Abstract

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling by -power law with Dirichlet boundary condition under the restriction . In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction , which arises from the nonflatness of domain.

1. Introduction

In this note, we consider the steady flows of non-Newtonian fluids in , , which is modeled by the following system: where is the velocity, the pressure, the external force, , a bounded domain, and a prescribed function.

This system arises from flows of electrorheological [1], thermorheological [2], chemically reacting non-Newtonian fluids [3].

For the existence of weak solutions to the problem (1), we refer to [1, 4].

Local higher differentiability for weak solutions to the problem (1) with has first shown in [5] for in 3D. This result was improved to in [6] by Wolf. Forin 3D,-regularity is proved (see [7]).

Global higher differentiability for weak solutions to the problem (1) with have been studied by several authors; for example, see [819] under the condition with and . It was first established in [8] by Beirao da Veiga. He developed a crucial device which was to denote the second-order derivatives of the velocity in the normal direction through ones (and the first-order derivatives of the pressure) in the tangential directions by using the very explicit form of the main equations. But in contrast to interior regularity, the interaction between pressure and nonlinearity of leading term results in the lower regularity for the second-order derivatives of the velocity (and for the first-order derivatives of the pressure) in the normal direction, in comparison to the tangential directions. His idea reveals to be quite fruitful in many subsequent papers. He [10, 11] studied global higher differentiability of weak solution to the problem with the boundary condition (3) for in 3D cubic domain.

With the help of the anisotropic Sobolev embedding theorem, Berselli [15] obtained an improved integrability of velocity gradient than in [11] in 3D cubic domain. His idea is that it is possible to apply the anisotropic Sobolev embedding theorem because of the difference in the regularity levels between the second-order derivatives of the velocity in the normal direction and the tangential ones. Beirao da Veiga [9] showed global -regularity for by combining the idea from [15] with a delicate estimate on the convective term in 3D cubic domain and then in [12] extended it to nonflat boundaries. In [19], Crispo proved the same type of results in cylindrical domains. In [14], the authors showed global -regularity for shear-thickening flows, i.e., in -D bounded smooth domain.

Recently, global higher differentiability of weak solution to the problem (1) in 3D smooth domain is studied by us in [20] by using a global higher integrability condition, which holds under the condition , where , , and . This is slightly stronger rather than the standard condition for the case . On the other hand, local higher differentiability of local weak solution to the problem (1) in 3D has been obtained in [21, 22] by relying on the local higher integrability result from [23].

In [24], the existence and uniqueness of -solution corresponding to small data are proved, without further restrictions on the bounds on .

For interior or boundary partial regularity, we refer to [23, 2527].

If one assumes the condition for the problem (1), then when applying difference quotient, due to the -dependence of leading term, the additional term will appear: which cannot be estimated in terms of a priori estimate on weak solutions. So for the system (1) with , the existence of strong solutions has been studied. In 3D, the existence of local strong solutions to the system (1) is first shown for in [1] (chapter 3) by Ruzicka. In [28], Ettwein and Růžička showed the existence of -solutions without the artificial upper bound . For 2D bounded domains, we refer to [29, 30].

In [31, 32], we gain the existence of strong solution for the system (1) under the standard assumption . But in that case, we consider the following the boundary condition: for , , and , where by -periodic, we mean periodic of period 1 both in and . This allows us to consider a bounded domain and simultaneously a flat boundary. Thus, it is natural to ask whether the sharp results proved in [31, 32] are valid for smooth domain. This is the aim of this note.

It seems to be possible to obtain the existence of -strong solution to the problem (1) in 2D by the result of this note and the same argument as in [32]. Very recently, we [32] show the result in the case of the boundary condition (3). For -regularity in 2D, we refer to [18, 29, 30, 3336].

Set

For , we define where is arbitrarily close to 0 for , and for , where is arbitrary real number such that and arbitrary close to 0.

The main results are as follows.

Theorem 1. Let . Assume that , , , and and for

Then, for from (5) and (6), there is a strong solution to the problem (1) satisfying where the constants depend on .

Moreover, for the problem (1) without the convective term, there is a unique strong solution satisfying (9) and (10) provided that

Remark 2. We note that if , then the condition (11) will be no longer needed (see [1214, 16, 20]).

But due to -dependence of the leading term, we cannot obtain -regularity of weak solution to (1) provided . So it is customary to consider an approximate problem. Fortunately, in [31, 32], the condition (11) does not appear but we consider the boundary condition (3). The condition (11) arises from the nonflatness of and Dirichlet boundary condition. More precisely, due to them, there appears the term in deriving of -regularity of the approximate solutions independent of parameter (see (41) and (45)). The term disappears in the case (3). For the estimate of the term, we need an additional condition (11). It is open whether the condition can be removed when one considers the approximate problem with Dirichlet boundary condition over nonflat domain. In fact, Kaplicky ([37], Lemma 4.2) showed (9) and (10) for the approximate problem to (1) with in 2D.

Due to -dependence of the extra stress tensor, we consider an approximate problem with for instead of . It is easy to see that the approximate solution belongs to owing to the trivial inequality . The main point is to derive the estimates about all derivatives of the approximation solutions in suitable Sobolev spaces, which are independent of parameter. This allows us to show convergence of approximation solutions to the one to problem (1).

The paper is organized as follows. In Section 2, we give preliminaries. Section 3 is devoted to prove the main results.

2. Preliminaries

2.1. Notations

By , we denote the conjugate function of . For , , define , . Let be a Sobolev conjugate exponent, i.e., when and for all when and otherwise.

For -matrices denote , . For two vectors and , and .

For , , the variable exponent Lebesgue space is defined by endowed with the norm . Then, we define the variable exponent Sobolev space by with the norm . We define as the closure of in . Let be dual of .

We do not distinguish between scalar, vector-valued, and tensor-valued function spaces in the notations. Define

Definition 3. We say that function is a weak solution to the problem (1) if and it satisfies We refer to the term strong solution as a weak solution which additionally satisfies for some .

2.2. Some Problems Related to Flattening of the Boundary

As before, our problem is reduced to a problem involving a flat boundary by a suitable change of variables. Here, we follow the arguments and notations in [14]. Since , for each point , there are local coordinates such that in these coordinates, we have and is locally described by a -function , where is the -dimensional cubic with center 0 and length (which is small enough and will be fixed later), with the following properties:

As is a compact, there exist a finite set of points and an open set such that . We construct a partition of unity , corresponding to this covering, such that for all and some suitable small . Let us fix some .

Set . For , and a function with , we define tangential translation through and tangential derivative through

Now, we give the two propositions below related to the tangential derivatives.

Proposition 4 (see [12, 14]). Let and . Then,

Proposition 5 (see [20]). Let , . Then, there holds

3. The Proof of Main Results

We use universal constants , which may vary in different occurrences. In particular, depends on , while on .

To begin with, let us define

As before, in order to prove the main result, i.e., Theorem 1, we will consider the following approximate problem: where . Let us denote

It is known that there exists a weak solution to the problem (23) by compactness method if . Furthermore, the following facts are valid.

Proposition 6. Let be a weak solution to the problem (23). Assume that and . Then, the following hold: where is from (8). For the problem (23) without convective term, these are valid for all .

Remark 7. In fact, the proposition above was proved in ([31], Lemma 4.1) and [32] for the boundary condition (3). But it is easy to see that these inequalities hold also for Dirichlet boundary condition.

Moreover, noting that we can prove by the same line in [20] that if , , then a weak solution of the problem (23) satisfies

However, the norms of in and in are dependent of .

Thus, from now on, we focus on the estimates about the derivatives of the approximation solutions in suitable Sobolev spaces, which are independent of parameter. This allows us to show convergence of the approximation solutions to the one to problem (1) in the spaces.

Hereafter, all constants are independent of parameter and introduce a shorthand notation if it will be clear from the context.

Let the assumptions of Theorem 1 hold.

To prove Theorem 1, we will combine the methods in [20, 31, 32].

The proof is divided into two cases: with and without the convective term.

3.1. The Proof of Theorem 1 without the Convective Term

Step 1. Estimates of the approximate solutions independent of parameter in tangential directions.
In this step, our aim is to prove that where is from (8), , and

Fix , and let , , and be as in Subsection 2.2. For simplicity we will omit the symbol “” in integration ones and denote the symbols “,, by “.

By ([20], Proposition 14.3.15) and Propositions 4 and 5, there exists a solution to the problem satisfying where the constant depends on .

Multiplying the first equations in (23) by , integrating by parts and using Proposition 4, we get

We apply Proposition 4 to the left hand side of (34) to get

It is clear that

From ([31], (3.6)), we have where . It is shown in ([31], (3.7)) that

Combining (37) with (38) yields that

The terms can be rewritten as

Hence these terms can be estimated as follows: by Korn’s inequality

Combining (39), (41) with (35), we arrive at

Since , the term can be estimated as by

Note that by Proposition 4

Hence the term also can be estimated as :

We use Hölder’s inequality to get

Using (33) and (25) yields that

Equation (34) together with (42)-(47) yields the desired estimate (29).

Next, let us prove (30).

In [32] (4.20), it is proved that

Indeed, though the formula (4.20) from [32] is proved in 2D for , it also holds for , . Moreover if neglecting the convective term, this holds for .

Let . Multiplying the first equations of (23) by and integrating by parts, we have

Since , we have

It follows that

The left hand of (49) can be rewritten as

Equation (49) together with (50)-(54) implies that

Thus, by the negative norm theorem ([38], Theorem 14.3.18) and (55), we have which is the just (30).

Step 2. Estimates of the approximate solutions independent of parameter in all directions.

Our aim in this step is to show (79) and (80).

Here, we follow the notations from [20]. By mimicking the derivation of [20], (4.51), we can get

Now, we want to derive some estimates on the first derivatives of from (22), independent of parameter . This allows us to prove boundedness of and and to improve regularity of solutions to problem (1).

Since (see, [1]), it follows from (57) that

Similarly, we have

Now, define

It follows from (58) that

Using Hölder’s inequality and the fact that , we have by Korn’s inequality

Taking into account the fact that , we have

From (30) and the fact that and in , it follows that

Gathering (61)-(64) and using (29), we obtain

On the other hand, it follows from (59) and (29) that

Define where is an arbitrary small positive real number. Applying the anisotropic embedding theorem (see [39], Theorem 2.5), we obtain that by (65) and (66),

It is shown in [31, 32] that

Now, let us estimate the term . Note that