Advances in Mathematical Physics
Volume 2018 (2018), Article ID 5807385, 7 pages
https://doi.org/10.1155/2018/5807385

Research Article

Decay of Strong Solutions for 4D Navier-Stokes Equations Posed on Lipschitz Domains

N. A. Larkin

Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790, Agência UEM, 87020-900, Maringá, PR, Brazil

Correspondence should be addressed to N. A. Larkin; nlarkine@uem.br

Received 14 August 2018; Accepted 27 November 2018; Published 13 December 2018

Academic Editor: Sergey Shmarev

Copyright © 2018 N. A. Larkin. 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.

Abstract.

Initial-boundary value problems for 4D Navier-Stokes equations posed on bounded and unbounded 4D parallelepipeds were considered. The existence and uniqueness of regular global solutions on bounded parallelepipeds and their exponential decay as well as the existence, uniqueness, and exponential decay of strong solutions on an unbounded parallelepiped have been established provided that initial data and domains satisfy some special conditions.

1. Introduction

This work concerns the existence and uniqueness of global strong solutions and sharp decay estimates of solutions to initial-boundary value problems for the 4D Navier-Stokes equations:

where is either a bounded or an unbounded parallelepiped in with the homogeneous Dirichlet condition on the boundary of .

The question of decay of the energy for weak solutions had been stated by J. Leray in [1] and attracts till now attention of many pure and applied mathematicians [2–9]. In all of these papers, the decay rate of was controlled by the first eigenvalue of the operator , where is the projection operator on the solenoidal subspace of . Obviously, this approach does not work in unbounded domains; see [6, 7, 9].

It is well known that solutions of the 2D Navier-Stokes equations posed on smooth bounded domains with the Dirichlet boundary conditions are globally regular [4, 6–9]. On the other hand, the question of regularity for 3D and 4D NSE with arbitrary initial data is till now an open problem even for smooth domains; see [6, 7, 9]. Small initial data help to solve this problem [6, 7, 9] as well as the so-called “thin” domains when some size of a domain is small [10, 11]. The question of regularity becomes more difficult while a domain is Lipschitzian [10, 12, 13].

In [6, 7, 9, 14], it has been proved that for 3D Lipschitz domains, bounded and unbounded and small initial data there exists a unique global strong solution

but it was not clear whether

at least for bounded Lipschitz domains.

Our goal here is making some geometrical restrictions, to prove the existence and uniqueness of strong global solutions in 4D Lipschitz domains for arbitrary regular initial data as well as exponential decay of solutions.

In this work, making use of ideas of [15], we have established that for a 4D bounded parallelepiped. The following inequality holds:

where

Our paper has the following structure: Section 1 is Introduction. Section 2 contains notations and auxiliary facts. In Section 3, existence, uniqueness, and decay of global strong solutions on a bounded 4D parallelepiped have been established. In Section 4, the existence, uniqueness, and decay of regular solutions on bounded 4D parallelepipeds and strong solutions on 4D unbounded parallelepipeds have been demonstrated.

2. Notations and Auxiliary Facts

Let and be a domain in Define as in [9], p.2-4

We denote for scalar functions the Banach space with the norm

For is a Hilbert space with the scalar product

The Sobolev space is a Banach space with the norm

When is a Hilbert space with the following scalar product and the norm:

Let or be the space of functions with compact support in or . The closure of functions in is denoted by and when

Define the auxiliary spaces which are projections for the solenoidal vector functions,

The space is equipped with the natural inner product. The space will be equipped with the scalar product

when is bounded. If is unbounded, we define the inner product as the sum of the inner products as follows:

We use the usual notations of Sobolev spaces , , and for vector functions and the following notations for the norms:

(i) For vector functions ,

The closures of in and in are the basic spaces in our study. We denote them by and , respectively.

Remark 1. By definition, is a proper subspace of

Define the operator

Lemma 2 (the Steklov inequality [16]). Let Then

Proof. Let , then by the Fourier series,

Inequality (17) follows by a simple scaling.

The next lemmas will be used in estimates.

Lemma 3 (see: [17] Theorem 7.1, p.14). Let , then

Lemma 4. Let , then

. If , then we can define the operator such that belongs to and

3. Existence Theorems

Let be a bounded 4D parallelepiped: which is a Lipschitz domain. Denote . Given , consider the following problem:

equivalent to the variational problem given by (see [7], [9] Problem 3.2, p. 191.)

where such that for all and such that

Theorem 5. Given and such that

there exists a unique strong solution to (22):

such that for all it satisfies the following identity:

Moreover, the following inequalities hold:

where

Proof. The estimates that follow may be established on Galerkin’s approximations (see [14], [6] p. 136-140, [7], [9], p. 192-197.).
Estimate I (). Multiply (23) by to obtain

It follows from here that

Making use of the Steklov inequalities, we get

Returning to (30), we obtain

which implies

This and (30) give

Estimate II (). Differentiating (23) and multiplying by , we get

We estimate

By Lemma 3,

and (35) becomes

Making use of (30), we find

Substituting this into (38), we get

By Lemma 2,

Substituting this into (40) and using (39), we find

Again by (39),

Rewrite this in the form

Taking into account conditions (25) of Theorem 5, we get that

hence (44) reduces to the form

By the Steklov inequality,

We estimate

This and (46), (47) imply that , and

Returning to (39), we get . This and (33), (49) prove validity of (27), (28), and consequently the existence part of Theorem 5.
Uniqueness of the Strong Solution. Let and be two strong solutions to (22) satisfying (23) and (27). Define . Then for we have

Taking , we come to the inequality

that can be rewritten as

Acting in the same manner as by the proof of Estimate II, we come to the inequality

By conditions of Theorem 5,

for Taking into account Estimates (28) and using standard arguments, we get for all

Hence, (53) becomes

This implies that proves uniqueness of the strong solution and completes the proof of Theorem 5.

4. More Regularity

Consider the Poisson problem in a bounded domain :

In [15] Theorem 11, p. 120-123, the following has been proved.

Lemma 6. Let

then the unique weak solution of (57) satisfies the following inequality:

It is possible to generalize this result for a bounded parallelepiped in

Theorem 7. The problem (57) posed in a parallelepiped , where , , has a solution . Moreover,

Returning to the original problem for the Navier-Stokes equations,

where is a vector function from into and is a real function from into , and making use of Galerkin approximations, we establish the following result.

Theorem 8. Given and a domain satisfying (25), then problem (61) has a unique regular solution such that

which for all satisfies the following integral identity:

Moreover,

where and depends on .

Proof (decay of -norm). Taking into account that conditions of Theorem 8 and of Theorem 5 are the same, by Theorem 5, we have a unique strong solution of (61). Hence, to prove Theorem 8, it is sufficient to establish that First write (22) as

We estimate

Hence by (33), (49), (51),

Returning to (23) and making use of (67), (49), we obtain

By the Theorem of de Rham (see [7, 18]), [9] Propositions 1.1, 1.2, p. 10, one can check that there exists such that (see [9])

and

Returning to (65) and having , we obtain, due to Theorem 7,

By the Sobolev theorems,

The proof of Theorem 8 is complete.

Remark 9. It follows from (72) that and consequently, and in (60). This means that we cannot achieve better regularity then In some sense, this is the superior regularity for the problem (61). It looks like is the critical case of the Navier-Stokes system.

Existence and Decay of a Strong Solution on an Unbounded 4D Domain. Define an unbounded four-dimensional parallelepiped and let

Given , consider in the following problem:

Theorem 10. Given and such that

there exists a unique strong solution to (73):

such that for all ,  , tending sufficiently rapidly to as , it satisfies the following identity:

Moreover, the following inequalities hold:

where

Proof. Obviously, the variational formulation of (73) is also (23). Repeating the proof of Theorem 5, we can prove the existence and uniqueness of the strong solution to problem (73). Using the Steklov inequalities with respect to variables , we obtain

hence, (23) becomes

This implies

Repeating the proof of Estimate II of Theorem 5, we find

and

Decay for Pressure. In order to obtain decay for , we start with

where is the dual to the space . Since

then by (81) and (83),

Moreover, by (69),

Jointly (80), (81), and (86) prove Theorem 10.

Conclusions. In our work, we tried to respond to some questions posed by J. Leray [1], namely, regularity of global solutions of the Navier-Stokes equations and their decay. Therefore, our results can be divided into two parts: the first one concerns decay of global regular solutions of the 4D Navier-Stokes equations posed on bounded 4D parallelepipeds. It is known that there exist global regular solutions for the 2D Navier-Stokes equations posed on smooth bounded domains [4, 6, 8, 9], but regularity in nonsmooth (Lipschitz) domains is not obvious. For bounded 4D parallelepipeds, we have established the existence of a unique global regular solution which decays exponentially as provided that initial data satisfies (25). We demonstrated that the decay rate is different for different norms; see (77), where is defined by the geometrical characteristics of a domain .

The second part of our work concerns decay of solutions for the 4D Navier-Stokes equations posed on an unbounded parallelepiped. In existing publications [3, 4, 6, 9], the decay rate of is controlled by the first eigenvalue of the operator , where is the projection operator on a solenoidal subspace of It is clear that this approach does not work in unbounded domains.

On the other hand, our approach based on the Steklov inequalities allowed us to estimate the decay rate of a strong solution for the 4D Navier-Stokes equations posed on an unbounded 4D parallelepiped.

We must emphasize that this estimate is the first one which gives an explicit value of the decay rate for unbounded 4D domains. Results established in our work can be used in constructing of numerical schemes for solving initial-boundary value problems for the Navier-Stokes equations appearing in Mechanics of viscous liquid. From the physical point of view, decay estimates show that the decay rate of perturbations of solutions caused by the initial data is bigger for bigger values of viscosity and smaller sizes of 4D parallelepipeds.

My interest for the 4D Navier-Stokes equations is purely mathematical and, on my opinion, can not be extended to higher dimensions beyond 4. I must also note that there are publications on the existence of weak solutions for 4D Navier-Stokes equations [7], [9] p.189-197.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research has been supported by Fundação Araucaria, Parana, Brazil, Convenio No. 307/2015, Protocolo No. 45.703.

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