On the Boundary Regularity for the 6D Stationary Navier-Stokes Equations

Transcription

1 On the Bounday Regulaity fo the D Stationay Navie-Stokes Equations Jitao LIU, Wendong WANG, and Zhouping XIN The Institute of Mathematical Sciences, The Chinese Univesity of Hong Kong, Shatin, N.T., Hong Kong School of Mathematical Sciences, Dalian Univesity of Technology, Dalian, 04, P. R. China lijt005@gmail.com, wendong@dlut.edu.cn and zpxin@ims.cuhk.edu.hk Septembe 3, 03 Abstact It is shown in this pape that suitable weak solutions to the D steady incompessible Navie-Stokes equations ae Hölde continuous nea bounday povided that eithe 3 B u(x) 3 dx o B u(x) dx is sufficiently small, which implies that the D Hausdoff measue of the set of singula points nea the bounday is zeo. This genealizes ecent inteio egulaity esults by Dong-Stain [4]. Keywods: Navie-Stokes equations, suitable weak solutions, bounday egulaity. 00 Mathematics Subject Classification: 35Q30, 7D03. Intoduction In this pape, we conside the following D steady incompessible Navie-Stokes equations on Ω R : (SNS) { u u u = π f, u = 0, (.) whee u epesents the fluid velocity field, π is a scala pessue, and the bounday condition of u is given as a no-slip condition, namely u = 0, on Ω. (.) The main inteests ae in the bounday patial egulaity fo suitable weak solutions to the equations (.). Recall the development of inteio and bounday egulaity citeia fo the Navie-Stokes equations in bief. Fo the thee dimensional time-dependent Navie-Stokes equations, patial egulaity of weak solutions satisfying the local enegy inequality was poved in a seies of papes by Scheffe [9, 30, 3]. Late, the notion of suitable weak solutions was fist intoduced in a celebated pape by Caffaelli-Kohn-Nienbeg [], showing that the set S of possible inteio singula points of a suitable weak solution is one-dimensional paabolic Hausdoff

2 measue zeo. Simplified poofs and impovements can be found in many woks by Lin [], Ladyzhenskaya-Seegin [], Tian-Xin [3], Escauiaza-Seegin-Šveák [5], Seegin [5], Gustafson-Kang-Tsai [7], Vasseu [38], Kukavica [0], Wang-Zhang [37] and the efeences theein. Some simila bounday egulaity esults ae also poved, see Seegin [4,, 7], Kang [8, 9], Gustafson-Kang-Tsai [] and so on. Moeove, it s woth to mention that geneal cuved bounday egulaity was obtained by Seegin-Shilkin-Solonnikov in [8]. Thee ae only fewe esults available in the liteatue fo the 4D and highe dimensional time-dependent Navie-Stokes equations. In [3], Scheffe showed that thee exists a weak solution in R 4 R, whose singula set has vanishing 3D Hausdoff measue. Late, Dong-Du [] poved that, fo any local-in-time smooth solution to the 4D Navie-Stokes equations, the D Hausdoff measue of the set of singula points at the fist potential blow-up time is equal to zeo, and we efe to [3] fo ecent esults with geneal suitable weak solutions. As it s commented in [4], fou (similaly, six fo stationay Navie-Stokes equations) is the highest dimension in which all the existing methods on patial egulaity could be applied. Now we tun to the steady Navie-Stokes equations. In a seies of papes by Fehse and Ruzicka [, 7, 8, 9], the existence on a class of special egula solutions of (.) was obtained fo the five-dimensional and highe dimensional case. Gehadt [] obtained the egulaity of weak solutions unde the fou-dimensional case. Fo N 5, it is not known yet whethe weak solutions ae egula. In [34, 35], Stuwe obtained patial egulaity fo N = 5 by egulaity methods of elliptic systems (c.f. Moey [3] and Giaqinta [3]). Late, the esult of Stuwe was extended to the bounday case by Kang [8]. In a ecent pape, Dong-Stain [4] extended the inteio egulaity esult of Stuwe to the six-dimensional space. Thei main idea is to apply an iteation method and a bootstap agument to get a suitable decay estimate of L 3/ nom of u, then the Moey lemma implies the equied egulaity. In this pape, we genealized the esult in [4] and poved the bounday egulaity of six-dimensional steady Navie-Stokes equations. At last, we efe to [0] by Fawig and Soh fo existence and egulaity citeia fo weak solutions to inhomogeneous Navie-Stokes equations. Let us intoduce the definition of suitable weak solutions nea the bounday. Definition. Let Ω R be an open domain, and Γ Ω be an open set. (u, π) is said to be a suitable weak solution to the steady Navie-Stoks equations (.) in Ω nea the bounday Γ, if the following conditions hold. (i) u H (Ω), π L 3 (Ω), π L 5 (Ω), f L (Ω); (ii) (u, π) satisfies the equations(.) in the sense of distibution sense and the bounday condition u Γ = 0 holds; (iii) u and π satisfy the local enegy inequality u [ φdx u φ u φ( u π) ] fuφdx (.3) Ω Ω fo any nonnegative C test function φ vanishing at the bounday Ω\Γ. The existence of such a suitable weak solution can be found in [8]. The majo concen of this pape is the egulaity and the main esults can be stated as follows: Theoem. Let (u, π) be a suitable weak solution to (.) in B nea the bounday {x B, x = 0}. Then 0 is a egula point of u, if thee exists a small positive constant ε such

3 that one of the following conditions holds, i) lim sup 3 0 ii) lim sup 0 B B u(x) 3 dx < ε, u(x) dx < ε. Remak.3 The bounday egulaity citeia above fo the D steady Navie-Stokes equations genealize ecent inteio egulaity esults by Dong-Stain [4] and bounday egulaity esults fo the 5D case by Kang in [9]. Hee we conside the flat bounday fo simplicity, the esults hold tue fo geneal C bounday, which follows fom the analysis hee and the standad techniques as in [8]. Theoem.4 Let (u, π) be a suitable weak solution to (.) in B nea the bounday {x B, x = 0}. Then the D Hausdoff measue of the set of singula points of (u, π) in B is equal to zeo. Remak.5 This theoem follows diectly fom Theoem. by the standad aguments fom the geometic measue theoy, which is explained fo example in []. As it will be shown late, Theoem. will follow fom the following patial egulaity citeia. Poposition. Let (u, π) be a suitable weak solution to (.) in B nea the bounday {x B, x = 0}. If thee exists 0 > 0 and a small positive constant ε such that Then 0 is a egula point of u. 3 0 u 3 L 3 (B 0 ) 0 π L /5 (B 0 ) 3 0 f 3 L 3 (B 0 ) < ε Remak.7 Thee ae seveal emaks in ode concening the poof of the main esults. Fist, it should be pointed out that it seems difficult to adapt the bounday egulaity theoy fo 5D steady Navie-Stokes system by Kang in [9] to ou case, since the key blow-up aguments thee (Lemma 4., [9]) ae based on the compact imbedding W, (B ) L 3 (B ) which fails in the D case. Ou analysis is motivated by a bootstap agument due to Dong-Stain [4] fo the inteio egulaity theoy. Howeve, due to the bounday, new difficulties aise. In paticula, thee ae slow decaying tems involving E / () in the pessue decomposition in the pesence of boundaies (see Lemma 3.), whee E() = u(x) dx. B This means that E / () and D () π L 5 (B ae the same ode in the standad ) iteative scheme, which seems impossible to obtain an effective iteative estimate by the local enegy inequality as in [4] (moe details see Remak 4.). To ovecome the difficulty, we fist deive a evised local enegy inequality, see Poposition.4, such that thee exists a fee paamete in the coefficient of the scaled enegy on the lage ball in the local enegy inequality, which plays an impotant ole in the equied estimate in Lemma 4. and yields an effective iteation scheme. 3

4 The est of the pape is oganized as follows. In section, we intoduce some notations and some technical lemmas about Stokes estimates and local enegy estimates. In section 3, we pove Theoem. unde the assumption of Poposition.. Section 4 is devoted to the poof of Poposition.. Notations and some technical lemmas Thoughout this aticle, C 0 denotes an absolute constant independent of u,, and may be diffeent fom line to line. Let (u, π) be a solution to the steady Navie-Stokes equations (.). Set the following scaling: u λ (x) = λu(λx), π λ (x) = λ π(λx), f λ (x) = λ 3 f(λx), (.) fo any λ > 0, then the family (u λ, π λ ) is also a solution of (.) with f eplaced by f λ (x). Now define some quantities which ae invaiant unde the scaling (.): A() = 4 u(x) dx, C() = 3 u(x) 3 dx; E() = B D() = 3 B B u(x) dx, D () = π L 5 (B ) ; B π π B 3 dx, πb = B B F () = 3 f(x) 3 dx, B whee B (x 0 ) is the semi-ball of adius centeed at x 0, and we denote B (0) by B. Moeove, a solution u is said to be egula at x 0 if u L (B (x 0 )) fo some > 0. The following two lemmas about L p estimates on solutions to bounday value poblems fo the Stokes equations ae well-known, fo example, seeing Solonnikov [33], Giga-Soh [5], Galdi [], Seegin [4] and Kang [8]. Lemma. Assume that Ω R n is a bounded open domain, n 3, < p <, and f L p (Ω). (v, q) satisfies the following conditions: v W,p (Ω), q W,p (Ω), v q = f, v = 0, (.) qdx = 0, v Ω = 0. Then thee holds Ω v W,p (Ω) q W,p (Ω) C(p, Ω) f L p (Ω). πdx; 4

5 Lemma. Assume that B Rn is a bounded open domain, n 3, < p 0 p <, and f L p (B ). (v, q) satisfies the following conditions: v W,p 0 (B ), q W,p 0 (B ), v q = f, v = 0, (.3) v {xn=0} = 0. Then thee holds v W,p (B / ) q W,p (B / ) C(p 0, p) ( f L p (B ) q L p 0 (B ) v W,p 0 (B ) ). Recall the inteio egula esult by Dong-Stain (Poposition 3.7 and (3.33), [4]), which is necessay in the poof of Theoem.. We wite it in a slightly diffeent fom. Poposition.3 Thee exists ε 0 > 0 satisfying the following popety. Suppose that (u, π) is a suitable weak solution of (.) in B (x 0 ), and fo some 0 (0, ), it holds that u 3 dx 3 0 π π B0 (x 0 ) 3 dx 3 0 f 3 dx ε. 3 0 B 0 (x 0 ) B 0 (x 0 ) B 0 (x 0 ) Then, fo 0 < < 0 /8, the following inequality will hold unifomly u 5 dx C , (.4) B (x 0 ) whee C 0 is a positive constant independent of. Next, modifying the analysis in [4] by choosing a new test function, we will deive the following evised local enegy inequality, which impoves the local enegy inequality in [4] and plays a cucial ole in ou poof late. Poposition.4 Let 0 < 4 <. It holds that k A() E() C 0 k 4( ) A() C0 k ( ) 3 [ C() C 3 ()D whee k and the constant C 0 is independent of k,,. ( 3 ()] C0 ) C 3 ()F 3 (), (.5) Poof. Let ζ be a cutoff function, which vanishes outside of B and equals in B /, satisfying Intoduce a smooth function as ζ C 0, ζ C 0. Γ(x) = (k x ), k, which clealy satisfies Γ = 4k (k x ) 4 < 0. 5

6 Taking the test function φ = Γζ in the local enegy inequality (.3), we obtain that u ζ Γdx u ζγdx B B [ u (Γ ζ Γ ζ) u φ( u π π B )] dx fuγζdx B It follows fom some staightfowad computations that ζγ(x, t) C 0 (k) 4, ζ Γ(x, t) C 0 (k) in B, φ Γ ζ Γ ζ C 0 (k) 5 in B, Γ ζ Γ ζ C 0 in B, fom which and the Hölde inequality, one deives that k A() E() C 0 k 4( ) A() C0 k ( ) 3 3 B ( u 3 u π π B )dx B C 0 ( B f 3 dx ) 3 ( B u 3 dx ) 3 C 0 k 4( ) A() C0 k ( ) 3 [ C() C 3 ()D The poof is completed. ( 3 ()] C0 ) C 3 ()F 3 (). Remak.5 If k = in (.5), then we obtain the local enegy inequality as the inteio case in [4]: A() E() ( C 0 ) A() C0 ( ) 3 [ C() C 3 ()D ( 3 ()] C0 ) C 3 ()F 3 (). (.) Howeve, it seems difficult fo us to constuct an effective iteation scheme based on (.) due to the fast decay of the second tem in the ight hand side of (.) (fo moe details see Remak 4.). Yet, by choosing a suitable k in Poposition.4, we can ovecome the difficulty, see the poof of Lemma Poof of Theoem. In this section, we ae going to pove Theoem. assuming that Poposition. holds tue, and the next section is devoted to the poof of Poposition.. In fact, due to the Sobolev inequality in Lemma 3., it suffices to assume that C() < ε, fo any 0 < < 0 <. (3.) Fist, we need some technical lemmas on the scaling quantities intoduced befoe. Lemma 3. Fo any 0 < < 0, thee holds C() C 0 E() 3, D() C0 D 3 (). (3.) Poof. These follow fom the Sobolev imbedding inequality in the six dimensional space.

7 Lemma 3. Fo any 0 < 4 < < 0, thee holds ( ) 3 D () C 0 whee p, and C 0 depends on p. p ( E () D () ) C 0 ( ) ( E ()C 3 () F 3 () ), (3.3) Poof. Choose a domain B with a smooth bounday such that B / B B. Let v and π be the unique solution to the following initial bounday value poblem fo the Stokes system: v π = f u u, divv = 0 in B, (π ) B = B π B dx = 0, v = 0 on B. Then by the estimates fo the steady Stokes system in Lemma., one can obtain (3.4) v L 5 ( B ) v L 5 ( B ) v L 5 ( B ) π π L 5 ( B ) L 5 ( B ) ( C 0 u u L f ) 5 ( B ) L 5 ( B ) C 0 ( E ) ()C 3 () F 3 () (3.5) with C 0 depending on B. On the othe hand, let w = u v, π = π π B π (3.) / Then w, π solve the following bounday value poblem: w π = 0, divw = 0 in B, w = 0 on B {x = 0}. Then the local estimate nea the bounday fo the steady Stokes system in Lemma. fo p yields that 3 p π L p (B ) C ( 0 /4 4 w L 5 (B / ) 3 w L 5 (B / ) 3 π ) L 5 (B. / ) It follows fom the Sobolev inequality, (3.), and (3.5) that 3 p π L p (B /4 ) C 0 ( 3 w L 5 (B / ) 3 π L 5 (B / ) ) C 0 ( 3 u L 5 (B / ) 3 v L 5 (B / ) π L 5 (B / ) 3 π L 5 (B / ) ) C 0 ( E () E ()C 3 () F 3 () D () ). (3.7) 7

8 Combining the two estimates (3.5) and (3.7) shows that D () = ( B C 0 ( C 0 ( C 0 p 5 dx ) 5 ( π L 5 (B ) 5 ) ( E ()C 3 () F ) 3 p π L p (B ) ) ( 3 () C0 p ( E () D () ) C 0 ( ) ) 3 p ( E () E ()C 3 () F 3 () D () ) ) ( E ()C 3 () F 3 () ). Thus the poof is completed. Poof of Theoem.: Take 0 < < 4 < κ < 0 and set G() = A() E() ε D () ε F 3 (). It follows fom (3.) and (3.3) fo p = that ) 5 ( D () C ( 0 E (κ) D (κ) ) (κ C 0 κ (κ) ( ) ( C 0 E(κ) F 3 (κ) C0 κ ) ( E (κ)c 3 (κ) F 3 (κ) ) ) 5 ( D (κ) ). (3.8) Then using the local enegy inequality in Poposition.4 fo k =, (3.) and (3.), one can obtain that ( ) A() () 3 [ () G() C 0 C0 C() C 3 ()D 3 C ()] C0 3 ()F 3 () C 0 ε D () C 0 ε F C 0 ( ) A() C0 ( C 0 ε D () C 0 ( 3 () ) 3ε ( 3 D () C 0 ) 3ε. ) ε 3 F 3 () C0 ε F 3 () Applying the inequality (3.8) twice yields (κ) 4 ( ) G(κ) () G() C 0 C0 ε 3 (κ) ( ) 3 E(κ) F 3 (κ) () C 0 ε 3 ( ) 5 ( 3 D (κ) ) () C 0 ε ( ) ( 3 F 3 (κ) C0 ε ) F 3 (κ) κ κ κ (κ) C 0 ε ( ) ( E(κ) F 3 (κ) C0 ε ) 5 ( D (κ) ) () 3ε C 0 κ [(κ) 4 ( ) ( ) 5 C 0 ( ) (κ) ]G(κ) C0 ε () 3 3G(κ) κ κ [() C 0 ε 3 (κ) () ( )] [() 4 G(κ) C0 ε 3 ( ) 5 ( κ) ] G(κ) κ κ (κ) C 0 ε G(κ) () C0 ε 3 ( ) 5 ( 3 C 0 ε ) 5 () 3ε. C 0 κ κ Set = θ 3, = θκ with 0 < θ < 8. The above inequality yields that G() C 0 [ θ ε 3 θ ε 4 θ ε θ 8 ε θ 8 ] G(κ) C 0 ε 3 θ 3 C0 ε θ 0 C 0 εθ 9. 8

9 Thus choosing θ small at fist, and then ε small, one can get C 0 [ θ ε 3 θ ε 4 θ ε θ 8 ε θ 8 ], C 0 ε 3 θ 3 C0 ε θ 0 C 0 εθ 9 ε 3, whee ε is the constant as in Poposition.. Thus we obtain the following iteative inequality On the othe hand, it is easy to see that G(θ 4 κ) G(κ) ε3. (3.9) G( 0 ) C 0, whee C 0 depends on 0, u H (B ), p L 5 and f (B ) L (B ). Then the standad iteation aguments fo (3.9) ensue that thee exists > 0 such that G() 3ε 3 fo all (0, ). (3.0) Next, due to Lemma 3., (3.0) and f L (B ), thee exists > 0 such that D () C 0 ( ) (ε 3/ D ()) C 0 ( ) ε 3, 0 < < < <, F () ε 3, 0 < < <. Then a standad iteation agumemnt yields that thee exists 3 > 0 such that D () F () C 0 ε 3/ fo 0 < < 3 <. Combining (3.0) and the above inequality, we get C() D () F () C 0 ε 3/ fo 0 < < 3. We can assume that C 0 ε / without loss of geneality. Hence the poof of Theoem. is completed due to Poposition.. 4 Poof of Poposition. This section is devoted to the poof of Poposition.. To ovecome the lowe ode tem E / () fom the pessue decomposition, we ll make full use of the evised local enegy inequality in Poposition.4 and the decay estimates (3.3) in ode to deive an effective iteation (fo moe details, see the following lemma and Remak 4.). Finally, we impove the decay of u by a simila bootstap agument as in [4], and cay out the bounday estimates as in [4]. 9

10 Lemma 4. Let > 0 be a positive constant. Then thee exists a θ 0, which is suitably small and independent of, such that 4 θ 0 A(θ 0 ) E(θ 0 ) θ 0 0 D(θ 0 ) [ θ 0 A() E() θ 0 0 D () ] C 0 ( E 3 () E () F 3 () ), (4.) whee C 0 is a positive constant independent of. Poof. Fo any θ (0, 4 ], Poposition.4 with = θ and (3.) yield that Hence k A(θ) E(θ) C 0 k 4 θ A() C 0 k θ 3[ C() C 3 ()D 3 ()] C0 θ C 3 ()F 3 (), C 0 k 4 θ E() C 0 k θ 3[ E 3 () E ()D ()] C 0 θ E ()F 3 (), (4.) By choosing p = 00 in Lemma 3., one hets Set D (θ) C 0 θ 3 ( 00 E () D () ) C 0 θ ( ) E() F 3 (). D (θ) C 0 θ 50 ( E() D () ) C 0 θ 4( E () F 3 () ). (4.3) G() = k A() E() γ D (), and = θ, with γ > 0 to be decided. It then follows fom (4.) and (4.3) that G(θ) C 0 k 4 θ E() C 0 k θ 3[ E 3 () E ()D ()] C 0 θ E ()F 3 () C 0 γ θ ( 50 E() D () ) C 0 γ θ 4( E ) () F 3 () C 0 k 4 θ E() C 0 k θ 3( E 3 () γ E() γ D () ) C 0 γe() C 0 γ θ ( 50 E() D () ) C 0 γ θ 4( E ) () F 3 () C 0 [k 4 θ k θ 3 γ γ γ θ 50 θ 50 ]G() C 0 k θ 3 E 3 () C0 γ θ 4( E () F 3 () ) Choosing k = θ 4, γ = θ 0, we get C 0 [k 4 θ k θ 3 γ γ γ θ 50 θ 50 ] C0 θ Finally, θ is chosen so small such that the equied inequality (4.) holds. The lemma is poved. 0

11 Remak 4. It should be noted that Poposition.4 is cucial in the poof of Lemma 4.. Recall the local enegy inequality (.) as in [4] and the pessue decomposition in Lemma 3.: ( ) 3 D () C 0 p ( E () D () ) C 0 ( ) ( E ()C 3 () F 3 () ), (4.4) fo 0 < 4 < < and p. The two inequalities indicate that A(), E(), and D () ae the same ode, hence it is natual to ty to deive an iteative inequality fo the following scaled quantity: G() = A() E() γ D(), = θ, with γ > 0 to be decided. It then follows fom (.) and (4.4) that (as in Lemma 4.) G(θ) C 0 [θ θ 3 γ γ γ θ p θ p ]G() C 0 θ 3 E 3 () C0 γ θ 4( E ) () F 3 () (4.5) In ode to obtain Poposition., one needs the smallness of C 0 [θ θ 3 γ γ γ θ p θ p ], howeve, which seems to be impossible. To ovecome the difficulty above, we make use of the fee paamete in the evised local enegy inequality (.5) such that the tem ( A() ) in (.) plays an impotant ole, and consequently we obtain Lemma 4.. Fo α 0 = log θ0 and any small δ > 0, we will pove that thee exists an intege m = m(δ) and an inceasing sequence of eal numbe {α k } m k= (α 0, ) with α m > δ. Fo these fixed δ and m(δ), we obtain the following poposition. Poposition 4.3 Fo 0 (0, ), thee exits a small constant ε 0 > 0 satisfying the following popety that if C( 0 ) D ( 0 ) F ( 0 ) ε 0, then, fo any given δ > 0, any 0 < < 0 /8 and x 0 0 /8 with x 0 = (x, x ) and x = 0, thee exists an inceasing sequence of eal numbes {α k } m k=0 [α 0, ) with α m > δ such that the following inequality holds unifomly A(, x 0 ) E(, x 0 ) C 3 (, x0 ) D (, x 0 ) C 0 α k, k = 0,,, m (4.) whee C 0 is a positive constant depending on δ and k, but independent of. Poof. We will pove this by indiction. Without loss of geneality, assume that x = 0. By Poposition.4 and (3.), it holds that A( 0 /8) E( 0 /8) C 0 A( 0 ) C 0 [ C(0 ) C 3 (0 )D ( 0 )] C 0 C 3 (0 )F 3 (0 ) C 0 C 3 (0 ) C 0 [ C(0 ) C 3 (0 )D ( 0 )] C 0 F (0 ) C 0 (ε 0 ε 0 ε 3 0 ε ).

12 Thus we can choose ε 3 > 0 fist, then ε 0 = ε 0 (ε 3 ) > 0 small enough such that and ( C 0 ε 3 ε ) 3 < /4, C0 (ε 0 ε 0 ε 3 0 ε ) < ε 3 /4, (4.7) ϕ( 0 ) = θ 0 A( 0 /8) E( 0 /8) θ 0 0 D ( 0 /8) ε 3. (4.8) It follows fom (4.7)-(4.8) and (4.) that Futhemoe, by (4.9) and (4.7), one has Then (4.9) and (4.) imply that ϕ(θ k 0 0 ) ε 3. (4.9) C 0 (E(θ k 0 0 /8) / E(θ k 0 0 /8)) < 4. ϕ(θ k 0 0 ) ϕ(θk 0 0 ) C (θ k 0 0 ) 4, whee C is a constant independent of k and we have used that fact that f L. One can iteate the above inequality to each ϕ(θ0 k 0 ) ( k )k ϕ( 0 ) C 4 0 ( )j (θ k j 0 ) 4 j=0 ( )k[ ϕ( 0 ) C θ ] ( )k[ ϕ( 0 ) C 4 ] 0. θ 0 Fo (0, 0 /8), we can find k such that θ0 k 0 8 < < θ0 k 0 8. Note that α 0 = log θ0, then we have θ 0 A() E() θ 0 0 D () C(θ 0 )ϕ(θ k 0 0 ) C(θ 0 )( )k[ ϕ( 0 ) C θ C(θ 0 )(θ k 0) α 0 [ ϕ( 0 ) C θ C(θ 0 ) α 0 [ ϕ( 0 ) C θ ] ] ] C0 α 0, whee C 0 = C 0 (θ 0, ϕ( 0 ), C, 0 ). By (3.) in Lemma 3., we can get simila estimates fo C(), thus the case k = 0 is poved. Assume that (4.) is tue fo m = k, i.e. When m = k, we will estimate A() E(). A() E() C 3 () D () C 0 α k. (4.0)

13 Let α β γ =, with α, β, γ (0, ) to be decided. By Poposition.4, (3.) and the assumption fo m = k, one can obtain that A() E() C 0 α A( βγ ) C 0 3α E( βγ ) 3 It follows fom (4.0) and f L that A() E() C 0 3α E( βγ ) D ( βγ ) C 0 α E( βγ ) F 3 ( βγ ) C 0 α A( βγ ) C 0 3α E( βγ ) 3 C0 3 α F ( βγ ) C 0 3α E( βγ ) D ( βγ ). C 0 α(βγ)α k C 0 3α 3(βγ)α k C 0 3 α3(βγ) C 0 3α E( βγ ) D ( βγ ) C 0 α k( α k )α C 0 α k( β γ α) 3α C 0 3 α3(βγ) C 0 3α E( βγ ) D ( βγ ). Fo the last tem, by Lemma 3. with p = and (4.0), one can deive that 3α E( βγ ) D ( βγ ) C 0 3α (βγ)α k ( 5β γα k β γα k β γ) C 0 3α (βγ)α k ( 5β γα k β γα ) k Since α k <, one can choose α = Now we define α k as : (4.) C 0 α k( α β ) 5 β 3α C 0 α k( γ α β ) 3α β. (4.) α k 00 5α k, β = 4α k 00 5α k, γ = α k. α k = min{α k ( α k )α, α k ( β γ α) 3α, 3 α 3(β γ), α k ( α β ) 5 β 3α, α k( γ α β ) 3α β} = min{ 0 4α k α k, 47 α k 300 α k, α k, 07 α k α k, 39 α k α k } 00 5α k 00 5α k 00 5α k 00 5α k 00 5α k = 0 4α k 00 5α k α k (α k, ). Thus it follows fom (4.) and (4.) that A() E() C 3 () C0 α k, (4.3) whee C() is estimated diectly by using Lemma 3.. Next, we estimate D (). Since α k <, by Lemma 3. with p =, the inequality ( D (γ) C 0 γ 5 E () D () ) C 0 γ ( E ) ()C 3 () F 3 () ( 5 C 0 γ D () γ 5 α k γ α k γ ) ( 5 C 0 γ D () γ α k ) 3

14 holds fo any γ, (0, ). Fo fixed (0, 0 8 ), let θ = 5 α k. Then we can iteate the above inequality to each D (γ m [ 5 ) C 0 γ D (γ m ) γ (γ m ) α k ] m [ 5 C 0 γ m D ( ) γ γ θj (γ m ) α k ] j=0 C 0 [ (γ m ) α k γ ( 5 α k )m α k α k [ C 0 (γ m ) α k γ 3 γ θ (γm ) αk ] C 0 (γ m ) α k, γ γ α k (γ m ) α k whee we have used α k <, and C 0 is independent of m. This yields the desied estimate fo D(). Combining this with (4.3) shows (4.0) fo the case k. Hence (4.) is poved. Finally, it follows fom the choice of α k, that α k is inceasing as k, and α k. Thus thee exists a limit ᾱ of α k satisfying 0 < ᾱ. By the definition of α k, we claim that ᾱ =. The poof is completed. Poof of Poposition.: It follows fom Poposition 4.3 that, fo any small δ > 0 and 0 < < 0 /8, thee hold u(x) C 0 δ, (4.4) By (4.4), thee exists (/, ) such that whee S = {x; x =, x 0}. Let v be the unique H solution to the Laplace equation B B B m j=0 γ θj] u(x) C 0 4 δ, (4.5) u(x) 3 dx C 0 3 δ. (4.) S u(x) dx C 0 5 δ, (4.7) v = 0, in B, v = 0, on B {x; x = 0}, v = u, on S. Now we extend v(x) fom B to B. Fo convenience, we still wite x = (x, x ) and define ṽ(x) to be the odd extension of v(x) fom B to B as: { ṽ(x v(x, x ) x 0,, x ) = v(x, x ) x < 0. 4

15 Moeove, we set Obviously, ṽ(x) satisfies ũ(x, x ) = { u(x, x ) x 0, u(x, x ) x < 0. { ṽ(x) = 0, in B, ṽ = ũ, on S = {x; x = }. Then by the standad estimates fo hamonic functions and (4.7), we get sup v sup ṽ C 0 ũ(x) dx B B / S / C 0 u(x) dx C 0 δ/. (4.8) S On the othe hand, we let w = u v H (B ), then w satisfies the stationay Stokes equation { w π = u u f, in B, w = 0, on B. Then by the classical L p estimates fo Stokes equations in Lemma., we have w L /5 (B ) C 0 u L 3 (B ) u L (B ) C 0 f L /5 (B ). This, togethe with the assumption that f L (B ) and (4.5)-(4.), yields that w L /5 (B ) C 0 5 δ C 0 5 C 0 5 δ. (4.9) Since u w v, combining (4.8) and (4.9), fo any (0, /4), we obtain that u 5 dx C0 5 δ C δ. B Taking = δ, we deive that B u 5 dx C0 α, (4.0) whee α = 5 δ 5 0 δ > 4 5 0, fo a sufficiently small δ > 0. It should be noted that (4.0) and Poposition.3 imply that u is Hölde continuity in a neighbohood of x 0, whee x 0 B 0 /8. In fact, the following aguments ae simila to that in [4] and we sketch its poof fo completeness. Let Γ = {x; x 0, x = 0}. Fo any x 0 B 0 /8, we define = dist{x 0, Γ} = dist{x 0, x } with x Γ. 5

16 Case I: 0 /3. Note that B (x 0 ) B 0 / (x ), then one has 3 B (x 0 ) u 3 dx 3 B (x 0 ) π π B (x 0 ) 3/ dx 3 C 0 ( C(0 ) D 3/ ( 0 ) F ( 0 ) ) C 0 (ε ε 3/ ) ε. Hence by (.4) in Poposition.3, we obtain, fo any 0 < < B (x 0 ) B (x 0 ) f 3 dx u 5 dx C (4.) Case II: 0 /3. Fist, fo / < < 0 /3, since B (x 0 ) B 4 (x ), the bounday estimate (4.0) yields that u 5 dx C0 u 5 dx C (4.) B (x 0 ) B B 4 (x ) On the othe hand, if /, then B (x 0 ) B 4 (x ). By (4.) in Poposition 4.3, one has 3 u 3 dx 3 π π B (x 0 ) 3/ dx 3 f 3 dx C 0 < ε, B (x 0 ) B (x 0 ) B (x 0 ) if 0 is sufficiently small. Again, one can use (.4) in Poposition.3 to deive that u 5 dx C , 0 < /. (4.3) B (x 0 ) Combining the inequalities (4.0)-(4.3) shows that fo any x 0 B 0 /8 and 0 < < 0/8, B (x 0 ) B u 5 dx C (4.4) Then the Moey lemma yields that u is Hölde continuity in a neighbohood of 0. Hence we complete the poof of Poposition.. Acknowledgments. This eseach is suppoted patially by Zheng Ge Ru Foundation, Hong Kong RGC Eamaked Reseach Gants CUHK4048/3P and CUHK404/P, a gant fom Couche Foundation, and a Focus Aea Gant fom The Chinese Univesity of Hong Kong, and Wang is patially suppoted also by the Fundamental Reseach Funds fo the Cental Univesities. Refeences [] L. Caffaelli, R. Kohn and L. Nienbeg, Patial egulaity of suitable weak solutions of the Navie-Stokes equations, Comm. Pue Appl. Math., 35(98), [] H. Dong and D. Du, Patial egulaity of solutions to the fou-dimensional Navie- Stokes equations at the fist blow-up time, Comm. Math. Phys., 73(007),

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