On the regularity of the axisymmetric solutions of the Navier-Stokes equations

Transcription

1 Math. Z. 9, () Digital Object Identifie (DOI).7/s97 On the egulaity of the axisymmetic solutions of the Navie-Stokes equations Dongho Chae, Jihoon Lee Depatment of Mathematics, Seoul National Univesity, Seoul 5-747, Koea ( {dhchae,zhlee}@math.snu.ac.k) Received: Apil ; in final fim: 6Novembe / Published online: 8 Febuay c Spinge-Velag Abstact. We obtain impoved egulaity citeia fo the axisymmetic weak solutions of the thee dimensional Navie-Stokes equations with nonzeo swil. In paticula we pove that the integability of single component of voticity o velocity fields, in tems of noms with zeo scaling dimension give sufficient conditions fo the egulaity of weak solutions. To obtain these citeia we deive new a pioi estimates fo the axisymmetic smooth solutions of the Navie-Stokes equations. Intoduction In this pape, we ae concened with the initial value poblem of the Navie- Stokes equations in [,T). u +(u )u u = p, t () div u =, () u(x, ) = u (x), () whee u =(u,u,u ) with u = u(x, t), and p = p(x, t) denote unknown fluid velocity and scala pessue, espectively, while u is a given initial velocity satisfying div u =. In the above we denote (u )u = j= u j u x j, u = j= u x. j

2 646 D. Chae, J. Lee Fo simplicity we assume that thee is no extenal foce tem in the ight hand side of (). Inclusion of the extenal foce does not povide essential difficulty in ou woks of this pape. We ecall that the Leay-Hopf weak solution of the Navie-Stokes equations is defined as a vecto field u L (,T; L ( )) L (,T; H ( )) satisfying div u =in the sense of distibution, the enegy inequality and T u(t) + t u(s) ds u a.e. t [,T] (u φ t +(u )φ u + u φ)dt + u (x) φ(x, ) =, fo all φ C (R [,T)) with div φ =. Fo given u L ( ) with div u =in the sense of distibution, a global weak solution u to the Navie-Stokes equations was constucted by Leay[4] and Hopf[], which we call the Leay-Hopf weak solution. Fo eview on the theoy of the Leay-Hopf weak solution see [7] and [5]. Regulaity of such Leay-Hopf weak solutions is one of the most outstanding open poblems in the mathematical fluid mechanics. Fo futhe discussion below we intoduce the Banach space L α, T, equipped with the nom ( T u L α, = u(t) α T dt) α, whee { ( u(t) = R u(x, t) ) if <. ess sup x u(x, t) if = In paticula, we use the same nom fo scala function u = u(x) and the vecto function u(x) =(u (x),u (x),u (x)). Taking the cul opeation of (), we obtain the evolution equation of the voticity ω = cul u, ω ω +(u )ω (ω )u =. (4) t As an appoach to the egulaity poblem Sein[](See also[]) studied the egulaity citeion of the Leay-Hopf weak solutions, and obtained that if a weak solution u belongs to L α, T whee α + <, <α, <, then u is egula, and becomes a smooth in space vaiables on (,T]. Afte Sein s wok, thee ae many impovements and developments egading the study of egulaity citeion.(see e.g. [8], [9], [], and [4].) It is found, in paticula, the Leay-Hopf weak solution becomes smooth in (x, t) if u L α, T with α +, α, <. On the othe hand, Beião

3 Axisymmetic solutions of the Navie-Stokes equations 647 da Veiga[] obtained the egulaity citeion by imposing the integability of the gadient of the velocity. Recently, one of the authos[] impoved Beião da Veiga s esult by imposing the same integability condition on two components of the voticity. Vey Recently, Neustupa et al[8] obtained egulaity citeion by imposing integability of single component of velocity field. The integability condition hee, howeve, is stonge than the Sein s one, and is not optimal in the sense of scaling consideations. Moeove, the weak solution concened in [8] is not the Leay-Hopf weak solutions, but the so called suitable weak solutions intoduced fist in []. We ae concened hee with the egulaity citeia of axisymmetic weak solutions of the Navie-Stokes equations. By an axisymmetic solution of the Navie-Stokes equations we mean a solution of the equations of the fom u(x, t) =u (, x,t)e + u θ (, x,t)e θ + u (, x,t)e in the cylindical coodinate system, whee we used the basis e =( x, x, ), e θ =( x, x, ), e =(,, ), = x + x. In the above u θ is called the swil component of the velocity u. Fo the axisymmetic solutions, we can ewite the equation () and () as follows. Du Dt ( + + )u + u uθ u θ + p =, Du θ Dt ( + + )u θ + uθ + uθ u =, Du Dt ( + + )u + p =, (u )+ (u )=, whee we denote D Dt = t + u + u. In the following, we will also use the notation fo the axisymmetic vecto field u ũ = u e + u e, and =(, ). Fo the axisymmetic vecto field u, we can compute the voticity ω = cul u as follows. ω = ω e + ω θ e θ + ω e,

4 648 D. Chae, J. Lee whee ω = u θ, ω = u θ + uθ, ωθ = u + u. Fo the study of axisymmetic solutions of the Navie-Stokes equations without swil, Ukhovskii and Yudovich [6],and independently Ladyzhenskaya [] poved the existence of genealized solutions, uniqueness and the egulaity. Recently, Leonadi, Málek, Nečas and Pokoný [] gave a efined poof. In the elated case of helical symmety, Mahalov, Titi and Leibovich [5] poved the global existence of the stong solution. Fo the axisymmetic Navie-Stokes equations with nonzeo swil component, howeve, the egulaity poblem is still open. Fo the axisymmetic Eule equations with swil, the azimuthal component of the voticity in the cylindical coodinates alone contols the blow-up of the velocity( See [5] and fo the othe studies on the axisymmetic solutions of the Eule equations, see [4], [6], [7], [], [6] and efeences theein.). Ou main esults in this pape ae the followings. Theoem Let u be an axisymmetic weak solution of the Navie-Stokes equations with u H (Ω) div u =and Q T = Ω [,T), whee Ω is bounded domain o.ifω θ is in L α, T (Q T ) whee α and satisfies <<, <α and α +, then the weak solution u is smooth in Q T = Ω (,T) whee Ω Ω. Remak. Compaing with the esult in [], we find that Theoem is an obvious impovement of the coesponding citeion theoem in [], which is, in tun, impovement of [] fo the axisymmetic case. The following theoem povides us the available a pioi estimate fo ω θ, which is new fo the axisymmetic solutions to the knowledge of the authos. Theoem If u is an axisymmetic smooth solution of the Navie-Stokes equations with initial data u L ( ) with div u =satisfying ω θ L ( ) and u θ L4 ( ), then ω θ L, T L (,T; H ( )). Fo a given axisymmetic function f on, we define the L nom by ( f L = f(, x ) d ). Theoem Let u be an axisymmetic weak solution of the Navie-Stokes equations with the initial data u satisfying u θ L4 ( ), ω θ L ( ) and u H ( ). (i) If u and u θ is in L α (,T; L (d )) whee <<, <α and α +, then the solution is smooth in R (,T).

5 Axisymmetic solutions of the Navie-Stokes equations 649 (ii) If ω θ is in L α (,T; L (d )) whee <<, <α and α +, then the solution is smooth in R (,T). Remak. One of the inteesting consequence of Theoem (ii) is that if ω θ L (,T; L (d )), then the solution becomes smooth. By the enegy inequality fo weak solutions, howeve, we know that ω L (,T; L (d )) fo the Leay-Hopf weak solutions. Thus thee is a discepancy between the available estimate and the citeion in the egion only nea the axis of symmety. We note that the nom u L α (,T ;L (d )) has zeo scaling dimension if α + =, and ω L α (,T ;L (d )) has zeo scaling dimension if α + =. The above citeia ae optimal in this sense. As an immediate coollay of Theoem (ii), we can epove the following esult on the egulaity of axisymmetic weak solutions in the case without swil, which was obtained peviously by Ukhovski and Yudovich [6] and Leonadi et al. [] using diffeent agument. Coollay Suppose u is an axisymmetic initial data without swil, satisfying the hypothesis of Theoem. Then thee exists a smooth solution on (,T). Next, we obtain egulaity citeia in tems of a single component of velocity field. Theoem 4 Let δ>be given, and set Γ δ = {x <δ}. Suppose that u is an axisymmetic weak solution, and satisfies the hypothesis of Theoem and one of the following conditions. Eithe u (i) Lα (,T; L (Γ δ )) whee α and satisfy <<, <α, and α +,o (ii) u L α (,T; L (Γ δ )) whee α and satisfy <<, <α, and α +. Then u is smooth in (,T). Remak. Compaing with the esult in [8] on the egulaity citeion in tems of single component of velocity fo the geneal case(without assumption of any symmety of solutions), we find that in Theoem 4 the integability assumption fo the single component is much weake, and optimal in the sense of scaling dimension of the noms. The oganization of this pape is the following: In Sect. we establish the Caldeon-Zygmund type of estimates fo use in the late sections. In Sect. we pove two a pioi estimates fo smooth

6 65 D. Chae, J. Lee axisymmetic solutions of the Navie-Stokes equations. Theoem is one of them. In Sect. 4, based on the pevious estimates in Sects. and, we pove Theoem, Theoem and Theoem 4 as well as Coollay. A esult simila to Theoem 4 (ii) was obtained independently at about the same time by Neustupa et al., see [9]. Kinematic estimates We ecall the following definition of the A p class and the weighted inequalities fo the singula integal opeato of the convolution type. (See Stein[] pp fo details.) Definition 5 Let p (, ). A eal valued function w(x) is said to be in A p class if it satisfies ( )( ) p sup w(x) w(x) p p p <, B B B B B whee the supemum is taken ove all balls B in. Hee p is the Hölde conjugate of p, i.e. p + p =. Fo function w(x) A p we can extend the Caldeon-Zygmund inequality fo the singula integal opeato with the integal having weight function w(x). Theoem 6 ([] p.5) Let p (, ). Suppose T is a singula integal opeato of the convolution type, and w(x) A p. Then fo f L p ( ), Tf(x) p w(x) C f(x) p w(x). Now we begin with the elementay lemma. Lemma Fo any p (, ) the function w(x) =, whee x = x +x (x,x,x ),isina p class. Poof. Let p be the Hölde conjugate of p. We need to show that ( )( ) p sup w(x) w(x) p p p <. B B B B B Let x be given. We set B = {x x x <}, and d is the distance between x and the x axis. If d, then d+ w(x) d

7 Axisymmetic solutions of the Navie-Stokes equations 65 fo all x in B. Thus, ( L := B ( 4π B )( w(x) B B B )( d 4π ) p w(x) p p p B ) p (d + ) p p p = d + d. If d<, then the cylinde {(x,x,x ) x + x < (d + ), x <} contains the ball B. x Thus it is easily seen that ( L 4π π x + d+ x ( 4π π x ( (d + ) ( ) p 7p p 7 p. +p x + d+ p p +p ) ρ ρdρ ) p ρ p p + p dρ ) p p (d + ) p + p The lemma is poved. Combining Lemma and Theoem 6, we immediately have the following. Coollay If T is a singula integal opeato of the convolution type, then Tf p C f p, whee = x + x. Lemma If u is an axisymmetic vecto field with div u =, and ω = cul u vanishes sufficiently fast nea infinity in, then the gadients of ũ and u θ e θ can be epesented as the singula integal fom. ũ(x) =Cω θ (x)e θ (x)+[k (ω θ e θ )](x), (u θ e θ (x)) = C ω(x)+[h ( ω)](x), whee the kenels (K(x)) and (H(x)) ae the matix valued functions homogeneous of degee, defining a singula integal opeato by convolution, and f g(x) = f(x y)g(y)dy denotes the standad convolution opeato. In the above, (C) and ( C) ae the constant matices.

8 65 D. Chae, J. Lee Poof. We obseve that div ũ =, and cul ũ = ω θ e θ. Similaly, div (u θ e θ ) =and cul (u θ e θ )=ω e + ω e. Then the conclusion is immediate. (See [5] and [6].) The following is a localized vesion of the well-known Caldeon-Zygmund type of inequality fo the velocity gadients and the voticity. Lemma Suppose p (, ) is given and u(x) is a divegence fee vecto field in, which is in L p (B R ) and cul u = ω is in L p (B R ). Then, we have the inequality u L p (B R ) Cu L p (B R ) + Cω L p (B R ). Poof. Choose cut off function ρ(x) C (R ) such that supp ρ B R, ρ(x) =if x R, and ρ C R. We have div (ρu) =ρdiv u + ρ u = ρ u, cul (ρu) =ρω + ρ u. Note that in geneal, fo any u W,p ( ), u L p ( ) C(div u L p ( ) + cul u L p ( )). Fom the above standad inequality we obtain the conclusion of the lemma. Lemma 4 Fo any ɛ>, ( )( ) u θ u θ lim ɛ =. Poof. We can pove this lemma simila to the poof of Coollay ( of )[]. Since we have uθ 6 C uθ Cu H, (uθ ) and u θ 4 ɛ ɛ belong to L (,T; L ( )).(We can pove this fact similaly with the poof of ( ) Lemma 4 of [] if we set g = u θ.) ( ) And we also get u δ θ 6 and ( ( )) δ u θ ae bounded fo small δ>. Wehave ( )( ) u θ u θ ɛ ( ( ) ) u θ ( ɛ ( ) u θ 6 ) ɛ ɛ. Then the lemma follows immediately.

9 Axisymmetic solutions of the Navie-Stokes equations 65 Estimates fo smooth solutions The following is athe well-known fact, and below we povide a poof of it, which could not find in the liteatue. Poposition Suppose that u is a smooth axisymmetic solution of the Navie-Stokes equations with initial data u L ( ). Let p [, ]. If u θ Lp, then u θ L (,T; L p ) and (u θ ) p L (,T; W, ). Poof. We obseve that azimuthal component of the axisymmetic Navie- Stokes equations educes to D Dt (uθ ) ( + + )(u θ )+ (u θ )=. (5) Multiplying the both sides of (5) by u θ p (u θ ) and integating ove, we obtain d u θ p + p dt R = 4(p ) p u θ p (u θ ) u θ p (u θ ) := I. (6) In (6), the integation by pats can be justified by use of the standad cut-off agument, the details of which we omitted fo simplicity. Befoe poceeding futhe, we note that f = f, fo an abitay axisymmetic function f. Since u θ is an axisymmetic smooth function which vanishes at infinity and on the x -axis, we obtain Thus we get I = 4π p u θ p d =. d u θ p + C dt u θ p =. By Gonwall s inequality, we have T sup u θ p + C u θ p dt C(u θ p ). (7) t T The case p = is immediate if we let p in the above. Using the Poposition, we now pove Theoem.

10 654 D. Chae, J. Lee Poof of Theoem. Below we denote Θ = u θ. Conside the e,e components of the Navie-Stokes equations. Du Dt ( + )u u + u = p + Θ, Du Dt ( + )u u = p. Applying the opeato (, ) to the above equations, the azimuthal component of the voticity equation is obtained as the following. Dω θ Dt +( u + u )ω θ ( + + )ω θ + ωθ = ( Θ ). (8) Suppose that ω θ L. Simila to the poof of Poposition, we fist assume ω θ decays sufficiently fast. Multiplying the both sides of (8) by 6 ω θ and integating ove,weget d ( ω θ ) + [(u + u )( ω θ )]( ω θ ) dt R u ( )ω θ ( ω θ ) u 5 (ω θ ) R ( ω θ )( + + )( ω θ ) + 4 (ω θ ) R + (ω θ ) ( + )( ) + ω θ ( )( ω θ ) R = ( Θ ) ω θ. Afte integation by pats and moe elementay computations, we have d ( ω θ ) + dt ( ω θ ) R =4 u 5 (ω θ ) +8 4 (ω θ ) R +6 ( ω θ ) ω θ + Θ ω θ := I + I + I + I 4. By use of Hölde s inequality, Young s inequality, and the Gagliado- Nienbeg inequality, we estimate I as follows.

11 Axisymmetic solutions of the Navie-Stokes equations 655 ) I u (R (ω θ ) 4 ( ) C ( ω θ ) (ω θ ) ( ) ( ) C (ω θ ) 6 ( ω θ ) 5 R ( ) (ω θ ) + C ( ω θ ) 5 5 R (ω θ ) + C ω θ 5 ( ω θ ) 9 5 R (ω θ ) + ɛ ( ω θ ) + C ɛ ω θ. Fo I and I, we estimate I C ( ω θ ) 4 (ω θ ) R ( ) ( ) C (ω θ ) ( ω θ ) R (ω θ ) + C ( ω θ ), and ( ) ( ) I C ( ( ω θ )) 4 (ω θ ) R ɛ ( ( ω θ )) + C ɛ 4 (ω θ ) R ɛ ( ( ω θ )) + (ω θ ) + C ( ω θ ). Finally, we obtain I 4 ( Θ ) + ( ω θ ). Combining the above estimates altogethe, and choosing ɛ to be sufficiently small, we obtain d ( ω θ ) + C dt ω θ ) R C ( ω θ ) + C (ω θ ) + C ( (Θ )).

12 656 D. Chae, J. Lee Applying Gonwall s inequality, we get T sup ω θ + C ( ω θ ) dt t T C(T ) ω θ + C T ω θ dt + C T Θ dt C(T, ω θ, u, u θ 4 ). (9) The last inequality of (9) is fom the enegy inequality and the Poposition. Similaly to the poof of Poposition, we can justify the integation by pats in the above computations the poof by using the standad cut-off function technique. 4 Poof of egulaity citeia We can constuct weak solutions in vaious ways.( See [6], [], and [6].) Fo example( [] and [6]), it is possible to constuct weak solutions by consideing the following egulaized equations. Let ρ δ be the standad mollifie in, ρ δ (x) = ρ( x δ δ ), whee ρ C (R ), ρ and supp ρ { x }. We egulaize the Navie-Stokes equations as follows. ( t +(ρ δ u δ ) )u δ + p δ =, div u δ =, u δ = ρ δ u. Fo each δ>, the above equations have a global axisymmetic smooth solution if u is axisymmetic, and belongs to L ( ).Asδ, we obtain an axisymmetic Leay-Hopf weak solution as defined in the intoduction. In the poof of Theoem below, we povide only a pioi estimates. The coesponding estimates fo weak solutions can be justified by the above egulaization pocedue. Poof of Theoem. Let u be an axisymmetic smooth solution of the Navie-Stokes equations. Taking cul on the both sides of the Navie-Stokes equations, then we obtain the following equations. ω t ω +(u )ω (ω )u =. We may assume Q T = B R [,T)={(x, t) t<t, x < R} whee R> and Q T = B R [,T)={(x, t) t<t, x < R} fo simplicity. Fo the geneal domain the poof is simila. Let η = η() be a smooth cut off function which has a suppot in B R, η, and η =

13 Axisymmetic solutions of the Navie-Stokes equations 657 on B R. By multiplying ωη k on the both sides of the above equations, and integating ove, we get the following equation. d ωη k + (ωη k ) dt B R B R +k ( j η)ω i ( j ω i )η k i,j= B R + ω η k ( η k ) + [u (ωη k )](ωη k ) B R B R (u η k )ω ωη k B R (η k ω )u ωη k =. B R Fom which it follows d ωη k + (ωη k ) dt B R B R = ( η k ) ω η k k ( j η)(ωη k ) j (ωη k ) B R j= B R +k ω η k (u )η + (η k ω )u ωη k B R B R := {} + {} + {} + {4}. Fist, {} is estimated easily. {} C ω, B R whee C = C(η). On the othe hand, {} and {} can be estimated by vitue of the Young inequality, the Hölde inequality and the Gagliado-Nienbeg inequality. {} C ωη k (ωη k ) B R ɛ (ωη k ) + C ɛ ω η k B R B R ( ) ( ) k ɛ (ωη k ) + C ɛ ω k ( ω η k ) k B R B R BR ɛ (ωη k ) + C ɛ ω + C ɛ ( ω η k ), B R B R B R

14 658 D. Chae, J. Lee and {} C u ω η k B R ( ) ( ) C u ω 4 η (k ) B R BR ( ) k ω + C ( ω η k ) (k ) k k B R B R ( ) k ω + C ( ω η k ) (k ) B R B R ( ) k ω η k ) (k ) B R C ω + ɛ ω η k + C ɛ ( ω η k ). B R B R B R To estimate {4}, we compute {4} = [(η k ω B R ηk ω θ θ + η k ω )(u e + u θ e θ + u e )] (η k ω e + η k ω θ e θ + η k ω e ) = η k ω ( u )η k ω B R ηk ω θ u θ η k ω +η k ω ( u )η k ω + η k ω ( u θ )η k ω θ + ηk ω θ u η k ω θ + η k ω ( u θ )η k ω θ +η k ω ( u )η k ω + η k ω ( u )η k ω := I I 8. Now we estimate I,..., I 8. I Cω θ L p (B R )η k ω L p p (B R ) Similaly, we get the following. Cω θ L p (B R ) (η k p ω) p ηk ω p p C ɛ ω θ p L p (B R ) ηk ω + ɛ (η k ω). p I, I 6, I 7, I 8 C ɛ ω θ p L p (B R ) ηk ω + ɛ (η k ω).

15 Axisymmetic solutions of the Navie-Stokes equations 659 Since uθ = u θ + ω,wehave I = η k ω θ ( u θ )η k ω + η k ω θ ω η k ω B R B R = ω θ ( (η k u θ ))η k ω k ( η)u θ ω θ ω η k B R B R + η k ω θ ω η k ω B R := I + I + I. In the poof of Lemma, we know that Thus we get (η k u) Cu + Cη k ω. p I, I, I C ɛ ω θ p L p (B R ) ηk ω + ɛ (η k ω). Similaly, I 4 and I 5 can be estimated. Putting togethe the above estimates, we get d ωη k + C (ωη k ) dt B R B R p Cω θ p L p (B R ) ηk ω L (Ω R ) + Cω L (B R ). Using Gonwall s inequality, we have sup t T C T ωη k (t) L (B R ) + C (ωη k ) L (B R ) dt ( ω + T ω L (B R ) dt ) exp ( T C ) ω θ p p L p (B R ) dt. Applying Lemma and Lemma, sup t [,T ) u(t) L (B R ) <Ci.e. u L (,T; L 6 (B R )). Thus we get the inteio egulaity by applying Sein s citeion. Remak 4. Fo late use, we note that if we set η in the poof of Theoem then we get the following a pioi estimate fo the whole domain. T ( T ) sup ω + C ω dt Cω exp C ω θ p p p dt. t T

16 66 D. Chae, J. Lee Fo the poof of Theoem and Theoem 4, we will use the standad continuation pinciple fo the local stong solution. Poof of Theoem. Poof of (i): Fist note that thee exists maximal time T such that thee is a unique classical solution u C((,T ); L s ( )), s> and u(τ) s C (T τ) s s with constant C, which is independent of T and s.( See [9].) We povide a pioi estimate fo the smooth axisymmetic solution. We wite the e and e θ components of the voticity equation as follows. Dω Dt ( + ) + ω + ω (ω + ω )u =. () Dω θ ( Dt + ) + ω θ + ωθ + uθ ω {(ω + ω ) u θ + ωθ u } =. () Multiplying the both sides of () and () by ω and ω θ espectively, and integating ove (, ) (, ), we get the following equalities. d (ω ) +(ω θ ) d + dt ω + ω θ d (ω θ ) (ω ) + d + d = ( ω )ω d [(u + u )ω ]ω d + [(ω + ω )u ]ω d + ( ω θ )ω θ d [(u + u )ω θ ]ω θ d + [(ω + ω )u θ ]ω θ d ω ω θ u θ d + ωθ u ω θ d = ω ω d u (ω ) d + ω u ω d + ω θ ω θ d + u u (ωθ ) θ d u ω d

17 Axisymmetic solutions of the Navie-Stokes equations 66 u θ u ω d + ω u θ ω θ d u θ u θ ω θ d u θ u θ ω θ d uθ ω ω θ d = ω ω d u (ω ) d + ω u ω d + ω θ ω θ d + u (ωθ ) d u θ u ω d u θ ω ω θ d u θ u ω d := I + I + I + I 4 + I 5 + I 6 + I 7 + I 8. () We will estimate I,...,I 8 as follows. Fist, it is easily seen fom Young s inequality that I (ω ) d + ( ω ) d, and I 4 (ω θ ) d + ( ω θ ) d. By means of the Hölde inequality, the Young inequality and the Gagliado- Nienbeg inequality, we get ( (ω ) ) ( ) I d (u ) (ω ) d (ω ) ɛ d + C ɛ u L ω L (ω ) ɛ d + C ɛ u L ω ( ) ω L 4 L (ω ) ɛ d + C ɛ u L ω L + ɛ ω L, I = u ( ω )ω d

18 66 D. Chae, J. Lee and ɛ ( ω ) d + ɛ ω L + C ɛ u L ω L, (ω θ ) I 5 ɛ d + C ɛ (u ) (ω θ ) d (ω θ ) ɛ d + ɛ ω θ L + C ɛ u L ωθ L. On the othe hand, we note that u Lp Cω θ Lp, which follows fom Coollay and Lemma. By use of the above fact, the Young inequality, the Hölde inequality, and the Gagliado-Nienbeg inequality again, we ae lead to (ω I 6 C ɛ u θ u ) L + ɛ d (ω C ɛ u θ L u ) + ɛ L d (ω C ɛ u θ L ω θ ) + ɛ L d C ɛ u θ L ω θ ( ) ω L θ 4 (ω L + ɛ ) d C ɛ u θ L ωθ L + ɛ ω (ω θ ) L + ɛ d. Similaly to the above, we estimate I 7 C ɛ u θ L ω L + ɛ ω (ω ) L + ɛ d. Since it is not easy to estimate I 8 diectly, we integate by pats fist. I 8 = u θ u ω d + u θ u ω d = ω u ω d u θ u ω d u θ u ω d = ω u ω d u θ u ω d + u θ u ω d := I 8 + I 8 + I 8.

19 Axisymmetic solutions of the Navie-Stokes equations 66 As peviously, thanks to the Young inequality, the Hölde inequality, and the Gagliado-Nienbeg inequality, we obtain I8 ɛ ( ω ) d + C ɛ (u ) (ω ) d ɛ ( ω ) d + ɛ ω L + C ɛ u L ω L, I8 ɛ ɛ ( ω ) d + C ɛ (u θ ) ( u ) d ( ω ) d + ɛ ω θ L + C ɛ u θ L ωθ L, and I8 ɛ ( ω ) d + ɛ ω θ L + C ɛ u θ L ωθ L. Choosing ɛ sufficiently small, we have d (ω ) +(ω θ ) d + C dt (ω θ ) +C + (ω ) d L ω + ω θ d C(u + u θ L )(ωθ L + ω L). () Using Gonwall s inequality, we get the following a pioi estimates. sup (ω (t) L + ω θ (t) L)+C t [,T ) ( ) +C T ω L + ω θ C(ω L + ω θ L) L ( + exp T ω L + ω θ Ldt dt (4) ( T C u L )) + u θ L dt. Applying Theoem and above a pioi estimates, we get ω θ = ω θ d + ω θ d ω θ d + ω θ 7 d. In the above inequality, ight hand side is bounded by some constant by Theoem and (5). By Remak 4 below the end of the poof of the Theoem

20 664 D. Chae, J. Lee and the Sobolev embedding theoem, we find that u L ((,T );L 6 ( )) is bounded by some constant. Thus we can continue ou local smooth solution until T by the standad continuation agument. Poof of (ii) : The θ-component of the axisymmetic Navie-Stokes equations is as follows. t u θ +(u + u )u θ ( + + )u θ + uθ + uθ u =. (5) Multiplying the both sides of (5) and (8) by u θ u θ and ω θ, and integating ove (, ) (, ),weget d u θ 4 d + d ω θ d dt dt + (u θ ) d + ω θ d u θ 4 (ω θ + d ) + d (u θ ) (u θ ) d + 5 u θ 4 u 4 d + ω θ ω θ d + u (ωθ ) d + (u θ ) ω θ d := I + I + I + I 4 + I 5. Using Young s inequality, we estimate I (u θ ) + L (u θ ) L, I ω θ + L (ω θ ) L, and I 5 (u θ ) L + ω θ. L On the othe hand, by vitue of the Gagliado inequality and Hölde inequality,wehave and I Cω θ L I 4 Cω θ (uθ ) L + ɛ (u θ ) L L ωθ L + ɛ ω θ L.

21 Axisymmetic solutions of the Navie-Stokes equations 665 Choosing ɛ sufficiently small and using Gonwall s inequality, we obtain that T sup (u θ 4 L4+ω θ L) C(u θ 4 L4+ω θ L) exp(c ω θ t T By the same agument below (5) in the poof of (i), we obtain (ii). L dt). Poof of Coollay. Fo the axisymmetic solution fo initial data without swil, u θ, and thus ω. In view of (ii) of Theoem, it suffices to pove T ω θ d dt <. (6) The main difficulty of the global existence is that we do not know if ωθ L (,T; L ( ))( See [].). Choose ωθ as a test function of the voticity equation and integate ove. Since ω, ω and u θ vanish in this case, we get the following inequality by integating by pats. d ω θ d + (ω θ ) dt d + ω θ d C (ω θ ) d. The Gonwall inequality gives us that ω θ L (,T; L ( )). Thus, (6) is now poved. Poof of Theoem 4. (i) Let T be the maximal time as in the poof of Theoem. We wish to multiply the both sides of (8) and (5) by ωθ and (uθ ), 4 espectively, and integate ove. Indicated in the poof of Coollay ( See also [].), howeve, we do not know if they belong to L (,T ; L ( ). To justify the pocedue, we multiply the both sides of (8) and (5) by ωθ ɛ and (uθ ) with sufficiently small ɛ>, espectively and integate ove. 4 ɛ Then we have the following by the integation by pats. ( ) d ω θ dt + d ɛ 4 dt ( ) + ω θ + ɛ ( u θ ɛ 4 ( u θ ɛ 4 ) 4 )

22 666 D. Chae, J. Lee ( ɛ ɛ ) ( ) ω θ 4 + ɛ ( ɛ ) R (u θ ) 4 ɛ 4 6 ɛ ɛ u ( ) ω θ ( ) u θ + ɛ ω θ ɛ ɛ +( ɛ) u ( ) u θ 4 := I ɛ + I + I. 4 Note that bounday tems vanish (see [] and Lemma 4). By vitue of the Hölde inequality, the Gagliado-Nienbeg inequality, and the Young inequality, we obtain I ɛ u ω θ and C u ( I u θ C ɛ 4 ω θ ɛ I ( ɛ) u C u ɛ ) ω θ ɛ + ( 4 ω θ ω θ ɛ + ( 8 u θ ( u θ ( u θ ɛ 4 ɛ 4 ɛ 4 ) ) ) ɛ + ( 8 u θ, ɛ 4 ), ) By the use of the Gonwall inequality, we have sup ω θ (t) t [,T ] ɛ + 4 sup u θ 4 (t) t [,T ] ɛ 4 4 T ( ) +C ω θ ( ) + ɛ u θ dt (7) ɛ 4 C ω θ ( ) u θ ( ( T + + exp u )) dt ɛ ɛ 4.

23 Axisymmetic solutions of the Navie-Stokes equations 667 If ɛ, then we have the following by the Lebesgue dominated convegence theoem. sup ω θ (t) t [,T ] + 4 sup u θ 4 (t) t [,T ] 4 T ( ) +C ω θ ( ) + u θ dt (8) C ω θ ( ) u θ ( ( T + + exp u )) dt In ode to complete the poof, we choose cut-off function η() such that η on and supp η { }. Multiplying the both sides of ( ) (8) and (5) by ωθ ɛ η4 and u θ, 4 ɛ η espectively, and integate ove. Since the pocedue is athe standad, we omit the details hee. then we can deduce the following inequality. sup ω θ (t) η + t [,T ] 4 sup u θ 4 (t) η t [,T ] 4 T ( ) +C ω θ ( ) η + u θ η dt C ω θ ( ) η u θ ( ( T + η + exp u )) dt +C T u θ 4 4dt + C T L (Γ ) ω θ dt. (9) The ight hand side of (9) is contolled by the initial datum u, u θ 4 and u L α (,T ;L (Γ, which is finite by hypothesis. Similaly to the poof )) of Theoem, we conside the following estimate. ω θ = ω θ d + ω θ d C (ω θ ) d + C ω θ 7 d. Thus it follows that u L (,T ; L 6 ( )), which implies that we can continue ou local smooth solution by the standad continuation agument.

24 668 D. Chae, J. Lee Poof of (ii) is simila to that of (i). Fo simplicity, we do not pesent the cut-off function technique. It will be shown that the integability of u θ is contolled by that of u. Fist we multiply the both sides of the azimuthal component of the Navie-Stokes equations by u θ u θ and integate ove, then we get d u θ dt 4 u θ R u θ 4 R + = u θ 4 u := I. () By means of the Young inequality, the Hölde inequality, and the Gagliado- Nienbeg inequality, it is established that ( u θ ) I ɛ + C u θ 4 (u ) R ( u θ ) ɛ + Cu ( u θ ) ( u θ ) ɛ + Cu u θ ( ) u θ 6 ( u θ ) ɛ + ɛ u θ + Cu u θ. Thus fom (), we have the following inequality. ( d u θ 4 + C 4 dt u u θ θ ) + C Cu (u θ ). Using Gonwall s inequality, we find that the following is immediate. T 4 sup u θ C ( u θ ) dt () t T T ( u θ ) ( T ) +C dt C(u θ 4 4) exp u dt. The ight hand side of () is contolled by u θ 4 and T u dt. Multiplying the both sides of (8) by ω θ and integating ove,weget ( ) d (ω θ ) + dt ω ω θ θ + R = ( (u θ ) ) ωθ R + u (ωθ ) := I + I. ()

25 Axisymmetic solutions of the Navie-Stokes equations 669 As peviously, by means of the Hölde inequality, the Young inequality, and the Gagliado-Nienbeg inequality, it can be obtained that ( ) ω θ I ɛ + C ɛ ( u θ ). Due to the hypothesis on the integability of u, it is deived that ( ) ω θ I ɛ + C ɛ (u ) (ω θ ) R ( ) ω θ ɛ + ɛ ω θ + C ɛ u ω θ. Choosing ɛ sufficiently small, the inequality () educes to the following. ( ) d (ω θ ) + C dt (ω ω θ ) θ + C C ( u θ ) + Cu ω θ. Fo the completion of the poof we choose η as in the poof of (i). Multiplying the both sides of (5) and of (i). Multiplying the both sides of (5) and (8) by u θ u θ η 4 and ω θ η 4, espectively, and poceeding similaly, we get T sup ω θ η + C (ω θ η ) t T T ( ω θ η ) +C dt ( T ) C ωη θ + ( u θ η) + ω θ dt ( T exp C ) u L (Γ ) dt. () The ight hand side of () is contolled by the initial datum u H, u θ 4 and the nom u L α (,T ;L (Γ )), which is finite by hypothesis. Similaly to the case (i), we conclude that u is egula. Acknowledgements. We deeply thank to the anonymous efeee fo vey caeful eading of the pape, and many helpful and constuctive citicism. This eseach is suppoted patially by KOSEF(K ) and BSRI-MOE.

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