain a decreasing integrand when calculating the L2 -norm
of this derivative it is necessary that M(|x|) is decreasing in |x| such that
N+2+δ
it is at least M(|x|) ∼ (1 + |x|)− 2 for some δ > 0. But ω is in L2 (RN )
as well, and the integrands of the above estimate are the same except for
the terms with positive powers of |x|. That is to say that |ω| ≤ CM(|x|) ∼
N+2+δ
(1+|x|)− 2 . Incorporate this estimate in Expression (20) of the solution
N +δ
v and one can readily see that |v| ∼ (1 + |x|)− 2 as desired, which in turn
confirms the boundedness of the energy E(t) for every t > 0 and implies
the decay of v as |x| → ∞.
However, since ω ∈ V 4 (RN ), then we actually have D4 ω ∈ L2 (RN ) for
every t > 0. This means that we can differentiate Formula (19) four times
and repeat the same argument as above to conclude that we actually have
N+8+δ N +6+δ
M(|x|) ∼ (1 + |x|)− 2 which in turn leads to |v| ∼ (1 + |x|)− 2 and
this completes the proof.

Remark. Better estimates for the decay of v can be obtained by repeating

the same procedure described above for higher order derivatives of ω which,
of course, entails higher assumptions on the data so that we can claim
the boundedness of the considered derivative apriori. On the other hand,
the assumptions on the smoothness of the forcing term f can be relaxed.
It is enough to assume that f ∈ C1 ([0, ∞]; C1 (RN ) ∩ H 1 (RN )) so that
one obtains g ∈ C([0, ∞]; L2 (RN )), which is sufficient to conclude that
ω ∈ L2 ((0, ∞]; V 2 (RN )). The boundedness of ∇ω in L2 (RN ) is enough to
deduce a sufficient decay estimate for v as shown above. The introduced
assumptions were chosen for consistency with the standard theory and the
Embedding Theorem, and in the same time to illustrate the estimation
procedure. For more information review [13, pp. 382-386].
We managed to prove that our solution is indeed a classical solution of
Model Equation (1). Here come some important questions, what if there
exists another solution in a more general form? Moreover, does the choice
of the domain or the choice of the boundary data play a role in the unique-
ness of the solution? The answer to these questions is addressed by the
statement of the next theorem.
Theorem 4 (Uniqueness). Let Ω ⊆ RN be an arbitrary domain. Suppose
v0 , v∗ and f are satisfying Conditions (3), (4) and (5) respectively. Then
Claim 1 is true and the unique classical solution of Model Equation (1) is
in the form v(x, t) = ψ(x, t)u(t). This solution is defined as per Theorems
2 and 3.
Proof. The proof here depends on our results in Theorems 2 and 3, and
also on [7, Theorems 3.4 and 3.6]. By virtue of the assumptions on the
data and our results in Theorems 2 and 3, it is clear that our solution v
satisfies all the conditions in [7] for every t ∈ (0, ∞].
Assume that there exists a more general solution than the proposed one
and denote it by vg . Such solution should definitely inherit the smoothness

9
W. S. Khedr
proved in [7] as well. That is to say that vg ∈ C1 ((0, ∞]; C2 ∩V m ) and such
that (vg , ωg , pg ) is the triad solution of Model Equation (1) with boundary
datum v∗ and initial profile v0 . Let w = v − vg and let q = p − pg . Hence,
w has zero boundary and initial data and it obeys the equation

wt − µ∆w + ∇q = (vg · ∇)vg . (21)

Dot product the above equation R by vg and integrate by parts

R over any
arbitrary domain Ω to get rCl Ω v · (wt − µ∆w + ∇q)dx = 12 ∂Ω |vg |2 vg ·
g

~ndxNR −1
= 12 ∂Ω |v|2 v · ~ndxN −1 = 0, where we used the Divergence theorem in the
right hand side, the facts that ∇ · vg = 0 and that vg = v∗ = v on the
boundary, and the results of Lemma 1. Now, recalling that ∇ · ωg = 0 and
using the vector identity

∇|vg |2 = 2(vg · ∇)vg + 2vg × ωg , (22)

Rone can dot product Equation (21) by ωg and integrate as above to get rCl
g 1 g 2 g ndx
R
Ω ωR · (wt − µ∆w + ∇q)dx = 2 ∂Ω |v | ω · ~ N −1
1 ∗ 2 ∗
= 2 R∂Ω |v | ∇ × v · ~ndxN −1
= 12 ∂Ω |v|2 ω · ~ndxN −1
= 0, where Lemma 1 is used again. Identities (3) and (3) imply one of
three possibilities. Either vg = ωg = 0 which is excluded for being trivial,
or wt − µ∆w + ∇q = 0 almost everywhere. The third possibility is wt −
µ∆w + ∇q being orthogonal to the space spanned by vg and ωg , which by
Equation (21) implies that (vg · ∇)vg is orthogonal to ωg and vg .
Let us start with wt − µ∆w + ∇q = 0 almost everywhere. Multiply this
equation by w, integrate by parts over Ω, employ the Divergence theorem
and recall that w = 0 on ∂Ω to get
Z Z Z
1d
|w|2 dx + µ |∇w|2 dx = − qw · ~n dxN −1 = 0.
2 dt Ω Ω ∂Ω

This readily implies that w = 0 almost everywhere. But by the results

obtained for v and vg , one concludes that at least w ∈ C2 (Ω) which implies
that w is identically zero. Hence, the solution v is the unique solution for
Model Equation (1) in this case.
Now, if vg and ωg are orthogonal to (vg · ∇)vg , then by Identity (22)
we deduce that ∇|vg |2 is orthogonal to the space spanned by vg and ωg
everywhere, which is equivalent to the nature of the solution v. This means
that ∇|v|2 and ∇|vg |2 are parallel to each other on the boundary. Since
both solutions coincide at the boundary and both are extended continuously
to the interior of the domain, then it is not hard to conclude that vg = v
everywhere, which is the aim of this proof. Assume not. Let θ = ω − ωg
so that the difference equation for θ takes the form

θt − µ∆θ = (vg · ∇)ωg − (ωg · ∇)vg . (23)

where (ωg · ∇)vg − (vg · ∇)ωg = −∇ × (vg · ∇)vg . Use the vector identity

∇(vg · ωg ) = (∇vg )T ωg + (∇ωg )T vg ,

and given the incompressibility of vg one also has

1 1
∇ωg ωg = ∇|ωg |2 − ωg × (∇ × ωg ) = ∇|ωg |2 + ωg × ∆vg .
2 2

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 W. S. Khedr
Use these two identities, bearing in mind the incompressibility, to find that
rCl vg · (∇ωg vg − ∇vg ωg ) = (∇ω)T vg · vg − 21 ∇|vg |2 · ωg
= ∇(vg · ωg ) · vg − (∇vg )T ωg · vg − 21 ∇|vg |2 · ωg
= ∇ · ((vg · ωg )vg − 12 |vg |2 ωg ) − ∇vg vg · ωg
= ∇ · ((vg · ωg )vg − 21 |vg |2 ωg )
− 12 ∇|vg |2 · ωg + (vg × ωg ) · ωg
= ∇ · ((vg · ωg )vg − |vg |2 ωg )
= ∇ · G1 (vg , ωg ). Now recall that we are discussing the possibility in which
∇|vg |2 is orthogonal to vg and ωg such that C∇|vg |2 = (vg × ωg ). Follow-
ing the same steps as above one can write rCl ωg · (∇ωg vg − ∇vg ωg ) =
∇ · (|ωg |2 vg − (vg · ωg )ωg ) − (ωg × ∆vg ) · vg
= ∇ · (|ωg |2 vg − (vg · ωg )ωg ) − (vg × ωg ) · ∆vg
= ∇ · (|ωg |2 vg − (vg · ωg )ωg ) − C∇|vg |2 · ∆vg
= ∇ · (|ωg |2 vg − (vg · ωg )ωg − C|vg |2 ∆vg )
= ∇ · G2 (vg , ωg ). Now, dot product Equation (23) by vg , use Identity (3),
integrate over Ω and use the Divergence theorem to reach
Z Z Z
g g g
v · (θt − µ∆θ)dx = G1 (v , ω ) · ~n dxN −1 = G1 (v, ω) · ~n dxN −1 .
Ω ∂Ω ∂Ω
(24)
Also dot product Equation (23) by ωg , use Identity (3), integrate over Ω
and use the Divergence theorem to get
Z Z Z
ωg · (θt − µ∆θ)dx = G2 (vg , ωg ) · ~n dxN −1 = G2 (v, ω) · ~n dxN −1 .
Ω ∂Ω ∂Ω
(25)
Now, reverse the Divergence theorem in the surface integrals that include
values of v and ω and reverse all the steps made to conclude G1 and G2
to obtain identities in the form
Z Z
g
v · (θt − µ∆θ)dx = v · ∇ × ((v · ∇)v)dx = 0, (26)
Ω Ω

and similarly
Z Z
g
ω · (θt − µ∆θ)dx = ω · ∇ × ((v · ∇)v)dx = 0. (27)
Ω Ω

Now, there are three other possibilities. The trivial solution; that is vg =
ωg = 0 and it is excluded. Another one is θt − µ∆θ = 0 and this one
is equivalent to wt − µ∆w + ∇q = 0 because the uniqueness of v implies
the uniqueness of ω and vice versa. It remains that ∇ × ((vg · ∇)vg )
is orthogonal to the space spanned by vg and ωg . But we already have
(vg · ∇)vg orthogonal to vg and ωg , which implies that (vg · ∇)vg and its
rotation (curl) are parallel to each other.
By definition, the curl operator is the unique vector field for which
(∇s−∇sT )a = (∇×s)×a for every vector field a [1, p. 32]. If s = λ(∇×s),
then let a = s = λ∇ × s to get ∇s = ∇sT which in turn implies that
∇ × s = 0 then s = 0 as well. Hence, it is necessary that (vg · ∇)vg = 0
which implies that vg = v. This completes the proof.

Remark. In the proof of uniqueness one can argue that the statement
of the proof was given in the sense of classical solutions, and such that
there still exists another solution in a weaker form. While this may sound
true, but in fact it is not. The basic idea of the proof is based on using

11
W. S. Khedr
the coincidence on the boundary and then moving back to the interior.
Assuming the existence of a weaker solution does not change the fact that
it is going to coincide with the proposed one on the boundary. However,
further investigation on this specific point will be introduced in a future
study.

4. Conclusions and Suggestions

A class of possible solutions for the incompressible Navier-Stokes equation
in bounded and unbounded regions of RN is proposed. It was demonstrated
that for such class of vector fields the flux of the energy of the flow is
orthogonal to the space spanned by v and its associated rotation ω. It
was also proved that v and ω are tangential to the boundary such that
v · ~n = ω · ~n = 0.
An investigation of the validity of this class of vector fields as a candidate
for a solution to the incompressible Navier-Stokes equation was carried out
in both bounded and unbounded domains. Given plausible assumptions on
the data and a forcing term in the case of unbounded domains, it turned
out that this class of solutions represents perfectly a classical solution of
the problem. A verification was established for the infinite differentiability,
the uniqueness and the boundedness of the energy in appropriate spaces
in light of well known and standard theories. An appropriate estimate was
also given for the minimum rate of decay of the solution v as |x| → ∞.
Moreover, global existence in time and the corresponding rate of decay were
quite obvious in the deduced formulas.
Finally, it was proved that this class of solutions represents actually the
unique classical solution of the incompressible Navier-Stokes equation. In
light of the non-convective nature of the proposed solution and the unique-
ness argument under arbitrary settings the incompressible Navier-Stokes
equation can safely be reduced to a linear equation. This point is quite
interesting and it motivates further investigations on a possible relation
between incompressibility and convection in fluid mechanics.

Acknowledgement
Sincere thanks to the members of JAMP for their professional performance,
and special thanks to managing editor Hellen XU for a rare attitude of high
quality.

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