Abstract and Applied Analysis

Volume 2012 (2012), Article ID 184674, 9 pages

doi:10.1155/2012/184674

Research Article

A Note on the Regularity Criterion of Weak Solutions of Navier-Stokes Equations in Lorentz Space

Xunwu Yin

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 3 July 2012; Accepted 7 August 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Xunwu Yin. 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.

This paper is concerned with the regularity of Leray weak solutions to the 3D Navier-Stokes equations in Lorentz space. It is proved that the weak solution is regular if the horizontal velocity denoted by satisfies The result is obvious and improved that of Dong and Chen (2008) on the Lebesgue space.

1. Introduction and Main Results

In this note, we consider the regularity criterion of weak solutions of the Navier-Stokes equations in the whole space

Here and denote the unknown velocity field and the unknown scalar pressure field. is a given initial velocity. For simplicity, we assume that the external force is zero, but it is easy to extend our results to the nonzero external force case. Here and in what follows, we use the notations for vector functions ,

For a given initial data , Leary in the pioneer work, [1] constructed a global weak solution

From that time on, although much effort has been made on the uniqueness and regularity of weak solutions, the question of global regularity or finite time singularity for weak solutions in is still open. One important observation is that the regularity can be derived when certain growth conditions are satisfied. This is known as a regularity criterion problem. The investigation of the regularity criterion on the weak solution stems from the celebrated work of Serrin [2]. Namely, Serrin's regularity criterion can be described as follows. A weak solution of Navier-Stokes equations is regular if the growth condition on velocity field

holds true.

It should be mentioned that the Serrin's condition (1.4) is important from the point of view of the relation between scaling invariance and regularity criteria of weak solutions; indeed, if a pair solves (1.1), then so does defined by

Scaling invariance means that

holds for all and this happens if and only if and satisfy (1.4).

Actually, the condition described by (1.4) which involves all components of the velocity vector field is known as degree growth condition (see Chen and Xin [3] for details), since

The degree growth condition is critical due to the scaling invariance property. That is, solves (1.1) if and only if is a solution of (1.1).

Moreover, this pioneer result [2] has been extended by many authors in terms of velocity , the gradient of velocity or vorticity in Lebesgue spaces or Besov spaces, respectively (refer to [4–7] and reference therein).

Actually, the weak solution remains regular when a part of the velocity components or vorticity is involved in a growth condition. On one hand, regularity of the weak solution was recently obtained by Dong and Chen [8] when two velocity components denoted by

satisfy the critical growth condition

It should be mentioned that the weak solution remains regular if the single velocity component satisfies the higher (subcritical) growth conditions (see Zhou [9], Penel and Pokorný [10], Kukavica and Ziane [11], and Cao and Titi [12]). One may also refer to some interesting regularity criteria [13–15] for weak solutions of micropolar fluid flows. It seems difficult to show regularity of weak solutions by imposing Serrin's growth condition on only one component of velocity field for both Navier-Stokes equations and micropolar fluid flows.

However, whether or not the result (1.9) can be improved to the critical weak spaces is an interesting and challenging problem, that is to say, when the weak critical growth condition is imposed to only two velocity components. The main difficulty lies in the lack of a priori estimates on two-velocity components due to the special structure of the nonlinear convection term in monument equations.

The aim of the present paper is to improve the two-component regularity criterion (1.9) from Lebesgue space to the critical Lorentz space (see the definitions in Section 2) which satisfies the scaling invariance property.

Before stating the main results, we firstly recall the definition of the Leray weak solutions.

Definition 1.1 (Temam, [16]). Let and . A vector field is termed as a Leray weak solution of (1.1) on if satisfies the following properties:

(i); (ii) in the distribution space ;(iii) in the distribution space ;(iv) satisfies the energy inequality

The main results now read as follows.

Theorem 1.2. Suppose , and in the sense of distributions. Assume that is a Leray weak solution of the Navier-Stokes equations (1.1) in . If the horizontal velocity denoted by satisfies the following growth condition:

then is a regular solution on .

Remark 1.3. It is easy to verify that the spaces (1.11) satisfy the degree growth conditions due to the scaling invariance property. Moreover, since the embedding relation , Theorem 1.2 is an important improvement of (1.9).

Remark 1.4. Unlike the previous investigations via two components of vorticity (see [17, 18]) in weak space, of which the approaches are mainly based on the vorticity equations and seem not available in our case here due to the special structure of convection term, the present examination is directly based on the momentum equations. In order to make use of the structure of the nonlinear convection term , we study every component of and estimate them one by one with the aid of the identities .

2. Preliminaries and A Priori Estimates

To start with, let us introduce the definitions of some functional spaces. with are usual Lebesgue space and Sobolev space.

To define the Lorenz space with , we consider a measurable function and define for the Lebesgue measure

of the set .

Then if and only if

Actually, Lorentz space may be alternatively defined by real interpolation (see Bergh and Löfström [19] and Triebel [20])

with

Especially, is equivalent to the norm

and thus it readily seen that

In order to prove Theorem 1.2, it is sufficient to examine a priori estimates for smooth solutions of (1.1) described in the following.

Theorem 2.1. Let , with . Assume that is a smooth solution of (1.1) on and satisfies the growth conditions (1.11). Then

holds true.

Proof of Theorem 2.1. Taking inner product of the momentum equations of (1.1) with and integrating by parts, one shows that

In order to estimate the right-hand side of (2.8), with the aid of the divergence-free condition and integration by parts, observe that

The estimation of the terms is now estimated one by one.

In order to estimate and , employing integration by parts deduces that

For , the divergence-free condition and integration by parts imply

Thus, plugging the above inequalities into (2.8) to produce

We now carry out the estimation of (2.12) based on the assumption described by (1.11).

Applying Hölder’s inequality and Young’s-inequality, we have for the right-hand side (RHS) of (2.12)

where we have used the following Hölder inequality’s in Lorentz space in the last line (refer to O'Neil [21, Theorems 3.4 and 3.5])

for

with

We now claim that the term in (2.13) can be estimated by applying the following Gagliardo-Nirenberg inequality in Lorentz space

Indeed, choosing and such that

and then applying Gagliardo-Nirenberg inequality, it follows that

Thus, applying the interpolation inequality (2.3), we have

that is to say,

and (2.17) is derived. Therefore, by employing (2.17) and Young’s inequality, the inequality (2.13) becomes

Inserting (2.22) into (2.12) to produce

Taking Gronwall’s inequality into account yields the desired estimate,

note that

This completes the proof of Theorem 2.1.

3. Proof of Theorem 1.2

According to a priori estimates of smooth solutions described in Theorem 2.1, the proofs of Theorem 1.2 are standard.

Since with , by the local existence theorem of strong solutions to the Navier-Stokes equations (see, e.g., Fujita and Kato [22]), there exist a and a smooth solution of (1.1) satisfying

Note that the Leray weak solution satisfies the energy inequality (1.10). It follows from Serrin's weak-strong uniqueness criterion [2] that

Thus, it is sufficient to show that

Suppose that . Without loss of generality, we may assume that is the maximal existence time for . Since on and by the assumptions (1.11), it follows from a priori estimate (2.7) that the existence time of can be extended after which contradicts with the maximality of .

Thus, we complete the proof of Theorem 1.2.

Acknowledgments

This work is partially supported by NNSF of China (11071185) and NSF of Tianjin (09JCYBJC01800).

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