1. Introduction
We consider the mixed problem for Navier–Stokes (N–S)-type equation with small parameter
where
are complex numbers,
,
are small positive parameters,
m is a positive integer with
and
A is a linear operator in a Banach space
E. Here,
and
are represent the
E-valued unknown velocity and pressure like functions, respectively,
and
a represent a given
E-valued external force and the initial velocity.
Remark 1. When we consider the N–S problem 1a–1c, it means that the solution u belongs to space . Using Lions and Peetre result (see foe example [1], § 1.8.2) then the trace operator is bounded from to. We assume A is to be such that for , where denotes real interpolation spaces between , (see e.g., [1], § 1.3.2),here, , , and will be defined in the sequel. Boundary value problems (BVPs) for differential-operator equations (DOEs) in classes of functions such as Lebesgue ones have been object of interest of a lots of scientists (see, for example [
1,
2,
3]). Presentations to differential-operator equations have been done by several authors [
4,
5,
6]. Regularity results for differential-operator equations are contained in [
7,
8,
9]. In the present note, authors study degenerate parameter-dependent Boundary Value Problems for arbitrary order differential-operator equations. These kinds of problems have been applied in several fields which are useful in lots of fluid mechanics models.
The focus of our work was to prove uniform existence and uniqueness of the stronger local and global solution of the Navier–Stokes problem with a small parameter (1a)–(1c). This problem is characterized by the presence of an abstract operator
A and a small parameter
that, respectively, corresponds to the inverse of a Reynolds number
that is very large for the N–S equations. Regularity results of N–S equations were obtained, for example, by the authors in [
4,
5,
6,
10,
11,
12,
13,
14,
15,
16,
17]. The N–S equations with small viscosity when the boundary is either characteristic or non-characteristic have been well-studied; see for example in the papers [
3,
14,
16]. In addition, regularity properties of abstract differential equation (ADE) were deeply studied in [
2,
7,
8,
9,
18,
19,
20,
21,
22]. Here, the authors study abstract N–S equations with a high elliptic part in a Banach space
E with operator coefficient
A. In [
22], we derived the
regularity properties of the abstract Stokes problem. For
,
,
the problem (1a)–(1c) state to be usual N–S problem. In this paper, the authors prove that the corresponding Stokes type problem
has a unique solution
for
and the following uniform estimate holds
with
independent of
f and
where
denote the real interpolation space between
and
defined by the
K-method (see e.g., [
1], §1.3.2). Then, by following Kato-Fujita [
13] method, by using (1e) we derive a local a priori estimates for solutions of (1a)–(1c), i.e., we prove that for
and
such that
there exists
independent of
such that
is continuous on
and satisfies
as
there exists a local solution of (1a)–(1c) such that
,
for some
and
as
uniformly in
for
with
Moreover, the solution of (1a)–(1c) is unique for some
with
. For sufficiently small date we show that there exists a global solution of (1a)–(1c). Particularly, we prove that there is a
such that if
, then there exists a global solution
of (1a)–(1c) so that
Moreover, the following uniform estimates hold
where
denotes the corresponding Stokes operator and
is a projection operator in
In application, we put
and
A to be differential operator in (1a) and (1b), with generalized Wentzell–Robin boundary condition defined by
where
b are complex-valued functions. Then, we obtain the existence, uniqueness and uniformly
estimates for solutions the following Wentzell–Robin type mixed problem for the N–S equation
Note that the regularity properties of Wentzell–Robin-type boundary value problems (BVP) for elliptic equations were studied e.g., in [
23,
24] and the references therein. Here,
denotes the space of all
-summable complex-valued functions with mixed norm i.e., the space of all measurable functions
f defined on
, for which
By using the general abstract result above, the existence, uniqueness and uniformly estimates for solution of the problem (1f)–(1h) is obtained.
Moreover, we choose
and
A to be degenerated differential operator in
defined by
where
for
is a contınous,
is a bounded function on
for a.e.
,
are complex numbers and
is a weighted Sobolev space defined by
Then, we obtain the existence, uniqueness and uniformly
estimates for solutions of the following mixed problem for degenerate N–S equation
Let
E be a Banach space and
denotes the space of strongly measurable
E-valued functions that are defined on the measurable subset
with the norm
The Banach space
E is called an
-space if the Hilbert operator
is bounded in
(see. e.g., [
19], § 4).
spaces include e.g.,
,
spaces and Lorentz spaces
p,
.
Let and be two Banach spaces. Let denote the space of all bounded linear operators from to . For it will be denoted by
Here,
denotes the set of natural numbers.
denotes the set of real numbers. Let
be the set of complex numbers and
A linear operator
A is said to be positive in a Banach space
E with bound
if
is dense on
E and
for any
where
I is the identity operator in
E (see e.g., [
1], §1.15.1). The positive operator
A is said to be
R-positive in a Banach space
E if the set
is
R-bounded (see [
19], § 4). The operator
is said to be positive in
E uniformly with respect to parameter
s with bound
if
is independent on
s,
is dense in
E and
for all
, where the constant
M does not depend on
s and
Assume
and
E are two Banach spaces and
is continuously and densely included into
E. Here,
is a measurable set in
and
m is a positive integer. Let
denote the space of all functions
that have the generalized derivatives
with the norm
Let
,
denotes the
valued fractional Sobolev space of order
s that is defined as:
with the norm
It clear that
It is known that if
E is a UMD space, then
for positive integer
m (see e.g., [
25], § 15).
denote the Fractional Sobolev-Lions type space i.e.,
2. Regularity Properties of Solutions for System of ADEs with Parameters
In this section, we will derive the maximal regularity properties of the BVP for system of ADE with small parameters in half-space
are complex numbers,
,
are small positive parameters,
m is a positive integer with
and
A is a linear operator in a Banach space
E. Here,
are represent the
E-valued unknown velocity and pressure like functions, respectively and
Let
denotes the class of
valued system of function
with norm
and let
with norm
By reasoning as in ([
9], Theorem 2) we have
Theorem 1. Let E be a UMD space and A be an R-positive operator in E. Assume m is a nonnegative number, . Then for all , with sufficiently large problem (2a) and (2b) has a unique solution u that belongs to and the following coercive uniform estimate holds Consider the operator generated by problem (2a) and (2b), i.e., From Theorem 1 we obtain the following results:
Result 1. Suppose the all conditions of Theorem 1 are satisfied. Then, there exists a resolvent
for
satisfying the following uniform estimate
It is clear that the solution (2a) and (2b) depend on parameters , i.e., In view of the Theorem 1, we derive the properties of solutions (2a) and (2b).
From Theorem 1 we obtain:
Result 2. For
there exists a resolvent
of the operator
satisfying the following uniform estimate
3. The Stokes System with Small Parameters
In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem
where
are complex numbers,
,
are small positive parameters,
m is a positive integer with
and
A is a linear operator in a Banach space
E. Here,
are represent the
E-valued unknown velocity and pressure like functions, respectively and
Here and hereafter
will denoted the conjugate of
E and
(resp.
)) denotes the duality pairing of functions on
(resp.
) and
will denote the dual spase of
where
. Let
denote the
E-valued solenoidal space. Let
A be a positive operator in
Let
. The spaces
,
will be denoted by
and
respectively. Let
becomes a Banach space with this norm. Consider the problem
By using Theorem 1, we obtain the following
Corollary 1. Let E be a UMD space and A be an R-positive operator in E. Assume m is a nonnegative number, . Then for all problem (3b) has a unique solution and the following estimate holds It is known that (see e.g., [
11,
12]) vector field
has a Helmholtz decomposition. In the following theorem we generalize this result for
valued function space
By reasoning as in [
11,
12] and ([
22], Theorem 3.1) we have decomposition result via operator
generated by problem (3b).
Theorem 2. Let E be an space and . Assume there exists a constant such that Then has a Helmholtz decomposition i.e., there exists a bounded linear projection operator from onto with null space In particular, all has a unique decomposition with so that For proving the Theorem 2 we need the following lemma:
By reasoning as in ([
12], Lemma 2) we get:
Lemma 1. is dense in
Here, <, > and denotes the duality pairing of abstract functions defined in and , respectively. From [22] we have Proposition 1. There exists a unique bounded linear operator , from , ontosuch thatand the following estimate holdswhere Proof. For
consider the linear form
□
By virtue of trace theorem in
the interpolation of intersection and dual spaces (see e.g., ([
22], §1.8.2, 1.12.1, 1.11.2)) and by localization argument we obtain that the operator
is a bounded linear and surjective from
onto
. Hence, we can find for each
an element
so that
Therefore, from (3e) we get
This implies the existence of an element
such that
and
where
Thus, we have proved the existence of the operators . The uniqueness follows from Lemma 1.
Now we are going to construct the projection operator
. Let
and
. Consider the boundary value problem
Since
, in view of Corollary 1, then for all
problem (3f) has a unique solution
and the following estimate holds
By Theorem 1, we obtain that for all
problem (3h) has a unique solution
and the following estimate holds
where
. For any
, we take the solution of (3f), then that of (3h) and put
. We define
Then by reasoning as in [
12,
16] we have
Lemma 2. Let E be an space and . Then, is a closed subspace of
Lemma 3. Let E be an space and . Then, the operator is a linear bounded operator in and if
Lemma 4. Assume E is an space, A is an R-positive operator in E and . Then the conjugate of is defined as and this operator is bounded linear in
LetFrom Lemmas 3 and 4 we obtain Lemma 5. Assume E is an space and . Then Lemma 6. Assume E is an space and . Then Now we are ready to prove the Theorem 2.
Proof of Theorem 2. From Lemmas 5 and 6 we get that
Then, by construction of
we have
By Lemmas 2 and 3, we obtain the estimate (3a). Moreover, by Lemma 5, is a close subspace of Then, it is known that the dual space of quotient space is By first assertion we have □
Theorem 3. Let E be an space, A is an R-positive operator in E, Then, problem (3a) and (3b) has a unique solution for , and the following coercive uniform estimate holdswith independent of , ..., , λ and Proof. By applying the operator
to problem (2a) and (2b) we get the Stokes problem (3a) and (3b). It is clear to see that
where
is the abstract Stokes operator generated by problem (3a) and (3b) and
is an abstrat elliptic operator in
defined by (2e). □
Then Theorem 2 we obtain the assertion.
Result 3. From the Theorem 3 we get that
is a positive operator in
and also generates a bounded holomorphic semigroup
for
In a similar way as in [
11] we show
Proposition 2. The following estimate holdsuniformly in for and Proof. From Theorem 3 we obtain that the operator
is uniformly positive in
, i.e., for
the following estimate holds
where the constant
M is independent of
and
Then, by using Danford integral and operator calculus (see e.g., in [
10]) we obtain the assertion.
Now we can prove the main result of this section □
Theorem 4. Let . Then, for and , there is a unique solution of the problem (1d) and the following uniform estimate holdswith independent of f and Proof. The problem (1d) can be expressed as the following abstract parabolic problem
By Proposition 2, operator
is uniform positive and generates holomorphic semigroup in
Moreover, by using ([
9], Theorem 3) we get that the operator
is
R-positive in
uniformly with respect to
Since
E is a UMD space, in a similar way as in ([
20], Theorem 4.2) we obtain that for all
and
there is a unique solution
of the problem (4b) so that the following uniform estimate holds
From the estimates (3k) and (3l) we obtain the assertion. □
Result 4. It should be noted that if we obtain maximal regularity properties of abstract Stokes problem without any parameters in principal part.
Remark 2. There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1d) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of by virtue of Theorem 3 and 4 we can obtain the maximal regularity properties of different class of stationary and instationary Stokes problems, respectively, which occur in numerous physics and engineering problems.
4. Existence and Uniqueness for N–S Equation with Parameters
In this section, we study the N–S problem (1a)–(1c) in
. The problem (1a)–(1c) can be expressed as
We consider the Equation (
4a) in integral form
For proving the main result we need the following lemma which is obtained from ([
11], Theorem 2).
Lemma 7. Let E be a UMD space, A an R-positive operator in E, and . For any the domain is the complex interpolation space [1], §1). Lemma 8. Let E be a UMD space, A an R-positive operator in E, and . For each the operator extends uniquely to a uniformly bounded linear operator from to
Proof. Since
is a positive operator, it has a fractional powers
From the Lemma 7, It follows that the domain
is continuously embedded in
for any
. Then by using the duality argument and due to uniform positivity of
we obtain the following uniform estimate
□
By reasoning as in [
10] we obtain the following
Lemma 9. Let E be a a UMD space, A an R-positive operator in E, and . Let . Then the following uniform estimate holds provided that , and Proof. Assume that
.Since
is continuously embedded in
and
is the same as
, by Sobolev embedding theorem we obtain that the operators
is bounded, where
By duality argument then, we get that the operator
is bounded from
to
where
Consider first the case
. Since
is bilinear in
, it suffices to prove the estimate on a dense subspace. Therefore assume that
u and
are smooth. Since div
0, we get
Taking
and using the uniform boundedness of
from
to
and Lemma 8 for all
we obtain the uniform estimate
By assumption we can take
r and
such that
Since
is continuously embedded in
then by Sobolev embedding we get
i.e., we have the required result for
. In particular, we get
Similarly we obtain
for
and
. The above two estimates show that the map
is a uniform bounded operator from
to
and from
to
By using the Lemma 7 and the interpolation theory of Banach spaces for
we obtain the uniform estimate
□
By using Lemma 9 and iteration argument, by reasoning as in Fujita and Kato [
13] we obtain the following.
Theorem 5. Let E be a UMD space, A an R-positive operator in E, and . Let be a real number and such that Suppose that , and that is continuous on and satisfies Then there is independent of ε and local solution of (4a) u such that ,
for some uniformly in ε as for all α with . Moreover, the solution of (4a) is unique if and uniformly in as for .
Proof. We introduce the following iteration scheme
By estimating the term
in (4c) and by using the Lemma 9 for
we get the uniform estimate
with
where
and
is the beta function. Here we suppose
. By induction assume that
satisfies the following
We shall estimate
by using (4b). To estimate the term
we suppose
so that the numbers
satisfy the assumptions of Lemma 9. Using Lemma 9 and (4d), we get
Therefore, we obtain
with
We get the uniform estimate. So, the remaining part of proof is obtained the same as in ([
10], Theorem 2.3). □
By reasoning as in [
13] we obtain
Lemma 10. Let the operator be uniform positive in a Banach space E and α be a positive number with . Then, the following uniform inequality holds for all
Proposition 3. Let E be a space satisfying a multiplier condition, A an R-positive operator in E, and . Let u be the solution given by Theorem 5 Then for is uniform Hölder continuous on every interval , for all parameters
Proof. It suffices to prove the Hölder continuity of
, where
Using the Lemma 10 we get the uniform estimate
Then by reasoning as in ([
10], Proposition 2.4) we obtain the assertion. □
Theorem 6. Let E be a UMD space, A an R-positive operator in E, and . Assume is Hölder continuous on each subinterval . Then, the solution of (4b) given by Theorem 5 satisfies Equation (4a) for all Moreover, for . Proof. It suffices to show Hölder continuity of
on each interval
It is clear to see that
and
Since
is continuous on
we get
The uniqueness of
ensured by Theorem 5, implies the following uniform estimates
where
So, by Proposition 4,
is continuous on every subinterval
Since we can choose
,
so that
□
Lemma 8 implies that is Hölder continuous on every interval
5. Regularity Properties
The purposes of this section is to show that the solutions of (1a) are smooth if the data are smooth. For simplicity, we assume . The proof when is the same. Consider first all of the Stokes problem (3d) and (3e).
By reasonıng as in ([
13], Lemma 2.14) we obtain
Lemma 11. Let E be a UMD space, A an R-positive operator in E, and .Let for some Then for every we have In a similar way as Lemmas 2, 5 and 6 in [10] we obtain, respectively: Lemma 12. Let E be a UMD space, A an R-positive operator in E, and . For the following hold:
(2) for there exists a constant such that Lemma 13. Let E be a UMD space, A an R-positive operator in E, and . Let be solution of (4b) for then and for Moreover, Lemma 14. Let E be a a UMD space, A an R-positive operator in E, and .Let be solution of (4b) for then for
Now, by reasoning as in ([
10], Proposition 3.5) we can state the following
Proposition 4. Let E be a UMD space, A an R-positive operator in E, and .Let E be Banach algebra, and Suppose that the solution of (4b) for given by Theorem 5 exists on Then
Proof. The solution
of (4b) for
given by Theorem 5 is expressed as
where
From (5a) we get
Since
and
we will examining only
. Integrating by parts, we obtain
Moreover, since
for all
,
, we have
where
Hence, by Lemma 7 we get the following uniform estimate
This estimates together with Lemma 13 shows that
Lemmas 11 and 12 now imply that
Since
Corollary 5.1, Lemmas 5.3, 5.4 and the identity
imply
Then the proof will be completed as in ([
10], Proposition 3.5) by using the induction. □
Now we can state the main result of this section
Theorem 7. Let E be a UMD space, A an R-positive operator in E, and .Let E be Banach algebra and Suppose that the solution of (4b) for given by Theorem 5 exists on Then
Proof. For
the assertion is obtained from the Proposition 4. Let us show that the assertion is valid for
Indeed, the solution
of (5b) for
given by Theorem 5 satisfies the Equation (
5a) on every subinterval
. Theorem 6 shows that
. Since
, we have
so that
for some
By
this means that we may assume
and
. □