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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 184674, 9 pages
http://dx.doi.org/10.1155/2012/184674
Research Article

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

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 ̃𝑢=(𝑢1,𝑢2,0) satisfies ̃𝑢(𝑥,𝑡)𝐿𝑞(0,𝑇;𝐿𝑝,(𝐑3))for2/𝑞+3/𝑝=1,3<𝑝<. 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 𝐑3𝜕𝑡𝑢+(𝑢)𝑢+𝜋=Δ𝑢,𝑢=0,𝑢(𝑥,0)=𝑢0.(1.1) Here 𝑢=(𝑢1,𝑢2,𝑢3) and 𝜋 denote the unknown velocity field and the unknown scalar pressure field. 𝑢0 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 𝑢,𝑣, (𝑢)𝑣=3𝑖=1𝑢𝑖𝜕𝑖𝑣𝑘(𝑘=1,2,3),𝑢=3𝑖=1𝜕𝑖𝑢𝑖.(1.2)

For a given initial data 𝑢0𝐿2(𝐑3), Leary in the pioneer work, [1] constructed a global weak solution 𝑢𝐿0,;𝐿2𝐑3𝐿20,𝑇;𝐻1𝐑3.(1.3) 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 𝐑3 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 𝑢 of Navier-Stokes equations is regular if the growth condition on velocity field 𝑢𝑢𝐿𝑝0,𝑇;𝐿𝑞𝐑3𝐿𝑝𝐿𝑞2,for𝑝+3𝑞1,3<𝑞,(1.4)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 (𝑢,𝑝) solves (1.1), then so does (𝑢𝜆,𝑝𝜆) defined by 𝑢𝜆(𝑥,𝑡)=𝜆𝑢𝜆𝑥,𝜆2𝑡,𝑝𝜆(𝑥,𝑡)=𝜆2𝑝𝜆𝑥,𝜆2𝑡.(1.5) Scaling invariance means that 𝑢𝐿𝑝(0,𝑇;𝐿𝑞(𝐑3))=𝑢𝜆𝐿𝑝(0,𝑇;𝐿𝑞(𝐑3))(1.6) holds for all 𝜆>0 and this happens if and only if 𝑝 and 𝑞 satisfy (1.4).

Actually, the condition described by (1.4) which involves all components of the velocity vector field 𝑢=(𝑢1,𝑢2,𝑢3) is known as degree 1 growth condition (see Chen and Xin [3] for details), since 𝑢𝜆,𝜆2𝐿𝑝𝐿𝑞=𝑢𝐿𝑝(0,𝜆2𝑇;𝐿𝑞(𝐑3))𝜆2/𝑝3/𝑞=𝑢𝐿𝑝(0,𝜆2𝑇;𝐿𝑞(𝐑3))𝜆1.(1.7) The degree 1 growth condition is critical due to the scaling invariance property. That is, (𝑢(𝑥,𝑡),𝑝(𝑥,𝑡)) solves (1.1) if and only if (𝑢𝜆(𝑥,𝑡),𝑝𝜆(𝑥,𝑡)) is a solution of (1.1).

Moreover, this pioneer result [2] has been extended by many authors in terms of velocity 𝑢(𝑥,𝑡), the gradient of velocity 𝑢(𝑥,𝑡) or vorticity 𝑤(𝑥,𝑡)=(𝑤1,𝑤2,𝑤3)=×𝑢 in Lebesgue spaces or Besov spaces, respectively (refer to [47] 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 𝑢̃𝑢=1,𝑢2,0(1.8) satisfy the critical growth condition ̃𝑢𝐿𝑝𝐿𝑞2,for𝑝+3𝑞=1,3<𝑞.(1.9)

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 [1315] 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 𝐿𝑝 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 ̃𝑢 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 𝑢0𝐿2(𝐑3) and 𝑢0=0. A vector field 𝑢(𝑥,𝑡) is termed as a Leray weak solution of (1.1) on (0,𝑇) if 𝑢 satisfies the following properties:(i)𝑢𝐿(0,𝑇;𝐿2(𝐑3))𝐿2(0,𝑇;𝐻1(𝐑3)); (ii)𝜕𝑡𝑢+(𝑢)𝑢+𝜋=Δ𝑢 in the distribution space 𝒟((0,𝑇)×𝐑3);(iii)𝑢=0 in the distribution space 𝒟((0,𝑇)×𝐑3);(iv)𝑢 satisfies the energy inequality (𝑢𝑡)2𝐿2+2𝑡0𝐑3||||𝑢(𝑥,𝑠)2𝑢𝑑𝑥𝑑𝑠02𝐿2,for0𝑡𝑇.(1.10)

The main results now read as follows.

Theorem 1.2. Suppose 𝑇>0, 𝑢0𝐻1(𝐑3) and 𝑢0=0 in the sense of distributions. Assume that 𝑢 is a Leray weak solution of the Navier-Stokes equations (1.1) in (0,𝑇). If the horizontal velocity denoted by ̃𝑢=(𝑢1,𝑢2,0) satisfies the following growth condition: 𝑇0̃𝑢(𝑡)𝑞𝐿𝑝,2𝑑𝑡<,for𝑞+3𝑝=1,3<𝑝<,(1.11) then 𝑢 is a regular solution on (0,𝑇].

Remark 1.3. It is easy to verify that the spaces (1.11) satisfy the degree 1 growth conditions due to the scaling invariance property. Moreover, since the embedding relation 𝐿𝑝𝐿𝑝,, 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 (𝑢)𝑢, we study every component of ((𝑢)𝑢,Δ𝑢) and estimate them one by one with the aid of the identities 𝑢=0.

2. Preliminaries and A Priori Estimates

To start with, let us introduce the definitions of some functional spaces. 𝐿𝑝(𝐑3),𝑊𝑘,𝑝(𝐑3) with 𝑘𝐑,1𝑝 are usual Lebesgue space and Sobolev space.

To define the Lorenz space 𝐿𝑝,𝑞(𝐑3) with 1𝑝,𝑞, we consider a measurable function 𝑓 and define for 𝑡0 the Lebesgue measure 𝑚(𝑓,𝑡)=𝑚𝑥𝐑3||𝑓||,(𝑥)>𝑡(2.1) of the set {𝑥𝐑3|𝑓(𝑥)|>𝑡}.

Then 𝑓𝐿𝑝,𝑞(𝐑3) if and only if 𝑓𝐿𝑝,𝑞=0𝑡𝑞(𝑚(𝑓,𝑡))𝑞/𝑝𝑑𝑡𝑡1/𝑞<for1𝑞<,𝑓𝐿𝑝,=sup𝑡0𝑡(𝑚(𝑓,𝑡))1/𝑝<for𝑞=.(2.2) Actually, Lorentz space 𝐿𝑝,𝑞(𝐑3) may be alternatively defined by real interpolation (see Bergh and Löfström [19] and Triebel [20]) 𝐿𝑝,𝑞𝐑3=𝐿𝑝1𝐑3,𝐿𝑝2𝐑3𝜃,𝑞,(2.3) with 1𝑝=1𝜃𝑝1+𝜃𝑝2,1𝑝1<𝑝<𝑝2.(2.4)

Especially, 𝑓𝐿𝑞, is equivalent to the norm sup0<|𝐸|<||𝐸||1/𝑞1𝐸||||𝑓(𝑥)𝑑𝑥,(2.5) and thus it readily seen that 𝐿𝑝𝐑3=𝐿𝑝,𝑝𝐑3𝐿𝑝,𝑞𝐑3𝐿𝑝,𝐑3,1<𝑝<𝑞<.(2.6)

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 𝑇>0, 𝑢0𝐻1(𝐑3) with 𝑢0=0. Assume that 𝑢(𝑥,𝑡) is a smooth solution of (1.1) on 𝐑3×(0,𝑇) and satisfies the growth conditions (1.11). Then sup0<𝑡<𝑇𝑢(𝑡)2𝐿2+𝑇0Δ𝑢(𝑡)2𝐿2𝑑𝑡𝑐𝑢02𝐿2exp𝑇0̃𝑢(𝑡)𝑞𝐿𝑝,𝑑𝑡(2.7) holds true.

Proof of Theorem 2.1. Taking inner product of the momentum equations of (1.1) with Δ𝑢 and integrating by parts, one shows that 12𝑑𝑑𝑡𝑢(𝑡)2𝐿2+Δ𝑢(𝑡)2𝐿23𝑖,𝑗,𝑘=1𝐑3𝑢𝑖𝜕𝑖𝑢𝑗𝜕𝑘𝑘𝑢𝑗𝑑𝑥.(2.8) In order to estimate the right-hand side of (2.8), with the aid of the divergence-free condition 3𝑖=1𝜕𝑖𝑢𝑖=0 and integration by parts, observe that 3𝑖,𝑗,𝑘=1𝐑3𝑢𝑖𝜕𝑖𝑢𝑗𝜕𝑘𝑘𝑢𝑗𝑑𝑥=3𝑖,𝑗,𝑘=1𝐑3𝜕𝑘𝑢𝑖𝜕𝑖𝑢𝑗𝜕𝑘𝑢𝑗=𝑑𝑥3𝑖,𝑗,𝑘=1𝐑3𝜕𝑘𝑢𝑖𝜕𝑖𝑢𝑗𝜕𝑘𝑢𝑗1𝑑𝑥+23𝑖,𝑗,𝑘=1𝐑3𝑢𝑖𝜕𝑖𝜕𝑘𝑢𝑗𝜕𝑘𝑢𝑗=𝑑𝑥3𝑖,𝑗,𝑘=1𝐑3𝜕𝑘𝑢𝑖𝜕𝑖𝑢𝑗𝜕𝑘𝑢𝑗=𝑑𝑥23𝑖=1𝑗,𝑘=1𝐑3𝜕𝑘𝑢𝑖𝜕𝑖𝑢𝑗𝜕𝑘𝑢𝑗𝑑𝑥+23𝑗=1𝑘=1𝐑3𝜕𝑘𝑢3𝜕3𝑢𝑗𝜕𝑘𝑢𝑗+𝑑𝑥3𝑘=1𝐑3𝜕𝑘𝑢3𝜕3𝑢3𝜕𝑘𝑢3𝑑𝑥=3𝑚=1𝐼𝑚.(2.9)
The estimation of the terms 𝐼𝑚 is now estimated one by one.
In order to estimate 𝐼1 and 𝐼2, employing integration by parts deduces that 𝐼1=23𝑖=1𝑗,𝑘=1𝐑3𝑢𝑖𝜕𝑘𝜕𝑖𝑢𝑗𝜕𝑘𝑢𝑗𝑑𝑥𝑐𝐑3||||||||||̃𝑢𝑢2𝑢||𝐼𝑑𝑥,2=23𝑗=1𝑘=1𝐑3𝑢𝑗𝜕3𝜕𝑘𝑢3𝜕𝑘𝑢𝑗𝑑𝑥𝑐𝐑3||||||||||̃𝑢𝑢2𝑢||𝑑𝑥.(2.10) For 𝐼3, the divergence-free condition 𝜕3𝑢3=𝜕1𝑢1𝜕2𝑢2 and integration by parts imply 𝐼3=3𝑘=1𝐑3𝜕𝑘𝑢3𝜕1𝑢1+𝜕2𝑢2𝜕𝑘𝑢3𝑑𝑥3𝑘=1𝐑3𝑢1𝜕1𝜕𝑘𝑢3𝜕𝑘𝑢3+𝑢2𝜕2𝜕𝑘𝑢3𝜕𝑘𝑢3𝑑𝑥𝑐𝐑3||||||||||̃𝑢𝑢2𝑢||𝑑𝑥.(2.11)
Thus, plugging the above inequalities into (2.8) to produce 𝑑𝑑𝑡𝑢(𝑡)2𝐿2+2Δ𝑢(𝑡)2𝐿2𝑐𝐑3||||||||||̃𝑢𝑢2𝑢||𝑑𝑥=RHS.(2.12)
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) ||||||||RHS𝑐̃𝑢𝑢𝐿22𝑢𝐿2||||||||𝑐(𝜀)̃𝑢𝑢2𝐿2+𝜀2𝑢2𝐿2𝑐̃𝑢2𝐿𝑝,𝑢2𝐿2𝑝/(𝑝2),2+12Δ𝑢2𝐿2,(2.13) 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]) 𝑓𝑔𝐿𝑝11,𝑞𝑐𝑓𝐿𝑝22,𝑞𝑔𝐿𝑝33,𝑞,(2.14) for 1𝑝1=1𝑝2+1𝑝3,1𝑞11𝑞2+1𝑞3,(2.15) with 1𝑝2,𝑝3,1𝑞2,𝑞3.(2.16)
We now claim that the term 𝑢𝐿2𝑝/(𝑝2),2 in (2.13) can be estimated by applying the following Gagliardo-Nirenberg inequality in Lorentz space 𝑓𝐿2𝑝/(𝑝2),2𝑐𝑓𝐿(𝑝3)/𝑝2Δ𝑓𝐿3/𝑝2.(2.17)
Indeed, choosing 𝑝1 and 𝑝2 such that 3<𝑝1<𝑝<𝑝22<,𝑝=1𝑝1+1𝑝2,(2.18) and then applying Gagliardo-Nirenberg inequality, it follows that 𝑓2𝑝𝑖/(𝑝𝑖2)𝑐𝑓(𝑝𝑖3)/𝑝𝑖𝐿2Δ𝑓3/𝑝𝑖𝐿2,𝑖=1,2.(2.19) Thus, applying the interpolation inequality (2.3), we have 𝐿2𝑝/(𝑝2),2𝐑3=𝐿2𝑝1/(𝑝12)𝐑3,𝐿2𝑝2/(𝑝22)𝐑31/2,2,(2.20) that is to say, 𝑓𝐿2𝑝/(𝑝2),2𝑐𝑓𝐿1/2112𝑝/(𝑝2)𝑓𝐿1/2222𝑝/(𝑝2)𝑐𝑓𝑝13/𝑝1𝐿2Δ𝑓3/𝑝2𝐿21/2𝑓(𝑝23)/𝑝2𝐿2Δ𝑓3/𝑝2𝐿21/2𝑐𝑓𝐿(𝑝3)/𝑝2Δ𝑓𝐿3/𝑝2,(2.21) and (2.17) is derived. Therefore, by employing (2.17) and Young’s inequality, the inequality (2.13) becomes RHS𝑐̃𝑢2𝐿𝑝,𝑢𝐿2(𝑝3)/𝑝2Δ𝑢𝐿6/𝑝2+12Δ𝑢2𝐿2𝑐̃𝑢𝐿2𝑝/(𝑝3)𝑝,𝑢2𝐿2+Δ𝑢2𝐿2.(2.22)
Inserting (2.22) into (2.12) to produce 𝑑𝑑𝑡𝑢(𝑡)2𝐿2+Δ𝑢(𝑡)2𝐿2𝑐̃𝑢𝐿2𝑝/(𝑝3)𝑝,𝑢2𝐿2.(2.23)
Taking Gronwall’s inequality into account yields the desired estimate, sup0<𝑡<𝑇𝑢(𝑡)2𝐿2+𝑇0Δ𝑢(𝑡)2𝐿2𝑑𝑡𝑐𝑢02𝐿2exp𝑇0̃𝑢(𝑡)𝑞𝐿𝑝,,𝑑𝑡(2.24) note that 2𝑝𝑝3=𝑞.(2.25) 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 𝑢0𝐻1(𝐑3) with 𝑢0=0, by the local existence theorem of strong solutions to the Navier-Stokes equations (see, e.g., Fujita and Kato [22]), there exist a 𝑇>0 and a smooth solution 𝑢 of (1.1) satisfying 𝑢𝐶0,𝑇;𝐻1𝐶10,𝑇;𝐻1𝐶0,𝑇;𝐻3,𝑢(𝑥,0)=𝑢0.(3.1) Note that the Leray weak solution satisfies the energy inequality (1.10). It follows from Serrin's weak-strong uniqueness criterion [2] that 𝑢𝑢on0,𝑇.(3.2) Thus, it is sufficient to show that 𝑇=𝑇.(3.3) Suppose that 𝑇<𝑇. Without loss of generality, we may assume that 𝑇 is the maximal existence time for 𝑢. Since 𝑢𝑢 on [0,𝑇) and by the assumptions (1.11), it follows from a priori estimate (2.7) that the existence time of 𝑢 can be extended after 𝑡=𝑇 which contradicts with the maximality of 𝑡=𝑇.

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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