Large Time Behavior for Weak Solutions of the 3D Globally Modified Navier-Stokes Equations

Abstract

This paper is concerned with the large time behavior of the weak solutions for three-dimensional globally modified Navier-Stokes equations. With the aid of energy methods and auxiliary decay estimates together with estimates of heat semigroup, we derive the optimal upper and lower decay estimates of the weak solutions for the globally modified Navier-Stokes equations as The decay rate is optimal since it coincides with that of heat equation.

1. Introduction

It is well known that the motion of the viscous incompressible fluids is governed by the following classic Navier-Stokes equations [1]: Here and denote the unknown velocity and pressure of the fluid motion, respectively. This motion essentially presumes that the derivatives of the components of the velocity are small.

In Leray’s pioneer work [2] in 1930’, for any initial data in , Navier-Stokes equations (1) exits a global weak solution satisfying However, the question of global existence for smooth solutions of the 3D Navier-Stokes equations is still a big open problem. In order to overcome this large difficulty, many efforts have been made to study some related modified Navier-Stokes equations (see [3, 4]). Recently, Caraballo et al. [5] (see also Kloeden et al. [6, 7]) introduced an interesting and important mathematical model which is the so-called global modification of the Navier-Stokes equations associated with where (for some ) is defined by

Let us give a profile analysis to this globally defined model. The modifying factor is a function of . Essentially, it prevents large gradients dominating flux and leading to explosions. What is the most important is that this model exhibits a unique global weak solution for the system (1) in bounded domain (see [5]).

However, it should be mentioned that although the presence of actually canceled some singularities of the nonlinear term , it cannot increase the effect of low frequency of the solutions of the system (3). Therefore, it is interesting to consider the time decay issue of this model which largely depended on the effect of low frequency of the solutions. In this paper, we are focused on the decay of weak solutions for the modified Navier-Stokes equations (3). To carry out this issue, it is necessary to recall some classic time decay results of the fluid dynamical models. decay of weak solutions for the Navier-Stokes equations was first studied by Schonbek [8] (see also [9]). She first posed interesting methods the so-called Fourier splitting methods and the finite energy weak solutions decay as Later on there are large good results to develop the Fourier splitting methods on the incompressible Navier-Stokes equations [10]. One may also refer to some interesting decay issues of the related fluid models [11–13].

Motivated by the upper and lower decay estimates of nonlinear fluid models [14], in this study, we will develop another technique to deal with the time decay problem of the weak solutions for the globally modified Navier-Stokes equations (3). Our trick is mainly based on the energy methods together with the estimates of heat semigroup in whole space . We can get the optimal time decay rate, since it coincides with that of linear equations.

2. Preliminaries and Main Result

In this paper, we denote by a generic positive constant which may vary from line to line.

with is denoted by the Lebesgue space associated with the norm

with is denoted by the fractional Sobolev space with the norm where is the Fourier transformation is the space of all measurable functions with the norm and when ,

To state the main results of this paper, we first give the definition of the weak solutions of the three-dimensional globally modified Navier-Stokes equations (3) [5].

Definition 1. is called a weak solution for three-dimensional globally modified Navier-Stokes equations (3) associated with if the following properties (i);(ii)for any with , (iii)the energy inequality hold true.

Our results read as follows.

Theorem 2. Suppose that is a weak solution for three-dimensional globally modified Navier-Stokes equations (3). Moreover, if the solution of the heat equation satisfies then the weak solution of (1) possesses the following optimal upper and lower decay rate:

Remark 3. The decay rate is optimal since it coincides with that of heat equation. The finding is mainly based on energy methods and auxiliary decay estimates together with estimates of heat semigroup.

3. Auxiliary Decay

In this section, we will first study auxiliary decay of weak solutions for three-dimensional globally modified Navier-Stokes equations (3).

Lemma 4. Suppose that is a weak solution of three-dimensional globally modified Navier-Stokes equations (3); then one has

Proof of Lemma 4. Taking the inner product of (3) with gives where we have used the following properties: due to the divergence free of the velocity fields.
Since then the right hand side of inequality (17) can be estimated after by applying Hölder inequality With the aid of the Gagliardo-Nirenberg inequality,
Plugging (21) into (20), one shows that Then inserting the above inequality into (17), one gets
Thus we rewrite inequality (23) as Now for any small , there exists a large , such that, for , Otherwise, there exists a positive constant , such that, for all from which and together with energy inequality we have which implies that
On the other hand, from energy inequality we have which contradicts (28).
Hence, we have
We now choose in (24) and apply (30) to yield from which and together with the energy inequality we have which implies that

4. Optimal Upper and Lower Decay Estimates

4.1. Upper Decay Estimate

Consider the integral equations of (3) where

Taking the norm of the integral equation and applying the estimates of heat equation, it follows that together with Hölder inequality where we have used the properties

According to Lemma 4, we let and it is obvious that

Thus we rewrite (36) as That is to say, with

It is easy to check that or for large ,

Thus we obtain the optimal upper decay estimates of the weak solution for three-dimensional globally modified Navier-Stokes equations (3) as

4.2. Lower Decay Estimate

From the integral equations (34), we will investigate the error estimates of solutions between three-dimensional globally modified Navier-Stokes equations (3) and the heat equation: For , employing the estimates of heat semigroup and upper decay estimates gives

For , similarly, Thus we have from the estimates and

Hence by the triangle inequality, one shows that

Combination of the upper and lower decay estimates for weak solutions of three-dimensional globally modified Navier-Stokes equations (3) completes the proof of Theorem 2.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partially supported by the NSF of China nos. 11361026, 11161022, and 71161013.