Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

Pathak, Santosh

doi:https://doi.org/10.1155/2021/6686526

Abstract and Applied Analysis

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Abstract Introduction Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6686526 | https://doi.org/10.1155/2021/6686526

Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

Santosh Pathak¹

Academic Editor: Janne Heittokangas

Received04 Jan 2021

Accepted06 Feb 2021

Published17 Feb 2021

Abstract

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in for with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in and respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

1. Introduction

We consider the Cauchy problem of the incompressible Navier-Stokes equations in : where and stand for the unknown velocity vector field of the fluid and its pressure, while is the given initial velocity vector field, with and . In what follows, we will use the same notations for the space of vector-valued and scalar functions for convenience in writing.

There is a large literature on the existence and uniqueness of solution of the Navier-Stokes equations in . For the given initial data, solutions of (1) have been constructed in various function spaces. For example, if for some with , then it is well known that there is a unique classical solution in some maximum interval of time: , where . But, for the uniqueness of the pressure, one requires as . See [1] and [2] for and [3] for . The solution is for .

It is well known that for , there is a unique, smooth, and local-in-time solution for the Navier-Stokes equations with where is the th Riesz operator. It is known that in , this solution can be extended globally in time. For , where , the existence of a regular solution follows from [4]. The solution is only unique if one puts some growth restrictions on the pressure as . A simple example of nonuniqueness is demonstrated in [5], where the velocity is bounded, but . In addition, an estimate with (see [6]) implies uniqueness. Also, the assumption (see [7]) implies uniqueness.

For , where , the existence of a local mild solution is proved by Bae and Jin in [8]. In the same paper, it is also proved that such mild solution is indeed a strong solution of the Navier-Stokes equations (1). Before the result of Bae and Jin, the local-in-time existence of mild (strong) solution of the halfspace problem was provided in [9] by Solonikov for continuous bounded initial data in .

In this paper, I am interested in obtaining estimates of the maximum norm of the derivatives of in terms of the maximum norm of the initial function , assuming that the solution exists, and it is for . The work of this paper is a continuation of the work of my papers [10] and [11] to the halfspace case for nondecaying initial data. Nonempty boundary in the domain in this paper makes this work different, in some aspects, and significantly more challenging in proving the key lemmas than the work in my previous works where the initial functions are in or .

We begin by transforming the momentum equations of (1) into the abstract ordinary differential equations: where is the Stokes operator and is the Leray projector, which is given by where . Note that where , if , and denotes the surface area of the unit sphere in which is given by .

The solution of (3) is formally expressed in the integral form:

Solonikov [9] has expressed the solution operator of the Stokes equations in in the integral form

where is given by

The function is the -dimensional Gaussian kernel defined by .

A solution formula of the Stokes equations (3) in has also been provided by Ukai in [12]. Such solution formula has been used in the setting, particularly for (see [13, 14]). For and estimates of the Stokes flow or its gradient, see [15, 16]. The solution formula provided by Solonikov [9] has mainly been used for framework (see [14, 17]).

To formulate the main result of this paper, we first introduce some notations as follows: and . In what follows, if , for any , then we will denote by . We also set

Clearly, measures all space derivatives of order in maximum norm. For later purposes, let us also introduce a few other notations:

Throughout this paper, will be understood as the derivative of order . In addition, denotes the characteristic function which is 1 on and 0 otherwise. is a translation operator defined by .

The goal of this paper is to prove the following theorem.

Theorem 1. Consider the Cauchy problem for the Navier-Stokes equations ((1)) whereandis understood in the sense of distribution. There is a constant, and for anywithwherethere is a constantso thatThe constants and are independent of and .

One of the important tools in the proof of Theorem 1 is the uniform estimates of the composite operator . But, obtaining such uniform estimates is complicated because of the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain; hence, and may not be commutative.

To overcome this difficulty, we will generalize the techniques of obtaining the uniform estimates on of the paper [8] by Bae and Jin to obtain our desired uniform estimates on . In their paper, they require the uniform estimates to prove the existence of the local solution of the Navier-Stokes equations in halfspace for bounded initial data.

This paper is organized in the following ways. In “Some Auxiliary Results,” we introduce some auxiliary results which will be labelled as propositions. In “Estimate of ,” we derive an important estimate on the composite operator . In “Estimates for the Navier-Stokes Equations,” we establish some estimates on the solution of the Navier-Stokes equations. In “Estimates for the Navier-Stokes Equations,” a proof of Theorem 1 will be provided. Finally, Appendices A, B, and C contain proofs of the propositions which are introduced in “Some Auxiliary Results.”

2. Some Auxiliary Results

Let us consider the Stokes problem in : where . Here, we note that each is quadratic in components of .

Solonikov in [9] has obtained the solution of (13) which is given by

Next, we state the following proposition.

Proposition 2. If, then we haveand

Proof. The proof is given in Appendix A.

Proposition 3. Letandbe any Hölder continuous function with the exponent: Then, for or , we have For or , we have

Proof. The proof is given in Appendix B.

Next, we define the Hardy space . Let . Let be the space of functions so that with the norm . Let be the space of functions so that there is with with the norm .

Next, we state a few well-known results related to the Hardy-norm estimates of the Gaussian kernel

Proposition 4. Fix. Thenwithand for .

We omit the proofs of well-known results of Proposition 4.

Proposition 5. Letand. Then we have

Proof. The proof is given in Appendix C.

3. Estimate of

Solonikov in [9] and Shimizu in [16] provide the following estimates: where , , and for each . Also, and vanish on the boundary. In addition, in paper [8] by Bae and Jin, they prove as a critical estimate to prove their desired result.

With all the above estimates in hand, we begin to obtain the uniform estimate on the composite operator . For that purpose, recall where is defined by (8). In the following, consider , and denote by the kernel tensor of the operator . For simplicity in computational purpose, we consider as a Schwartz class function in vanishing on the boundary for each . Thus, we begin by writing where

With integration by parts, we obtain

Use for to write the following:

Use for and Proposition 5 to justify the following expression:

Therefore, (29) can be rewritten as

where

First, we estimate for . For that purpose, recall . Therefore,

Clearly, is the derivative of the th component of the solution of the Stokes equations. So, using estimate (23), we get the desired estimate on as below

Next, we estimate for . In [9], is given as where

Next, we use the estimate where , provided in [18], to obtain the desired estimate on for .

Therefore, we use modified for and rewrite as

The estimate for follows as

Applying Proposition 4, we obtain

To estimate , we use the estimate of (38) and obtain

Finally, we get

We obtain

Therefore, from (35) and (44), we obtain

Next, we estimate : For that, let us begin by rewriting after dropping the summation notations and negative signs for convenience in writing.

Equivalently, we write where

Using expression for from (8) for , we obtain where

To estimate , let us proceed by writing

It is well known that and are in Hardy space , for any fixed . Since the Calderon-Zygmund type transforms are bounded in Hardy space, we obtain that

Using the estimates of Proposition 4, we arrive at

With exactly the same argument as for , we also obtain

It remains to obtain an estimate for . We use Proposition 3 for by replacing by and also use to rewrite as

By the same argument as for , we can obtain

Let us rewrite as

Set

By Proposition 5,

Notice that

We also recall that and are in the Hardy space , for any fixed . Since the Calderon-Zygmund type transforms are bounded in Hardy space, after using , we arrive at

Let us recall a result of Proposition 4: for any , and is bounded by for . Hence, in similar way as for , we obtain

Therefore,

Using (56) and (64) leads us to obtain

Since , with the use of (53), (54), and (65), we obtain

Finally, using (45) and (66) with fact that commutes with , we have proved the following important lemma.

Lemma 6. For anywith, and, there exists a constantindependent ofandsuch thatfor , for some .

Corollary 7. Letbe as in the previous lemma, then the solution ofsatisfies for some .

Proof. The solution of (38) is given by and Applying the estimate (24), we obtain Hence, we obtain

4. Estimates for the Navier-Stokes Equations

Recall the transformed abstract ordinary differential equation (3):

Solution of (74) with given initial and boundary condition as in (1) is given by

Using the solution (75) along with the use of estimates (23), (24), and (25), we prove the following important lemma.

Lemma 8. SetThere is a constant , independent of and , so that

Proof. Using estimate (23) for the solution of the Stokes equations in (75), we obtain From (74), after using estimate (24), with the fact that is quadratic in gives us Since , which is independent of , we arrive at the following estimate Next, apply to in the integral form to obtain and estimate for : Let us estimate the integral in the above expression as below. We use the estimate (25) again with the fact that is quadratic in to obtain Therefore, we have the following estimate for The combination of (80) and (85), proves Lemma 8.

Lemma 9. Letandbe same as in Lemma 8for some. SetThen and

Proof. We prove this lemma by contradiction after recalling the definition of in (76). Suppose that (88) does not hold, then denote by the smallest time with . Use (77) to obtain Thus Therefore, . This contradiction proves (88) and .

5. Proof of Theorem 1

Lemma 9 proves Theorem 1 for for . Now, we apply induction on to prove Theorem 1. Suppose and assume

Apply to with the fact that commutes with . Also let to obtain

The solution of above system can be written as

Since , we can write

Using integral form of from above, we can write

Our goal is to prove . For that, let us start with the following where .

Using the estimate (23) in the first term of the above expression, we obtain where

To estimate uniformly, we proceed as

Using Lemma 6, we obtain

We use simple integration, and the fact that is quadratic in to arrive at

Next, we estimate . For that, we proceed in the following way:

Since the order of the derivatives of is , for convenience in writing, we use to estimate . Since is quadratic in ; therefore

By induction hypothesis (91) we obtain

Apply estimate (25) to the integral (103) with the use of (104) to obtain where

Since , where is independent of , and the using the estimate of (105), we obtain . For , let us begin as below.

Therefore

We use these bounds to bind the integral in (97). We have . Then, maximizing the resulting estimate for over all derivatives of order and setting and from (98), we obtain the following estimate:

Since , then . Therefore

Let us fix so that the above estimate holds and set

First, let us prove the following:

Suppose there is a smallest time such that with . Then, using (88), we obtain

Thus

which contradicts the assertion. Therefore, we proved the estimate

If then we start the corresponding estimate at . Using Lemma 9, we have and obtain

Finally, for any satisfying (118)

and (119) yield

This completes the proof of Theorem 1.

In the following appendices, we provide proofs of the propositions that are introduced in “Some Auxiliary Results.” However, these proofs have also been provided in [8]. For the reader’s convenience, we provide them with more details in this paper as well.

Appendix

A. Proof of Proposition 2

We first let the case . Differentiate with respect to to obain

Observe that for

This proves the desired result of Proposition 2 for . For the case . We start with the expression for some appropriately chosen function : and

Therefore, we arrive at

Since

therefore, after differentiating with respect to variable, we have

B. Proof of Proposition 3

Define a smooth cut-off function such that if and if with . For and , also define

Then, is compactly supported in . Let us define

Differentiating with respect to for yields

If , we have

Let us set

It is clear that

Let us denote where and

Observe that as , since where

Next, we will show that

Let us apply a change of variables and let , we get

If or , then by the symmetry of in terms of variables, we obtain

If or , then by the antisymmetry of in terms of variables, we obtain

This completes the proof of Proposition 3.

C. Proof of Proposition 5

Denote

Then

Notice that

Then, we have

Since, for we have

Next, we want to show . For that, notice and

Here, , and is the th component of the unit outer normal vector. If , then where is the unit sphere in is the outward unit normal vector to , and

If , then since

Similarly

This implies that

Hence, we finally show that .

Data Availability

I have provided all the essential references that I have used in this research article in the reference section.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Santosh Pathak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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