Acta Appl Math (014) 19:175 181 DOI 10.1007/s10440-013-9834-3 A New Regularity Criterion for the 3D Navier-Stokes Euations via Two Entries of the Velocity Gradient Tensor Zujin Zhang Dingxing Zhong Lin Hu Received: 17 July 01 / Accepted: 17 May 013 / Published online: 1 June 013 Springer Science+Business Media Dordrecht 013 Abstract We consider the Cauchy problem for the incompressible Navier-Stokes euations in R 3, and provide a new regularity criterion involving only two entries of the Jacobian matrix of the velocity field. Keywords Incompressible Navier-Stokes euations Regularity criterion Global regularity Weak solutions Strong solutions Mathematics Subject Classification (010) 35Q30 76D03 76D05 1 Introduction This paper is concerned with the following Cauchy problem for the three-dimensional Navier-Stokes euation (NSE): t u u + (u )u + p = 0, in R 3 (0,T), u = 0, in R 3 (0,T), (1) u = u 0, on R 3 {t = 0}, where T>0 is a given time, u = (u 1,u,u 3 ) is the velocity field, p is a scalar pressure, and u 0 is the initial velocity field satisfying u 0 = 0 in the sense of distributions. Here we assume the kinematic viscosity ν = 1, for convenience of presentation. Z. Zhang ( ) D. Zhong College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi, P.R. China e-mail: zhangzujin361@163.com D. Zhong e-mail: zhongdingxing678@sina.com L. Hu School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, Jiangxi, P.R. China e-mail: littleleave05@163.com
176 Z. Zhang et al. The global existence of a weak solution u to (1) with initial data of finite energy is wellknown since the work of Leary [14], see also Hopf [10]. However, the issue of uniueness and regularity of u was left open, and is still unsolved up to date. Pioneered by the work of Serrin [0, 1] and Prodi [19], many interesting sufficient conditions have been provided to ensure the smoothness of u. The readers are referred to [1,, 6 9, 11 13, 15 18, 3, 4, 6 30, 33], and references therein. Observe that some of the regularity criteria established in the above cited literatures involve only some components of the velocity u, the velocity gradient u, the vorticity ω = curl u, the pressure p, the pressure gradient p, and the alike. Here we just list some finest results up to now, at the best of the author s knowledge. Zhou and Pokorný [3] showed that the condition u 3 L p( 0,T; L ( R 3)), p + 3 3 4 + 1,>10 3, () ensures the smoothness of u. Due also to the above-mentioned authors, we have the following regularity criterion [31]: u 3 L p( 0,T; L ( R 3)), p + 3 3, 3. (3) 1 Sufficient conditions on 3 u is given in [4], which reads 3 u L p( 0,T; L ( R 3)), p + 3 7, 3. (4) 16 In fact, the author claimed that the range of in (4) is(7/16, ], which the author could not verify, see also [1]. Surprisingly enough, Cao and Titi [5] were able to establish the following conditional regularity: and 1 u 3 L p( 0,T; L ( R 3)), 3 u 3 L p( 0,T; L ( R 3)), p + 3 1 + 3, 3 <, (5) p + 3 3 4 + 3, <, (6) which involve only one entry of the velocity gradient tensor. Most recently, the author showed an unbelievable Serrin-type regularity criterion via u 3 and 3 u 3 only [5], and some anisotropic-in-space regularity criteria via 1 u 3, u 3 ;or u 3, 3 u 3 [7]. Motivated by the regularity criteria (4), (5) and(6), we consider in this paper a new regularity condition involving 3 u 1, 3 u only. Notice that our regularity criteria reuires one more component than that in [5], but one less component than that in [4]. Also, it is interesting that the regularity criteria via only two entries of the velocity gradient tensor are very sensitive to the places where they live: If the two components are diagonally placed in the Jacobian matrix of the velocity field 1 u 1 1 u 1 u 3 u =[ i u j ]= u 1 u u 3, 3 u 1 3 u 3 u 3
A New Regularity Criterion for the 3D Navier-Stokes Euations 177 then it is of Serrin-type [18, 3]. If the two components are vertically placed, it is proved in [7] that the scaling dimensions vary according to different positions. If the two components are horizontally placed, this is our result, see Theorem. Before stating the precise result, let us recall the weak formulation of (1). Definition 1 Let u 0 L (R 3 ) with u 0 = 0, and T>0. A measurable R 3 -valued vector u is said to be a weak solution of (1) if the following conditions hold: 1. u L (0,T; L (R 3 )) L (0,T; H 1 (R 3 ));. u solves (1) 1, in the sense of distributions; and 3. the energy ineuality, that is, t u(t) + u(s) ds L L u(t0 ), (7) L t 0 for almost every t 0 (including t 0 = 0) and every t t 0. Our main result now reads: Theorem Let u 0 L (R 3 ) with u 0 = 0, and T>0. Suppose u is a given weak solution on [0,T] of (1) with initial data u 0. If additionally, 3 u 1, 3 u L p (0,T; L (R 3 )) with then u is smooth on (0,T). 1 p + 1 = 1, <<, (8) Before proving this theorem in Sect., we collect here some notations used throughout this paper, and make some remarks on our result. The usual Lebesgue spaces L a (R 3 )(1 a ) is endowed with the norm a.fora Banach space (X, ), we do not distinguish it with its vector analogues X 3, thus the norm in X 3 is still denoted by ; however, all vector- and tensor-valued functions are printed boldfaced. Remark 3 It is well-known that dimensional analysis applies to the Navier-Stokes euations [3], which says that we can assign each uantity in (1) 1 a scaling number as: Thus (8) can be rewritten as u : 1, p:, t :, x : 1. 3 u 1, 3 u L p( 0,T; L ( R 3)) p + 3 = 1 + 1, <<. Remark 4 Observe that ( lim 1 + 1 ) = 3 +. We are still a little bit far away from Serrin-type regularity criterion.
178 Z. Zhang et al. Remark 5 One verifies easily from our proof in Sect. that the following regularity criterion holds: k u i L p i ( 0,T; L i ( R 3)), k u j L p j ( 0,T; L j ( R 3)), p i + 3 i = 1 + 1 i, < i <, p j + 3 j = 1 + 1 j, < j <, for some 1 k,l 3, 1 i j 3. Proof of the Main Result In this section, we shall prove Theorem. First, let us recall a technical lemma [5]. Lemma 6 For f,g,h Cc (R3 ), {i, j, k}={1,, 3}, <<, we have ( 1) 3 3 3 3 3 fghdx 1 dx dx 3 C f R 3 i f g j g k g h. (9) Proof Without loss of generality, we may assume i = 3, j = 1, k =, and prove ( 1) 3 3 3 3 3 fghdx 1 dx dx 3 C f R 3 3 f g 1 g g h. While this follows readily by Hölder ineuality, Minkowski ineuality and Gagliardo- Nirenberg ineuality as fghdx 1 dx dx 3 R 3 [ ( ) 1/ max f g dx 3 R x 3 R [ R ( max x 3 f ) 3 ( ) 1/ h dx 3 ]dx 1 dx R ] [( ) 3 ] ( ) 3 dx 1 dx g (3 ) 1/ dx 3 h dx 1 dx dx 3 R R3 [ ] [ ( ) ] C f ( 1) 3 3 f dx 1 dx dx 3 g (3 ) 3 1/ dx 1 dx dx3 h R 3 R R ( 1) 3 C f 3 f 3 3 3 3 g 1 g g h. Proof of Theorem For any ε (0,T), due to the fact that u L (0,T; L (R 3 )), we may find a δ (0,ε), such that u(δ) L (R 3 ).Takethisu(δ) as initial data, there exists an ũ C([δ,Γ ), V ) L (0,Γ ; H (R 3 )), where[δ,γ ) is the life span of the uniue strong solution, see []. Moreover, ũ C (R 3 (δ, Γ )). According to the uniueness result, ũ = u on [δ,γ ).IfΓ T,wehavealreadythatu C (R 3 (0,T)), due to the arbitrariness of ε (0,T). In case Γ <T, our strategy is to show that u(t) remains bounded independently of t Γ. The standard continuation argument then yields that
A New Regularity Criterion for the 3D Navier-Stokes Euations 179 [δ,γ ) can not be the maximal interval of existence of ũ, and conseuently Γ T.This concludes the proof. To bound u, taking the inner product of (1) 1 with u in L (R 3 ), we obtain 1 d dt u + u = [ ] (u )u udx R 3 = u i i u udx + u 3 3 u udx R 3 R 3 I + J. (10) The term I can be directly estimated by Lemma 6 and Young ineuality as I ( ) 3 u i 3 u i 3 C u 4 3 (u 1,u ) u + 1 u. 3 3 3 i u 1 i u i u u Meanwhile, to dominate J, we first integrate by parts (using the divergence free condition u = 0) J = u 3 3 u udx R 3 = = = i,j=1 j=1 R 3 i u 3 3 u i udx R 3 i u 3 3 u j i u j dx 1 R 3 3 u 3 3 u j 3 u j dx R 3 i u 3 3 u 3 i u 3 dx + 1 R 3 u 3 3 i u i udx i,j=1 R 3 u 3 3 i u dx R 3 i u 3 3 u i i u j dx R 3 3 u 3 i u dx. Then replacing 3 u 3 by 1 u 1 u in the last euality, and a further integration by parts gives J C This may be bounded as I, and thus (10) becomes R 3 u i u u dx. d dt u + u C u 4 3 (u 1,u ) u.
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