1. Introduction
The NSEs express the conservation of mass and momentum in incompressible Newtonian fluid dynamics ranging from large-scale atmospheric motions to ball-bearing lubrication. The equation is a generalization of the equation proposed by Swiss mathematician Leonhard Euler in the 18th century to describe the flow of incompressible and frictionless fluids. Later on in 1821 French engineer Claude-Louis Navier work on it. In the middle of the 19th century, British physicist and mathematician Sir George Gabriel Stokes improved on this work. They are sometimes accompanied by an equation of state relating to pressure, temperature and density. They arise from applying Isaac Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that NSEs take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g., they are never completely integrable). The NS equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The NSEs in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell’s equations, they can be used to model and study magnetohydrodynamics. The NSEs are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., they are infinitely differentiable at all points in the domain. This is called the NS existence and smoothness problem. More information can be found in Cannone’s [
1] and Varnhorn’s [
2] monographs (see, for example, Lemarie-Rieusset [
3] and Von Wahl [
4]); there are so many phenomena for a system that explaining their existence, regularity, and boundary conditions requires the complete strength of the mathematical theory.
It is worth noting that Leray first discovered that the boundary-value problem for time-dependent NSEs has a unique smooth solution for specific time intervals if the data are sufficiently smooth. Many authors have investigated the existence of mild, weak, and strong solutions for NSEs since then; for example, Heck et al. [
5], Chemin and Gallagher [
6], Choe [
7], Giga [
8], Raugel [
9], Almeida and Ferreira [
10], wabuchi and Takada [
11], Koch et al. [
12], Masmoudi and Wong [
13], Amrouche and Rejaiba [
14], Chemin et al. [
15], Danchin [
16] and Kozono [
17].
Fractional calculus has grown in popularity in recent decades, owing to its demonstrated applications in a variety of seemingly diverse and large-ranging fields of science and engineering, such as fluid flow, rheology, dynamical processes, porous structures, diffusive transport akin to diffusion, control theory of dynamical systems and viscoelasticity, etc., for example [
18,
19,
20,
21,
22]. The models given by partial differential equations with fractional derivatives are the most important. Not only physicists, but even pure mathematicians, are interested in such models.
According to recent theoretical and experimental findings, the classical diffusion equation fails to characterize diffusion phenomena in heterogeneous porous media with fractal properties. What changes are made to the classical diffusion equation to make it suitable for describing anomalous diffusion phenomena? For researchers, this is an interesting challenge. Since it has been acknowledged as one of the greatest methods for characterizing long memory processes, fractional calculus helps model anomalous diffusion processes. As a result, presenting the generalized NSEs with a Caputo fractional derivative operator, which can be used to model anomalous diffusion in fractal media, is logical and significant. Its evolutions act in a far more complex manner than standard inter-order evolutions, making study more difficult.
The most effort has been paid to attempts to acquire numerical and analytical solutions to time-fractional NSEs [
23,
24,
25]. We are only aware of a few conclusions about mild solutions of existence and regularity for time-fractional NSEs. Carvalho-Neto [
26] recently discussed the existence-uniqueness of global and local mild solutions for time-fractional NSEs. Niazi et al. [
27], Iqbal et al. [
28] and Shafqat et al. [
29] investigated the existence-uniqueness of the fuzzy fractional evolution equation.
Zhou and Peng [
30] worked on time-fractional NSEs in an open set:
where
is Caputo fractional derivative of order
represents velocity field at point
and time
is pressure,
v is viscosity,
is external force and
is initial velocity.
In this paper, we investigate the below time-fractional NSEs in an open set
, which is motivated by the above discussion:
where
is Caputo fractional derivative of order
represents velocity field at a point and time
is pressure,
v is viscosity,
is gravitational force and
is initial velocity. We will suppose that the boundary of
is smooth.
To begin, the pressure term is removed by using the Helmholtz projector
P to Equation (
2), which transforms Equation (
2) into Equation (
3) as:
The operator
with Dirichlet boundary conditions is effectively the Stokes operator
in the divergence-free function space under consideration. Then, in the abstract form illustrated below, we rewrite (
2).
where
. The solution to Equation (
2) is also the solution to Equation (
3) if the Helmholtz projection
P and the Stokes operator
make sense.
The purpose of this research is to demonstrate that global and local mild solutions to Equation (
2) in
exist and are unique. We further show that if
is Hölder continuous, there exists a single classical solution
such that
and
are Hölder continuous in
. The following is a breakdown of the structure of the paper. In
Section 2, we go through numerous notations, definitions, and background information. Before moving on to the local mild solution in
,
Section 3 looks at existence and uniqueness of global mild solution in
of issue (
3). In
Section 4, we use the iteration method to determine the existence and regularity of a classical solution to the issue (
2) in
. Finally, in
Section 4, a conclusion is provided.
2. Preliminaries
We establish notations, definitions and introductory facts in this section, which will be used throughout the work.
Assume
to be open subset of
, where
. Assume
. Then there is bounded projection
P called the Hodge projection on
, whose range is the closure of:
and whose null space is the closure of:
For notational convenience, let , which is a closed subspace of . be a Sobolev space with norm .
represents a Stokes operator in
whose domain is
; here,
It is well known that the closed linear operator forms the bounded analytic semigroup on .
So as to state our results, we need to introduce the definitions of the fractional spaces associated with
. For
and
, define:
Then
is a bounded, one-to-one operator on
. Let
be the inverse of
. For
, we denote the space
by the range of
with the norm:
It is easy to check that
extends (or restricts) to a bounded analytic semigroup on
. For more details, we refer to Van Wahl [
4].
Define
by equation
d is the Hausdorff metric for non-empty compact sets in
.
is a metric in . By using the following results:
- (i)
is complete metric space;
- (ii)
;
- (iii)
and ;
- (iv)
.
Let be a Banach space and be an interval of . denotes the set of all continuous -valued functions. For stands for the set of all functions which are Holder continuous with the exponent .
Assume
and
. The fractional integral of order
with the lower limit zero for the function
v is defined as:
provided the right hand-side is pointwise defined on
, where
denotes the RL kernel:
and
is the usual
function. In case
, we denote
; the Dirac measure is concentrated at the origin.
Further, for a function
, the Caputo derivative of order
is defined by:
We refer the reader to Kilbas et al. [
31] for further information. Let us look at the Mittag–Leffler special functions in general:
Definition 1 ([
32])
. The Wright function is defined by:where with . Proof. (i) In view of
and Fubini theorem, we get:
where
is a suitable integral path;
(ii) A similar argument shows that:
□
We also have the results below.
Lemma 1 ([
33])
. For and are continuous in the uniform operator topology. Moreover, for every , the continuity is uniform on . Lemma 2 ([
33])
. Let . Then,- (i)
for all ;
- (ii)
for all and ;
- (iii)
for all ;
- (iv)
for .
Before presenting the definition of a mild solution of the problem (
3), we give the following lemma for a given function
. For more details we refer to Zhou [
32,
34].
Lemma 3. If is solution of Equation (3) for , then is given.holds, then:where We adopt the following definitions of the mild solution to the problem (
3), which were inspired by the previous section.
Definition 2. A function is called a global mild solution of problem (3) in , if and for , Definition 3. Assume . A function (or ) is called a local mild solution of problem (3) in (or ), if (or ) and u satisfies (5) for . Two operators, ϕ and , are defined as follows for convenience: The following fixed point result is used in further proofs.
Lemma 4 ([
1])
. Assume that is a Banach space, that is a bilinear operator, and that is a positive real number:Then for any with , the equation has unique solution .
4. Global Existence in
In this section, our aim is to find the global mild solution of the problem (
3) in
. For convenience, we denote:
where
is given later.
Theorem 1. Assume and hold. For every , let thatwhere . If , then there is a and unique function satisfying: - (i)
is a continuous and ;
- (ii)
is a continuous and ;
- (iii)
U satisfies (5) for .
Proof. Let . Define as the space of all curves such that:
- (1)
is bounded and a continuous;
- (2)
is bounded and a continuous, in addition, ;
It is obvious that is a non-empty complete metric space.
We know that
is bounded bilinear map because of Weissler [
36], so there exists
such that for
.
Step 1. Suppose
. We demonstrate that the operator
belongs to
as well as
. For arbitrary
fixed and
enough small, consider
, we have:
Each of the four terms is estimated separately. In view of Lemma 6, we derive
,
According to the characteristics of the
function, there exists a
small enough that for
,
as a result,
tends to zero as
. For
, since:
noting that
Then, we get the dominated convergence theorem of Lebesgue:
one can deduce
. For
, since:
Using Lebesgue’s dominated convergence theorem once more, the fact that the operator
has uniform continuity owing to Lemma 4 indicates:
For
, by immediate calculation, we estimate:
according to the
function’s properties. As a result, it follows:
The continuity of the operator evaluated in follows by a similar discussion to that above. So, we omit the details.
Step 2. We show that the operation
is continuous bilinear operator. By Lemma 6, we have:
it follows that
Hence, and .
Step 3. We verify that
holds. Let
. Since:
The first two integrals and the last integral trend to 0 as
and
due to the properties of the
function. In light of Lemma 4, the third integral also equals 0 as
, implying that:
The same argument applies to the evaluation of in .
as owing to assumption . This implies that and ,
and .
This, together with Lemma 6, implies that for all
,
According to (
6), the inequality,
holds, which yields that
has a unique fixed point.
Step 4. To demonstrate that
in
as
. We need to verify:
in
. It is obvious that
owing to (
8). In addition,
□
Local Existence in
In this section, the local mild solution of the problem (
3) in
is investigated.
Theorem 2. Assume and hold. Let Then there is a such that for every there exists and a unique function satisfying:
- (i)
is a continuous and ;
- (ii)
is a continuous and ;
- (iii)
U satisfies (5) for .
Proof. Let . Assume that is the space containing all curves , and that:
- (1)
is a continuous;
- (2)
is a continuous and ;
It is simple to show that
is a continuous linear map and
, similar to the proof of Theorem 1. It is clear from Lemma 4 that for all
,
It follows from Lemma 6 that:
As a result, suppose
is sufficiently small such that:
which implies that, due to Lemma 4,
has a unique fixed point. □
6. Regularity
In this section, the regularity of solution
u that fulfils the problem (
3) is examined. We will assume the following throughout this section:
is Höilder continuous with an exponent
, which means:
Definition 4. A function , if with , which takes values in and satisfies (3) for all is termed a classical solution of problem (3). Lemma 7. Let be satisfied. Ifthen and . Proof. From Lemma 6 and
, we have for fixed
:
then
We get
by the closeness of
. It is necessary to demonstrate that
is Hölder continuous. Since:
then
Given that
, we can deduce that:
According to mean value theorem, we have, for every
,
Assume that
,
, then
Each of the three terms is evaluated individually. From (
18) and
, we have
.
We apply Lemma 6 and
for
.
Moreover, for
, by Lemma 4 and
, we now have:
Combining (
20), (
21) with (
22), we conclude that
is Hölder continuous. □
Theorem 4. Assume that assumptions of Theorem 3 are fulfilled. If holds, then the mild solution of (3) is a classical one for every . Proof. In the case of
. Then, Lemma 2(ii) ensures that
is a classical solution to the below problem:
Step 1. We prove that:
is a classical solution to the problem:
Theorem 3 states that
. We rewrite
, where
Lemma 7 states that
. To show that
has the same conclusion. We may see from Lemma 2(ii) that:
Given that
is true, it follows that:
thus
After that, we verify
. We have
because of Lemma 2(iv) and
.
It is still necessary to demonstrate that
is continuous differentiable in
. When
is used, the following results are obtained:
The dominated convergence theorem is then used to obtain:
From Lemmas 4 and 6 and
,
Additionally, Lemma 2(i) gives that
. Hence,
We deduce that is differentiable at and . Similarly, is differentiable at and .
We demonstrate that . In fact, it is obvious that due to Lemma 2(iii), which is continuous in view of Lemma 4. Moreover, according to Lemma 7 we know that is also continuous. Consequently, .
Step 2. Assume
u be a mild solution of (
3). To demonstrate that
, in view of (
6), we prove that
is Hölder continuous in
. Take
such that
.
Denote
, by Lemmas 2(iv) and
7, then:
Thus, .
Take
h for every small
, such that
, so:
Using Lemma 6 and
, we have:
We use the inequality to estimate
.
this yields that
thus
which ensures that
. Therefore,
due to arbitrary
.
Since , where is continuous and bounded in . We can get the Hölder continuity of in using a similar argument. Therefore, we have .
Since is proved, according to Step 2, this yields that and . As a result, we are able to achieve that and . We reach the conclusion that U is a classical solution. □
Theorem 5. Suppose that holds. If U is a classical solution of (3), then and . Proof. If
U is a classical solution of (
3), then
. It is still necessary to demonstrate that
; it suffices to show that
for every
. In fact, take
h that is
, by Lemma 2(iii):
We write
in the same way as Lemma 7,
for
. From Lemma 7 and Equation (
23), it follows that
, respectively.
Since , the result for function is also proven by a similar argument, implying that . As a result, and . The proof is completed. □