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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 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: YonghongΒ Yao

Copyright Β© 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 ̃𝑒=(𝑒1,𝑒2,0) satisfies ̃𝑒(π‘₯,𝑑)βˆˆπΏπ‘ž(0,𝑇;𝐿𝑝,∞(𝐑3))for2/π‘ž+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 𝐑3πœ•π‘‘π‘’+(π‘’β‹…βˆ‡)𝑒+βˆ‡πœ‹=Δ𝑒,βˆ‡β‹…π‘’=0,𝑒(π‘₯,0)=𝑒0.(1.1) Here 𝑒=(𝑒1,𝑒2,𝑒3) and πœ‹ denote the unknown velocity field and the unknown scalar pressure field. 𝑒0 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 𝑒,𝑣, (π‘’β‹…βˆ‡)𝑣=3𝑖=1π‘’π‘–πœ•π‘–π‘£π‘˜(π‘˜=1,2,3),βˆ‡β‹…π‘’=3𝑖=1πœ•π‘–π‘’π‘–.(1.2)

For a given initial data 𝑒0∈𝐿2(𝐑3), Leary in the pioneer work, [1] constructed a global weak solution π‘’βˆˆπΏβˆžξ€·0,∞;𝐿2𝐑3ξ€Έξ€Έβˆ©πΏ2ξ€·0,𝑇;𝐻1𝐑3.ξ€Έξ€Έ(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 𝑒 of Navier-Stokes equations is regular if the growth condition on velocity field π‘’π‘’βˆˆπΏπ‘ξ€·0,𝑇;πΏπ‘žξ€·π‘3ξ€Έξ€Έβ‰‘πΏπ‘πΏπ‘ž2,for𝑝+3π‘žβ‰€1,3<π‘žβ‰€βˆž,(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 (𝑒,𝑝) solves (1.1), then so does (π‘’πœ†,π‘πœ†) defined by π‘’πœ†ξ€·(π‘₯,𝑑)=πœ†π‘’πœ†π‘₯,πœ†2𝑑,π‘πœ†(π‘₯,𝑑)=πœ†2π‘ξ€·πœ†π‘₯,πœ†2𝑑.(1.5) Scaling invariance means that ‖𝑒‖𝐿𝑝(0,𝑇;πΏπ‘ž(𝐑3))=β€–β€–π‘’πœ†β€–β€–πΏπ‘(0,𝑇;πΏπ‘ž(𝐑3))(1.6) holds for all πœ†>0 and this happens if and only if 𝑝 and π‘ž satisfy (1.4).

Actually, the condition described by (1.4) which involves all components of the velocity vector field 𝑒=(𝑒1,𝑒2,𝑒3) is known as degree βˆ’1 growth condition (see Chen and Xin [3] for details), since β€–β€–π‘’ξ€·πœ†β‹…,πœ†2β‹…ξ€Έβ€–β€–πΏπ‘πΏπ‘ž=‖𝑒‖𝐿𝑝(0,πœ†2𝑇;πΏπ‘ž(𝐑3))πœ†βˆ’2/π‘βˆ’3/π‘ž=‖𝑒‖𝐿𝑝(0,πœ†2𝑇;πΏπ‘ž(𝐑3))πœ†βˆ’1.(1.7) The degree βˆ’1 growth condition is critical due to the scaling invariance property. That is, (𝑒(π‘₯,𝑑),𝑝(π‘₯,𝑑)) solves (1.1) if and only if (π‘’πœ†(π‘₯,𝑑),π‘πœ†(π‘₯,𝑑)) is a solution of (1.1).

Moreover, this pioneer result [2] has been extended by many authors in terms of velocity 𝑒(π‘₯,𝑑), the gradient of velocity βˆ‡π‘’(π‘₯,𝑑) or vorticity 𝑀(π‘₯,𝑑)=(𝑀1,𝑀2,𝑀3)=βˆ‡Γ—π‘’ 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 𝑒̃𝑒=1,𝑒2ξ€Έ,0(1.8) satisfy the critical growth condition Μƒπ‘’βˆˆπΏπ‘πΏπ‘ž2,for𝑝+3π‘ž=1,3<π‘žβ‰€βˆž.(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 PokornΓ½ [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 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 ̃𝑒 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 𝑒0∈𝐿2(𝐑3) and βˆ‡β‹…π‘’0=0. A vector field 𝑒(π‘₯,𝑑) is termed as a Leray weak solution of (1.1) on (0,𝑇) if 𝑒 satisfies the following properties:(i)π‘’βˆˆπΏβˆž(0,𝑇;𝐿2(𝐑3))∩𝐿2(0,𝑇;𝐻1(𝐑3)); (ii)πœ•π‘‘π‘’+(π‘’β‹…βˆ‡)𝑒+βˆ‡πœ‹=Δ𝑒 in the distribution space π’Ÿξ…ž((0,𝑇)×𝐑3);(iii)βˆ‡β‹…π‘’=0 in the distribution space π’Ÿξ…ž((0,𝑇)×𝐑3);(iv)𝑒 satisfies the energy inequality (‖𝑒𝑑)β€–2𝐿2ξ€œ+2𝑑0ξ€œπ‘3||||βˆ‡π‘’(π‘₯,𝑠)2‖‖𝑒𝑑π‘₯𝑑𝑠≀0β€–β€–2𝐿2,for0≀𝑑≀𝑇.(1.10)

The main results now read as follows.

Theorem 1.2. Suppose 𝑇>0, 𝑒0∈𝐻1(𝐑3) and βˆ‡β‹…π‘’0=0 in the sense of distributions. Assume that 𝑒 is a Leray weak solution of the Navier-Stokes equations (1.1) in (0,𝑇). If the horizontal velocity denoted by ̃𝑒=(𝑒1,𝑒2,0) satisfies the following growth condition: ξ€œπ‘‡0‖̃𝑒(𝑑)β€–π‘žπΏπ‘,∞2𝑑𝑑<∞,forπ‘ž+3𝑝=1,3<𝑝<∞,(1.11) then 𝑒 is a regular solution on (0,𝑇].

Remark 1.3. It is easy to verify that the spaces (1.11) satisfy the degree βˆ’1 growth conditions due to the scaling invariance property. Moreover, since the embedding relation 𝐿𝑝β†ͺ𝐿𝑝,∞, 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 (π‘’β‹…βˆ‡)𝑒, we study every component of ((π‘’β‹…βˆ‡)𝑒,Δ𝑒) and estimate them one by one with the aid of the identities βˆ‡β‹…π‘’=0.

2. Preliminaries and A Priori Estimates

To start with, let us introduce the definitions of some functional spaces. 𝐿𝑝(𝐑3),π‘Šπ‘˜,𝑝(𝐑3) with π‘˜βˆˆπ‘,1β‰€π‘β‰€βˆž are usual Lebesgue space and Sobolev space.

To define the Lorenz space 𝐿𝑝,π‘ž(𝐑3) with 1≀𝑝,π‘žβ‰€βˆž, we consider a measurable function 𝑓 and define for 𝑑β‰₯0 the Lebesgue measure π‘šξ€½(𝑓,𝑑)∢=π‘šπ‘₯βˆˆπ‘3∢||𝑓||ξ€Ύ,(π‘₯)>𝑑(2.1) of the set {π‘₯βˆˆπ‘3∢|𝑓(π‘₯)|>𝑑}.

Then π‘“βˆˆπΏπ‘,π‘ž(𝐑3) if and only if ‖𝑓‖𝐿𝑝,π‘ž=ξ‚΅ξ€œβˆž0π‘‘π‘ž(π‘š(𝑓,𝑑))π‘ž/𝑝𝑑𝑑𝑑1/π‘ž<∞for1β‰€π‘ž<∞,‖𝑓‖𝐿𝑝,∞=sup𝑑β‰₯0𝑑(π‘š(𝑓,𝑑))1/𝑝<∞forπ‘ž=∞.(2.2) Actually, Lorentz space 𝐿𝑝,π‘ž(𝐑3) may be alternatively defined by real interpolation (see Bergh and LΓΆfstrΓΆm [19] and Triebel [20]) 𝐿𝑝,π‘žξ€·π‘3ξ€Έ=𝐿𝑝1𝐑3ξ€Έ,𝐿𝑝2𝐑3ξ€Έξ€Έπœƒ,π‘ž,(2.3) with 1𝑝=1βˆ’πœƒπ‘1+πœƒπ‘2,1≀𝑝1<𝑝<𝑝2β‰€βˆž.(2.4)

Especially, β€–π‘“β€–πΏπ‘ž,∞ is equivalent to the norm sup0<|𝐸|<∞||𝐸||1/π‘žβˆ’1ξ€œπΈ||||𝑓(π‘₯)𝑑π‘₯,(2.5) and thus it readily seen that 𝐿𝑝𝐑3ξ€Έ=𝐿𝑝,𝑝𝐑3ξ€ΈβŠ‚πΏπ‘,π‘žξ€·π‘3ξ€ΈβŠ‚πΏπ‘,βˆžξ€·π‘3ξ€Έ,1<𝑝<π‘ž<∞.(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, 𝑒0∈𝐻1(𝐑3) with βˆ‡β‹…π‘’0=0. Assume that 𝑒(π‘₯,𝑑) is a smooth solution of (1.1) on 𝐑3Γ—(0,𝑇) and satisfies the growth conditions (1.11). Then sup0<𝑑<π‘‡β€–βˆ‡π‘’(𝑑)β€–2𝐿2+ξ€œπ‘‡0‖Δ𝑒(𝑑)β€–2𝐿2β€–β€–π‘‘π‘‘β‰€π‘βˆ‡π‘’0β€–β€–2𝐿2ξ‚»ξ€œexp𝑇0‖̃𝑒(𝑑)β€–π‘žπΏπ‘,βˆžξ‚Όπ‘‘π‘‘(2.7) holds true.

Proof of Theorem 2.1. Taking inner product of the momentum equations of (1.1) with Δ𝑒 and integrating by parts, one shows that 12π‘‘π‘‘π‘‘β€–βˆ‡π‘’(𝑑)β€–2𝐿2+‖Δ𝑒(𝑑)β€–2𝐿2β‰€βˆ’3𝑖,𝑗,π‘˜=1ξ€œπ‘3π‘’π‘–πœ•π‘–π‘’π‘—πœ•π‘˜π‘˜π‘’π‘—π‘‘π‘₯.(2.8) In order to estimate the right-hand side of (2.8), with the aid of the divergence-free condition βˆ‘3𝑖=1πœ•π‘–π‘’π‘–=0 and integration by parts, observe that βˆ’3𝑖,𝑗,π‘˜=1ξ€œπ‘3π‘’π‘–πœ•π‘–π‘’π‘—πœ•π‘˜π‘˜π‘’π‘—π‘‘π‘₯=3𝑖,𝑗,π‘˜=1ξ€œπ‘3πœ•π‘˜ξ€·π‘’π‘–πœ•π‘–π‘’π‘—ξ€Έπœ•π‘˜π‘’π‘—=𝑑π‘₯3𝑖,𝑗,π‘˜=1ξ€œπ‘3πœ•π‘˜π‘’π‘–πœ•π‘–π‘’π‘—πœ•π‘˜π‘’π‘—1𝑑π‘₯+23𝑖,𝑗,π‘˜=1ξ€œπ‘3π‘’π‘–πœ•π‘–ξ€·πœ•π‘˜π‘’π‘—πœ•π‘˜π‘’π‘—ξ€Έ=𝑑π‘₯3𝑖,𝑗,π‘˜=1ξ€œπ‘3πœ•π‘˜π‘’π‘–πœ•π‘–π‘’π‘—πœ•π‘˜π‘’π‘—=𝑑π‘₯23𝑖=1𝑗,π‘˜=1ξ€œπ‘3πœ•π‘˜π‘’π‘–πœ•π‘–π‘’π‘—πœ•π‘˜π‘’π‘—π‘‘π‘₯+23𝑗=1ξ“π‘˜=1ξ€œπ‘3πœ•π‘˜π‘’3πœ•3π‘’π‘—πœ•π‘˜π‘’π‘—+𝑑π‘₯3ξ“π‘˜=1ξ€œπ‘3πœ•π‘˜π‘’3πœ•3𝑒3πœ•π‘˜π‘’3𝑑π‘₯=3ξ“π‘š=1πΌπ‘š.(2.9)
The estimation of the terms πΌπ‘š is now estimated one by one.
In order to estimate 𝐼1 and 𝐼2, employing integration by parts deduces that 𝐼1=23𝑖=1𝑗,π‘˜=1ξ€œπ‘3π‘’π‘–πœ•π‘˜ξ€·πœ•π‘–π‘’π‘—πœ•π‘˜π‘’π‘—ξ€Έξ€œπ‘‘π‘₯≀𝑐𝐑3||||||||||βˆ‡Μƒπ‘’βˆ‡π‘’2𝑒||𝐼𝑑π‘₯,2=23𝑗=1ξ“π‘˜=1ξ€œπ‘3π‘’π‘—πœ•3ξ€·πœ•π‘˜π‘’3πœ•π‘˜π‘’π‘—ξ€Έξ€œπ‘‘π‘₯≀𝑐𝐑3||||||||||βˆ‡Μƒπ‘’βˆ‡π‘’2𝑒||𝑑π‘₯.(2.10) For 𝐼3, the divergence-free condition πœ•3𝑒3=βˆ’πœ•1𝑒1βˆ’πœ•2𝑒2 and integration by parts imply 𝐼3=3ξ“π‘˜=1ξ€œπ‘3πœ•π‘˜π‘’3ξ€·πœ•1𝑒1+πœ•2𝑒2ξ€Έπœ•π‘˜π‘’3𝑑π‘₯β‰€βˆ’3ξ“π‘˜=1ξ€œπ‘3𝑒1πœ•1ξ€·πœ•π‘˜π‘’3πœ•π‘˜π‘’3ξ€Έ+𝑒2πœ•2ξ€·πœ•π‘˜π‘’3πœ•π‘˜π‘’3ξ€œξ€Έξ€Έπ‘‘π‘₯≀𝑐𝐑3||||||||||βˆ‡Μƒπ‘’βˆ‡π‘’2𝑒||𝑑π‘₯.(2.11)
Thus, plugging the above inequalities into (2.8) to produce π‘‘π‘‘π‘‘β€–βˆ‡π‘’(𝑑)β€–2𝐿2+2‖Δ𝑒(𝑑)β€–2𝐿2ξ€œβ‰€π‘π‘3||||||||||βˆ‡Μƒπ‘’βˆ‡π‘’2𝑒||𝑑π‘₯∢=RHS.(2.12)
We now carry out the estimation of (2.12) based on the assumption described by (1.11).
Applying HΓΆlder’s inequality and Young’s-inequality, we have for the right-hand side (RHS) of (2.12) β€–β€–||||||||β€–β€–RHSβ‰€π‘Μƒπ‘’βˆ‡π‘’πΏ2β€–β€–βˆ‡2𝑒‖‖𝐿2β€–β€–||||||||‖‖≀𝑐(πœ€)Μƒπ‘’βˆ‡π‘’2𝐿2β€–β€–βˆ‡+πœ€2𝑒‖‖2𝐿2≀𝑐‖̃𝑒‖2𝐿𝑝,βˆžβ€–βˆ‡π‘’β€–2𝐿2𝑝/(π‘βˆ’2),2+12‖Δ𝑒‖2𝐿2,(2.13) where we have used the following HΓΆlder inequality’s in Lorentz space in the last line (refer to O'Neil [21, Theorems 3.4 and 3.5]) ‖𝑓𝑔‖𝐿𝑝11,π‘žβ‰€π‘β€–π‘“β€–πΏπ‘22,π‘žβ€–π‘”β€–πΏπ‘33,π‘ž,(2.14) for 1𝑝1=1𝑝2+1𝑝3,1π‘ž1≀1π‘ž2+1π‘ž3,(2.15) with 1≀𝑝2,𝑝3β‰€βˆž,1β‰€π‘ž2,π‘ž3β‰€βˆž.(2.16)
We now claim that the term β€–βˆ‡π‘’β€–πΏ2𝑝/(π‘βˆ’2),2 in (2.13) can be estimated by applying the following Gagliardo-Nirenberg inequality in Lorentz space β€–βˆ‡π‘“β€–πΏ2𝑝/(π‘βˆ’2),2β‰€π‘β€–βˆ‡π‘“β€–πΏ(π‘βˆ’3)/𝑝2‖Δ𝑓‖𝐿3/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 β€–βˆ‡π‘“β€–2𝑝𝑖/(π‘π‘–βˆ’2)β‰€π‘β€–βˆ‡π‘“β€–(π‘π‘–βˆ’3)/𝑝𝑖𝐿2‖Δ𝑓‖3/𝑝𝑖𝐿2,𝑖=1,2.(2.19) Thus, applying the interpolation inequality (2.3), we have 𝐿2𝑝/(π‘βˆ’2),2𝐑3ξ€Έ=𝐿2𝑝1/(𝑝1βˆ’2)𝐑3ξ€Έ,𝐿2𝑝2/(𝑝2βˆ’2)𝐑3ξ€Έξ€Έ1/2,2,(2.20) that is to say, β€–βˆ‡π‘“β€–πΏ2𝑝/(π‘βˆ’2),2β‰€π‘β€–βˆ‡π‘“β€–πΏ1/2112𝑝/(π‘βˆ’2)β€–βˆ‡π‘“β€–πΏ1/2222𝑝/(π‘βˆ’2)ξ‚€β€–β‰€π‘βˆ‡π‘“β€–π‘1βˆ’3/𝑝1𝐿2‖Δ𝑓‖3/𝑝2𝐿21/2ξ‚€β€–βˆ‡π‘“β€–(𝑝2βˆ’3)/𝑝2𝐿2‖Δ𝑓‖3/𝑝2𝐿21/2β‰€π‘β€–βˆ‡π‘“β€–πΏ(π‘βˆ’3)/𝑝2‖Δ𝑓‖𝐿3/𝑝2,(2.21) and (2.17) is derived. Therefore, by employing (2.17) and Young’s inequality, the inequality (2.13) becomes RHS≀𝑐‖̃𝑒‖2𝐿𝑝,βˆžβ€–βˆ‡π‘’β€–πΏ2(π‘βˆ’3)/𝑝2‖Δ𝑒‖𝐿6/𝑝2+12‖Δ𝑒‖2𝐿2≀𝑐‖̃𝑒‖𝐿2𝑝/(π‘βˆ’3)𝑝,βˆžβ€–βˆ‡π‘’β€–2𝐿2+‖Δ𝑒‖2𝐿2.(2.22)
Inserting (2.22) into (2.12) to produce π‘‘π‘‘π‘‘β€–βˆ‡π‘’(𝑑)β€–2𝐿2+‖Δ𝑒(𝑑)β€–2𝐿2≀𝑐‖̃𝑒‖𝐿2𝑝/(π‘βˆ’3)𝑝,βˆžβ€–βˆ‡π‘’β€–2𝐿2.(2.23)
Taking Gronwall’s inequality into account yields the desired estimate, sup0<𝑑<π‘‡β€–βˆ‡π‘’(𝑑)β€–2𝐿2+ξ€œπ‘‡0‖Δ𝑒(𝑑)β€–2𝐿2β€–β€–π‘‘π‘‘β‰€π‘βˆ‡π‘’0β€–β€–2𝐿2ξ‚»ξ€œexp𝑇0‖̃𝑒(𝑑)β€–π‘žπΏπ‘,βˆžξ‚Ό,𝑑𝑑(2.24) note that 2π‘π‘βˆ’3=π‘ž.(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 𝑒0∈𝐻1(𝐑3) with βˆ‡β‹…π‘’0=0, by the local existence theorem of strong solutions to the Navier-Stokes equations (see, e.g., Fujita and Kato [22]), there exist a π‘‡βˆ—>0 and a smooth solution 𝑒 of (1.1) satisfying π‘’βˆˆπΆξ€·ξ€Ί0,π‘‡βˆ—ξ€Έ;𝐻1ξ€Έβˆ©πΆ1ξ€·ξ€·0,π‘‡βˆ—ξ€Έ;𝐻1ξ€Έβˆ©πΆξ€·ξ€Ί0,π‘‡βˆ—ξ€Έ;𝐻3ξ€Έ,𝑒(π‘₯,0)=𝑒0.(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 𝑒≑𝑒on0,π‘‡βˆ—ξ€Έ.(3.2) Thus, it is sufficient to show that π‘‡βˆ—=𝑇.(3.3) Suppose that π‘‡βˆ—<𝑇. Without loss of generality, we may assume that π‘‡βˆ— is the maximal existence time for 𝑒. Since 𝑒≑𝑒 on [0,π‘‡βˆ—) and by the assumptions (1.11), it follows from a priori estimate (2.7) that the existence time of 𝑒 can be extended after 𝑑=π‘‡βˆ— which contradicts with the maximality of 𝑑=π‘‡βˆ—.

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).

References

  1. J. Leray, β€œSur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934. View at Publisher Β· View at Google Scholar
  2. J. Serrin, β€œOn the interior regularity of weak solutions of the Navier-Stokes equations,” Archive for Rational Mechanics and Analysis, vol. 9, pp. 187–195, 1962. View at Zentralblatt MATH
  3. Z. M. Chen and Z. Xin, β€œHomogeneity criterion for the Navier-Stokes equations in the whole spaces,” Journal of Mathematical Fluid Mechanics, vol. 3, no. 2, pp. 152–182, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. Q. Chen and Z. Zhang, β€œSpace-time estimates in the Besov spaces and the Navier-Stokes equations,” Methods and Applications of Analysis, vol. 13, no. 1, pp. 107–122, 2006. View at Zentralblatt MATH
  5. Z.-M. Chen and W. G. Price, β€œBlow-up rate estimates for weak solutions of the Navier-Stokes equations,” Proceedings of the Royal Society A, vol. 457, no. 2015, pp. 2625–2642, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. Z.-M. Chen and W. G. Price, β€œMorrey space techniques applied to the interior regularity problem of the Navier-Stokes equations,” Nonlinearity, vol. 14, no. 6, pp. 1453–1472, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. B. Dong, G. Sadek, and Z. Chen, β€œOn the regularity criteria of the 3D Navier-Stokes equations in critical spaces,” Acta Mathematica Scientia Series B, vol. 31, no. 2, pp. 591–600, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. B.-Q. Dong and Z.-M. Chen, β€œRegularity criterion for weak solutions to the 3D Navier-Stokes equations via two velocity components,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 1–10, 2008. View at Publisher Β· View at Google Scholar
  9. Y. Zhou, β€œA new regularity criterion for weak solutions to the Navier-Stokes equations,” Journal de Mathématiques Pures et Appliquées, vol. 84, no. 11, pp. 1496–1514, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. P. Penel and M. Pokorný, β€œSome new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,” Applications of Mathematics, vol. 49, no. 5, pp. 483–493, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. I. Kukavica and M. Ziane, β€œOne component regularity for the Navier-Stokes equations,” Nonlinearity, vol. 19, no. 2, pp. 453–469, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. C. Cao and E. S. Titi, β€œRegularity criteria for the three-dimensional Navier-Stokes equations,” Indiana University Mathematics Journal, vol. 57, no. 6, pp. 2643–2661, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. B.-Q. Dong and Z.-M. Chen, β€œRegularity criteria of weak solutions to the three-dimensional micropolar flows,” Journal of Mathematical Physics, vol. 50, no. 10, Article ID 103525, 2009. View at Publisher Β· View at Google Scholar
  14. B.-Q. Dong, Y. Jia, and Z.-M. Chen, β€œPressure regularity criteria of the three-dimensional micropolar fluid flows,” Mathematical Methods in the Applied Sciences, vol. 34, no. 5, pp. 595–606, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. B.-Q. Dong and W. Zhang, β€œOn the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 7, pp. 2334–2341, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, The Netherlands, 1977.
  17. H. Kozono and N. Yatsu, β€œExtension criterion via two-components of vorticity on strong solutions to the 3D Navier-Stokes equations,” Mathematische Zeitschrift, vol. 246, no. 1-2, pp. 55–68, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. Z. Zhifei and C. Qionglei, β€œRegularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in R3,” Journal of Differential Equations, vol. 216, no. 2, pp. 470–481, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. J. Bergh and J. Löfström, Interpolation Spaces, Springer, New York, NY, USA, 1976.
  20. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.
  21. R. O'Neil, β€œConvolution operators and L(p,q) spaces,” Duke Mathematical Journal, vol. 30, pp. 129–142, 1963. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  22. H. Fujita and T. Kato, β€œOn the Navier-Stokes initial value problem. I,” Archive for Rational Mechanics and Analysis, vol. 16, pp. 269–315, 1964.