On the regularity of the axisymmetric solutions of ... - Semantic Scholar

Math. Z. 239, 645–671 (2002) Digital Object Identifier (DOI) 10.1007/s002090100317

On the regularity of the axisymmetric solutions of the Navier-Stokes equations Dongho Chae, Jihoon Lee Department of Mathematics, Seoul National University, Seoul 151-747, Korea (e-mail : {dhchae,zhlee}@math.snu.ac.kr) Received: 11 April 2000; in final firm: 26 November 2000 / c Springer-Verlag 2002 Published online: 28 February 2002 –

Abstract. We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations.

1 Introduction In this paper, we are concerned with the initial value problem of the NavierStokes equations in R3 × [0, T ). ∂u + (u · ∇)u − ∆u = −∇p, ∂t div u = 0, u(x, 0) = u0 (x),

(1) (2) (3)

where u = (u1 , u2 , u3 ) with u = u(x, t), and p = p(x, t) denote unknown fluid velocity and scalar pressure, respectively, while u0 is a given initial velocity satisfying div u0 = 0. In the above we denote (u · ∇)u =

3 j=1

∂u uj , ∂xj

∆u =

3 ∂2u j=1

∂x2j

.

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D. Chae, J. Lee

For simplicity we assume that there is no external force term in the right hand side of (1). Inclusion of the external force does not provide essential difficulty in our works of this paper. We recall that the Leray-Hopf weak solution of the Navier-Stokes equations is defined as a vector field u ∈ L∞ (0, T ; L2 (R3 )) ∩ L2 (0, T ; H 1 (R3 )) satisfying div u = 0 in the sense of distribution, the energy inequality t 2 ∇u(s)22 ds ≤ u0 22 a.e. t ∈ [0, T ] u(t)2 + 2 0

and T 0

R3

(u · φt + (u · ∇)φ · u + u · ∆φ)dxdt +

R3

u0 (x) · φ(x, 0)dx = 0,

for all φ ∈ C0∞ (R3 × [0, T ))3 with div φ = 0. For given u0 ∈ L2 (R3 ) with div u0 = 0 in the sense of distribution, a global weak solution u to the Navier-Stokes equations was constructed by Leray[14] and Hopf[10], which we call the Leray-Hopf weak solution. For review on the theory of the Leray-Hopf weak solution see [7] and [25]. Regularity of such Leray-Hopf weak solutions is one of the most outstanding open problems in the mathematical fluid mechanics. For further discussion below we introduce the Banach space Lα,γ T , equipped with the norm uLα,γ = T

where u(t)γ =

0

T

u(t)αγ dt

α1

,

1 |u(x, t)|γ dx γ if 1 ≤ γ < ∞ . ess supx∈R3 |u(x, t)| if γ = ∞ R3

In particular, we use the same norm for scalar function u = u(x) and the vector function u(x) = (u1 (x), u2 (x), u3 (x)). Taking the curl operation of (1), we obtain the evolution equation of the vorticity ω = curl u, ∂ω − ∆ω + (u · ∇)ω − (ω · ∇)u = 0. ∂t

(4)

As an approach to the regularity problem Serrin[22](See also[20]) studied the regularity criterion of the Leray-Hopf weak solutions, and obtained that 2 3 if a weak solution u belongs to Lα,γ T where α + γ < 1, 2 < α ≤ ∞, 3 < γ ≤ ∞, then u is regular, and becomes a smooth in space variables on (0, T ]. After Serrin’s work, there are many improvements and developments regarding the study of regularity criterion.(See e.g. [8], [9], [11], and [24].) It is found, in particular, the Leray-Hopf weak solution becomes smooth in (x, t) if u ∈ 2 3 Lα,γ ao T with α + γ ≤ 1, 2 ≤ α ≤ ∞, 3 < γ ≤ ∞. On the other hand, Beir˜

Axisymmetric solutions of the Navier-Stokes equations

647

da Veiga[1] obtained the regularity criterion by imposing the integrability of the gradient of the velocity. Recently, one of the authors[3] improved Beirão da Veiga’s result by imposing the same integrability condition on two components of the vorticity. Very Recently, Neustupa et al[18] obtained regularity criterion by imposing integrability of single component of velocity field. The integrability condition here, however, is stronger than the Serrin’s one, and is not optimal in the sense of scaling considerations. Moreover, the weak solution concerned in [18] is not the Leray-Hopf weak solutions, but the so called suitable weak solutions introduced first in [2]. We are concerned here with the regularity criteria of axisymmetric weak solutions of the Navier-Stokes equations. By an axisymmetric solution of the Navier-Stokes equations we mean a solution of the equations of the form u(x, t) = ur (r, x3 , t)er + uθ (r, x3 , t)eθ + u3 (r, x3 , t)e3 in the cylindrical coordinate system, where we used the basis er = (

x1 x2 x2 x1 , , 0), eθ = (− , , 0), e3 = (0, 0, 1), r = r r r r

x21 + x22 .

In the above uθ is called the swirl component of the velocity u. For the axisymmetric solutions, we can rewrite the equation (1) and (2) as follows. ˜ r Du 1 1 1 − (∂r2 + ∂32 + ∂r )ur + 2 ur − uθ uθ + ∂r p = 0, Dt r r r θ ˜ 1 1 1 Du − (∂r2 + ∂32 + ∂r )uθ + 2 uθ + uθ ur = 0, Dt r r r ˜ 1 Du3 − (∂r2 + ∂32 + ∂r )u3 + ∂3 p = 0, Dt r ∂r (rur ) + ∂3 (ru3 ) = 0, where we denote

˜ D ∂ = + ur ∂r + u3 ∂3 . Dt ∂t In the following, we will also use the notation for the axisymmetric vector field u u ˜ = ur er + u3 e3 , and

˜ = (∂r , ∂3 ). ∇

For the axisymmetric vector field u, we can compute the vorticity ω = curl u as follows. ω = ω r er + ω θ eθ + ω3 e3 ,

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D. Chae, J. Lee

where ω r = −∂3 uθ ,

ω3 = ∂r uθ +

uθ , r

ω θ = −∂r u3 + ∂3 ur .

For the study of axisymmetric solutions of the Navier-Stokes equations without swirl, Ukhovskii and Yudovich [26] ,and independently Ladyzhenskaya [12] proved the existence of generalized solutions, uniqueness and the regularity. Recently, Leonardi, Málek, Neˇcas and Pokorný [13] gave a refined proof. In the related case of helical symmetry, Mahalov, Titi and Leibovich [15] proved the global existence of the strong solution. For the axisymmetric Navier-Stokes equations with nonzero swirl component, however, the regularity problem is still open. For the axisymmetric Euler equations with swirl, the azimuthal component of the vorticity in the cylindrical coordinates alone controls the blow-up of the velocity( See [5] and for the other studies on the axisymmetric solutions of the Euler equations, see [4], [16], [17], [21], [26] and references therein.). Our main results in this paper are the followings. Theorem 1 Let u be an axisymmetric weak solution of the Navier-Stokes equations with u0 ∈ H 2 (Ω) div u0 = 0 and QT = Ω × [0, T ), where Ω is bounded domain or R3 . If ω θ is in Lα,γ T (QT ) where α and γ satisfies 3 2 3 < γ < ∞, 1 < α ≤ ∞ and + 2 α γ ≤ 2, then the weak solution u is smooth in QT = Ω × (0, T ) where Ω ⊂⊂ Ω. Remark 1. Comparing with the result in [3], we find that Theorem 1 is an obvious improvement of the corresponding criterion theorem in [3], which is, in turn, improvement of [1] for the axisymmetric case. The following theorem provides us the available a priori estimate for ω θ , which is new for the axisymmetric solutions to the knowledge of the authors. Theorem 2 If u is an axisymmetric smooth solution of the Navier-Stokes equations with initial data u0 ∈ L2 (R3 ) with div u0 = 0 satisfying r3 ω0θ ∈ ∩ L2 (0, T ; H 1 (R3 )). L2 (R3 ) and ruθ0 ∈ L4 (R3 ), then r3 ω θ ∈ L∞,2 T ˜ γ norm by For a given axisymmetric function f on R3 , we define the L f L˜ γ =

∞

−∞

0

∞

γ

|f (r, x3 )| drdx3

1 γ

.

Theorem 3 Let u be an axisymmetric weak solution of the Navier-Stokes equations with the initial data u0 satisfying ruθ0 ∈ L4 (R3 ), r3 ω0θ ∈ L2 (R3 ) and u0 ∈ H 2 (R3 ). (i) If ur and uθ is in Lα (0, T ; Lγ (drdx3 )) where 2 < γ < ∞, 2 < α ≤ ∞ and α1 + γ1 ≤ 12 , then the solution is smooth in R3 × (0, T ).


649

(ii) If ω θ is in Lα (0, T ; Lγ (drdx3 )) where 1 < γ < ∞, 1 < α ≤ ∞ and 1 1 3 α + γ ≤ 1, then the solution is smooth in R × (0, T ). Remark 2. One of the interesting consequence of Theorem 3 (ii) is that if ω θ ∈ L2 (0, T ; L2 (drdx3 )), then the solution becomes smooth. By the energy inequality for weak solutions, however, we know that ω ∈ L2 (0, T ; L2 (rdrdx3 )) for the Leray-Hopf weak solutions. Thus there is a discrepancy between the available estimate and the criterion in the region only near the axis of symmetry. We note that the norm uLα (0,T ;Lγ (drdx3 )) has zero scaling dimension if = 12 , and ωLα (0,T ;Lγ (drdx3 )) has zero scaling dimension if α1 + γ1 = 1. The above criteria are optimal in this sense. As an immediate corollary of Theorem 3 (ii), we can reprove the following result on the regularity of axisymmetric weak solutions in the case without swirl, which was obtained previously by Ukhovski and Yudovich [26] and Leonardi et al. [13] using different argument. 1 1 α+γ

Corollary 1 Suppose u0 is an axisymmetric initial data without swirl, satisfying the hypothesis of Theorem 3. Then there exists a smooth solution on R3 × (0, T ). Next, we obtain regularity criteria in terms of a single component of velocity field. Theorem 4 Let δ > 0 be given, and set Γδ = {x ∈ R3 | r < δ}. Suppose that u is an axisymmetric weak solution, and satisfies the hypothesis of Theorem 3 and one of the following conditions. Either r (i) ur ∈ Lα (0, T ; Lγ (Γδ )) where α and γ satisfy 32 < γ < ∞, 1 < α ≤ ∞, and α2 + γ3 ≤ 2, or (ii) ur ∈ Lα (0, T ; Lγ (Γδ )) where α and γ satisfy 3 < γ < ∞, 2 < α ≤ ∞, and α2 + γ3 ≤ 1. Then u is smooth in R3 × (0, T ). Remark 3. Comparing with the result in [18] on the regularity criterion in terms of single component of velocity for the general case(without assumption of any symmetry of solutions), we find that in Theorem 4 the integrability assumption for the single component is much weaker, and optimal in the sense of scaling dimension of the norms. The organization of this paper is the following: In Sect. 2 we establish the Calderon-Zygmund type of estimates for use in the later sections. In Sect. 3 we prove two a priori estimates for smooth

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axisymmetric solutions of the Navier-Stokes equations. Theorem 2 is one of them. In Sect. 4, based on the previous estimates in Sects. 2 and 3, we prove Theorem 1, Theorem 3 and Theorem 4 as well as Corollary 1. A result similar to Theorem 4 (ii) was obtained independently at about the same time by Neustupa et al., see [19]. 2 Kinematic estimates We recall the following definition of the Ap class and the weighted inequalities for the singular integral operator of the convolution type. (See Stein[23] pp. 194–217 for details.) Definition 5 Let p ∈ (1, ∞). A real valued function w(x) is said to be in Ap class if it satisfies sup

B⊂R3

1 |B|

B

w(x)dx

1 |B|

− pp

B

w(x)

p p

dx

< ∞,

where the supremum is taken over all balls B in R3 . Here p is the Hölder conjugate of p, i.e. p1 + p1 = 1. For function w(x) ∈ Ap we can extend the Calderon-Zygmund inequality for the singular integral operator with the integral having weight function w(x). Theorem 6 ([23] p.205) Let p ∈ (1, ∞). Suppose T is a singular integral operator of the convolution type, and w(x) ∈ Ap . Then for f ∈ Lp (R3 ),

p

R3

|T f (x)| w(x)dx ≤ C

R3

|f (x)|p w(x)dx.

Now we begin with the elementary lemma. Lemma 1 For any p ∈ (1, ∞) the function w(x) = √ (x1 , x2 , x3 ), is in Ap class.

1 , x21 +x22

where x =

Proof. Let p be the Hölder conjugate of p. We need to show that sup

B⊂R3

1 |B|

B

w(x)dx

1 |B|

B

− pp

w(x)

dx

p p

< ∞.

Let x0 ∈ R3 be given. We set B = {x ∈ R3 ||x − x0 | < r}, and d is the 1 1 distance between x0 and the x3 axis. If d ≥ 2r , then d+r ≤ w(x) ≤ d−r


for all x in B. Thus,

651

p p 1 − pp L := w(x)dx w(x) dx |B| B B p p p 3 3 1 dx ≤ (d + r) p dx 3 3 4πr B d − r 4πr B d+r ≤ 3. = d−r 1 |B|

If d < 2r, then the cylinder {(x1 , x2 , x3 ) ∈ R3 |x21 + x22 < (d + r)2 , |x3 − x03 | < r} contains the ball B. Thus it is easily seen that

x0 +r d+r 3 3 1 · ρdρdx3 L≤ 2π 4πr3 ρ x03 −r 0 p

x0 +r d+r p 3 p 3 +1 p 2π ρ dρdx × 3 4πr3 x03 −r 0 p p p 3 3 p +2 (d + r) p 2r ≤ 3 (d + r)2r 3 2r 2r p + 2p p p 27p ≤ 27 . p + 2p The lemma is proved.

Combining Lemma 1 and Theorem 6, we immediately have the following. Corollary 2 If T is a singular integral operator of the convolution type, then 1 p1 |T f | dx ≤ C |f |p dx, r r R3 R3 2 2 where r = x1 + x2 . Lemma 2 If u is an axisymmetric vector field with div u = 0, and ω = curl u vanishes sufficiently fast near infinity in R3 , then the gradients of u ˜ and uθ eθ can be represented as the singular integral form. ∇˜ u(x) = Cω θ (x)eθ (x) + [K ∗ (ω θ eθ )](x), ˜ (x) + [H ∗ (˜ ω )](x), ∇(uθ eθ (x)) = C˜ ω where the kernels (K(x)) and (H(x)) are the matrix valued functions homogeneous of degree −3, defining a singular integral operator by convolution, and f ∗ g(x) = R3 f (x − y)g(y)dy denotes the standard convolution op˜ are the constant matrices. erator. In the above, (C) and (C)

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D. Chae, J. Lee

Proof. We observe that div u ˜ = 0, and curl u ˜ = ω θ eθ . Similarly, div (uθ eθ ) θ r = 0 and curl (u eθ ) = ω er + ω3 e3 . Then the conclusion is immediate. (See [5] and [16].) The following is a localized version of the well-known Calderon-Zygmund type of inequality for the velocity gradients and the vorticity. Lemma 3 Suppose p ∈ (1, ∞) is given and u(x) is a divergence free vector field in R3 , which is in Lp (B2R ) and curl u = ω is in Lp (B2R ). Then, we have the inequality ∇uLp (BR ) ≤ CuLp (B2R ) + CωLp (B2R ) . Proof. Choose cut off function ρ(x) ∈ C0∞ (R3 ) such that supp ρ ⊂ B2R , ρ(x) = 1 if |x| ≤ R, and |∇ρ| ≤ C R. We have div (ρu) = ρdiv u + ∇ρ · u = ∇ρ · u, curl (ρu) = ρω + ∇ρ × u. Note that in general, for any u ∈ W 1,p (R3 ), ∇uLp (R3 ) ≤ C(div uLp (R3 ) + curl uLp (R3 ) ). From the above standard inequality we obtain the conclusion of the lemma. Lemma 4 For any ) > 0, lim

∞

r→0 −∞

∂r

uθ r

uθ r

3

r dx3 = 0.

Proof. We can prove this lemma similar to the proof of Corollary 1 of[13]. θ (uθ )3 uθ uθ 1 Since we have r 6 ≤ C∇ r 2 ≤ CuH 2 , r4− and r1− ∂r ur be-

long to L2 (0, T ; L2 (R3 )).(We can prove this fact similarly with the proof of

θ 6

θ 3 ∞ dx3 Lemma 4 of [13] if we set g = ur .) And we also get −∞ rδ ur ∞ δ uθ 2 and −∞ r ∂r r dx3 are bounded for small δ > 0. We have θ 3 u uθ ∂r r dx3 r r −∞ 1 1

θ 2 2 2 ∞ ∞ θ 6 u u r 2 dx3 ∂r 2 dx ≤ r r2. 3 r r −∞ −∞

∞

Then the lemma follows immediately.


653

3 Estimates for smooth solutions The following is rather well-known fact, and below we provide a proof of it, which could not find in the literature. Proposition 1 Suppose that u is a smooth axisymmetric solution of the Navier-Stokes equations with initial data u0 ∈ L2 (R3 ). Let p ∈ [2, ∞]. If p ruθ0 ∈ Lp , then ruθ ∈ L∞ (0, T ; Lp ) and (ruθ ) 2 ∈ L2 (0, T ; W 1,2 ). Proof. We observe that azimuthal component of the axisymmetric NavierStokes equations reduces to ˜ D 1 2 (ruθ ) − (∂r2 + ∂32 + ∂r )(ruθ ) + ∂r (ruθ ) = 0. Dt r r

(5)

Multiplying the both sides of (5) by |ruθ |p−2 (ruθ ) and integrating over R3 , we obtain 1 d 4(p − 1) θ p ˜ θ | p2 |2 dx |ru | dx + |∇|ru 2 p dt R3 p R3 2 ∂r (ruθ )|ruθ |p−2 (ruθ )dx := I. =− (6) r 3 R In (6), the integration by parts can be justified by use of the standard cut-off argument, the details of which we omitted for simplicity. Before proceeding further, we note that ˜ 2 , ∇f 2 = ∇f for an arbitrary axisymmetric function f . Since ruθ is an axisymmetric smooth function which vanishes at infinity and on the x3 -axis, we obtain 4π ∞ ∞ ∂r |ruθ |p drdx3 = 0. I=− p −∞ 0 Thus we get

d θ p ˜ θ | p2 |2 dx |ru | dx + C |∇|ru dt R3 R3 = 0.

By Gronwall’s inequality, we have T θ p sup |ru | dx + C 0≤t≤T

R3

0

R3

p

˜ θ | 2 |2 dxdt ≤ C(ruθ0 p ). |∇|ru

The case p = ∞ is immediate if we let p → ∞ in the above. Using the Proposition 1, we now prove Theorem 2.

(7)

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D. Chae, J. Lee

Proof of Theorem 2. Below we denote Θ = ruθ . Consider the er , e3 components of the Navier-Stokes equations. ˜ r Du 1 1 Θ2 − (∂r2 + ∂32 )ur − ∂r ur + 2 ur = −∂r p + 3 , Dt r r r ˜ 3 1 Du − (∂r2 + ∂32 )u3 − ∂r u3 = −∂3 p. Dt r Applying the operator (∂3 , −∂r ) to the above equations, the azimuthal component of the vorticity equation is obtained as the following. 2 ˜ θ Dω Θ 1 1 θ r θ 2 2 θ + (∂r u + ∂3 u3 )ω − (∂r + ∂3 + ∂r )ω + 2 ω = ∂3 . (8) Dt r r r3 Suppose that r3 ω θ ∈ L2 . Similar to the proof of Proposition 1, we first assume ω θ decays sufficiently fast. Multiplying the both sides of (8) by r6 ω θ and integrating over R3 , we get 1d (r3 ω θ )2 dx + [(ur ∂r + u3 ∂3 )(r3 ω θ )](r3 ω θ )dx 2 dt R3 3 R r 3 θ 3 θ − u ∂r (r )ω (r ω )dx − ur r5 (ω θ )2 dx 3 3 R R 1 3 θ 2 2 3 θ (r ω )(∂r + ∂3 + ∂r )(r ω )dx + r4 (ω θ )2 dx − r 3 R3 R 1 + r3 (ω θ )2 (∂r2 + ∂r )(r3 )dx + 2 ∂r ω θ ∂r (r3 )(r3 ω θ )dx r R3 R3 (∂3 Θ2 )r3 ω θ dx. = R3

After integration by parts and more elementary computations, we have 1d 2 dt

˜ 3 ω θ )|2 dx |∇(r r 5 θ 2 =4 u r (ω ) dx + 8 r4 (ω θ )2 dx 3 3 R R 3 θ 2 θ +6 ∂r (r ω )r ω dx + ∂3 Θ2 r3 ω θ dx R3

R3

3 θ 2

(r ω ) dx +

:= I1 + I2 + I3 + I4 .

R3

R3

By use of Hölder’s inequality, Young’s inequality, and the GagliardoNirenberg inequality, we estimate I1 as follows.


r

|I1 | ≤ u 2

R3

≤C

R3

R3

≤ ≤ ≤

10 3

3

R

3

2

1

2 3

θ

(r ω ) (ω ) dx 1

(ω ) dx + C

R3

2

1

6

(ω θ )2 dx

θ 2

R

1

θ 4

r (ω ) dx

3 θ

≤C

10

655

R3

3 θ 5

2

(r ω ) dx 1

R3

3

(r3 ω θ )5 dx

5

9

˜ 3 ω θ ) 5 (ω θ )2 dx + Cr3 ω θ 25 ∇(r 2 ˜ 3 ω θ )22 + C r3 ω θ 22 . (ω θ )2 dx + )∇(r

For I2 and I3 , we estimate 4 2 (r3 ω θ ) 3 (ω θ ) 3 dx |I2 | ≤ C R3

1 3

θ 2

3 θ 2

2 3

(ω ) dx (r ω ) dx ≤C R3 R3 θ 2 ≤ (ω ) dx + C (r3 ω θ )2 dx, R3

and

R3

3 θ

2

1 2

4

θ 2

1 2

(∂r (r ω )) dx r (ω ) dx |I3 | ≤ C R3 R3 ≤) (∂r (r3 ω θ ))2 dx + C r4 (ω θ )2 dx 3 3 R R 3 θ 2 θ 2 ≤) (∂r (r ω )) dx + (ω ) dx + C (r3 ω θ )2 dx. R3

Finally, we obtain |I4 | ≤

R3

R3

(∂3 Θ2 )2 dx +

R3

R3

(r3 ω θ )2 dx.

Combining the above estimates altogether, and choosing ) to be sufficiently small, we obtain d 3 θ 2 ˜ 3 ω θ )|2 dx (r ω ) dx + C |∇(r dt R3 R3 3 θ 2 θ 2 ≤C (r ω ) dx + C (ω ) dx + C (∂3 (Θ2 ))2 dx. R3

R3

R3

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D. Chae, J. Lee

Applying Gronwall’s inequality, we get T ˜ 3 ω θ )|2 dxdt |∇(r sup r3 ω θ 22 + C 0≤t≤T

≤

R3

0

C(T )r3 ω0θ 22

+C

0

T

ω θ 22 dt

≤ C(T, r3 ω0θ 2 , u0 2 , ruθ0 4 ).

+C

0

T

∂3 Θ2 22 dt (9)

The last inequality of (9) is from the energy inequality and the Proposition 1. Similarly to the proof of Proposition 1, we can justify the integration by parts in the above computations the proof by using the standard cut-off function technique.

4 Proof of regularity criteria We can construct weak solutions in various ways.( See [6], [13], and [26].) For example( [2] and [6]), it is possible to construct weak solutions by considering the following regularized equations. Let ρδ be the standard mollifier in ∞ 3 R3 , ρδ (x) = δ13 ρ( |x| δ ), where ρ ∈ C0 (R ), ρ ≥ 0 and supp ρ ⊂ {|x| ≤ 1}. We regularize the Navier-Stokes equations as follows.   (∂t + (ρδ ∗ uδ ) · ∇ − ∆)uδ + ∇pδ = 0, div uδ = 0,  δ u0 = ρδ ∗ u0 . For each δ > 0, the above equations have a global axisymmetric smooth solution if u0 is axisymmetric, and belongs to L2 (R3 ). As δ → 0, we obtain an axisymmetric Leray-Hopf weak solution as defined in the introduction. In the proof of Theorem 1 below, we provide only a priori estimates. The corresponding estimates for weak solutions can be justified by the above regularization procedure. Proof of Theorem 1. Let u be an axisymmetric smooth solution of the Navier-Stokes equations. Taking curl on the both sides of the Navier-Stokes equations, then we obtain the following equations. ∂ω − ∆ω + (u · ∇)ω − (ω · ∇)u = 0. ∂t We may assume QT = B3R × [0, T ) = {(x, t)|0 ≤ t < T, |x| < 3R} where R > 0 and QT = B2R × [0, T ) = {(x, t)|0 ≤ t < T, |x| < 2R} for simplicity. For the general domain the proof is similar. Let η = η(r) be a smooth cut off function which has a support in B2R , 0 ≤ η ≤ 1, and η = 1


657

on BR . By multiplying ωη 2k on the both sides of the above equations, and integrating over R3 , we get the following equation. 1d k 2 |ωη | dx + |∇(ωη k )|2 dx 2 dt B2R B2R 3 (∂j η)ωi (∂j ωi )η 2k−1 dx +2k + −

i,j=1 B2R

B2R B2R

−

B2R

2 k

k

|ω| η (∆η )dx +

B2R

[u · ∇(ωη k )](ωη k )dx

(u · ∇η k )ω · ωη k dx (η k ω · ∇)u · ωη k dx = 0.

From which it follows 1d k 2 |ωη | dx + |∇(ωη k )|2 dx 2 dt B2R B2R 3 k 2 2k−2 (∆η )|ω| η dx − 2k =− +k

B2R

B2R

|ω|2 η 2k−1 (u · ∇)ηdx +

B2R

:= {1} + {2} + {3} + {4}. First, {1} is estimated easily.

j=1

{1} ≤ C

B2R

B2R

(∂j η)(ωη k−1 )∂j (ωη k )dx

(η k ω · ∇)u · ωη k dx

|ω|2 dx,

where C = C(η). On the other hand, {2} and {3} can be estimated by virtue of the Young inequality, the Hölder inequality and the Gagliardo-Nirenberg inequality. {2} ≤ C |ωη k−1 ||∇(ωη k )|dx B2R |∇(ωη k )|2 dx + C |ω|2 η 2k−2 dx ≤) B2R

≤)

B2R

≤)

B2R

B2R

k

2

|∇(ωη )| dx+ C |∇(ωη k )|2 dx + C

2

B2R

B2R

1 k

|ω| dx

|ω|2 dx + C

B2R

B2R

k−1 k (|ω|η ) dx k 2

(|ω|η k )2 dx,

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D. Chae, J. Lee

and

{3} ≤ C

B2R

|u||ω|2 η 2k−1 dx

≤C

B2R

≤

B2R

|ω| dx + C

k 2

B2R

B2R

4 2(2k−1)

B2R

|ω| η k

B2R

(|ω|η )

|ω| dx + C

≤C

|u| dx

2

B2R

×

1 2

2

≤

2

|∇|ω|η ) dx |ω|2 dx + )

k 2

B2R

2(2k−1) k−1

(|ω|η ) dx

1 2

dx

k−1 dx

2k−1

k−2 2(2k−1)

3k 2(2k−1)

B2R

|∇|ω|η k |2 dx + C

B2R

(|ω|η k )2 dx.

To estimate {4}, we compute 1 {4} = [(η k ω r ∂r − η k ω θ ∂θ + η k ω3 ∂3 )(ur er + uθ eθ + u3 e3 )] r B2R × (η k ω r er + η k ω θ eθ + η k ω3 e3 )dx 1 = η k ω r (∂r ur )η k ω r − η k ω θ uθ η k ω r r B2R

+η k ω3 (∂3 ur )η k ω r + η k ω r (∂r uθ )η k ω θ 1 + η k ω θ ur η k ω θ + η k ω3 (∂3 uθ )η k ω θ r +η k ω r (∂r u3 )η k ω3 + η k ω3 (∂3 u3 )η k ω3 dx := I1 + ... + I8 . Now we estimate I1 , ..., I8 . |I1 | ≤ Cω θ Lp (B3R ) η k ω2

2p

L p−1 (B2R ) 3

2p−3 p

≤ Cω θ Lp (B3R ) ∇(η k ω)2p η k ω2 2p

≤ C ω θ L2p−3 η k ω22 + )∇(η k ω)22 . p (B 3R ) Similarly, we get the following. 2p

η k ω22 + )∇(η k ω)22 . |I3 |, |I6 |, |I7 |, |I8 | ≤ C ω θ L2p−3 p (B 3R )


659

uθ = ∂r uθ + ω3 , we have r k θ θ k r η ω (∂r u )η ω dx + η k ω θ ω3 η k ω r dx I2 = B2R B2R ω θ (∂r (η k uθ ))η k ω r dx − k (∂r η)uθ ω θ ω r η 2k−1 dx = B2R B2R η k ω θ ω3 η k ω r dx +

Since −

:=

I21

B2R + I22

+ I23 .

In the proof of Lemma 3, we know that ∇(η k u)2 ≤ Cu2 + Cη k ω2 . Thus we get 2p

η k ω22 + )∇(η k ω)22 . |I21 |, |I22 |, |I23 | ≤ C ω θ L2p−3 p (B 3R ) Similarly, |I4 | and |I5 | can be estimated. Putting together the above estimates, we get 1d k 2 |ωη | dx + C |∇(ωη k )|2 dx 2 dt B2R B2R 2p

≤ Cω θ L2p−3 η k ω2L2 (Ω2R ) + Cω2L2 (B2R ) . p (B 3R ) Using Gronwall’s inequality, we have T k 2 ∇(ωη k )2L2 (B2R ) dt sup ωη (t)L2 (B2R ) + C 0≤t≤T

≤C

ω0 22

+

0

0

T

ω2L2 (B2R ) dt

exp C

0

T

θ

ω

2p 2p−3 Lp (B

3R )

dt .

Applying Lemma 2 and Lemma 3, sup ∇u(t)L2 (BR ) < C i.e. u ∈ t∈[0,T )

L∞ (0, T ; L6 (BR )). Thus we get the interior regularity by applying Serrin’s criterion. Remark 4. For later use, we note that if we set η ≡ 1 in the proof of Theorem 1 then we get the following a priori estimate for the whole domain R3 . T T 2p 2 2 2 θ 2p−3 ∇ω2 dt ≤ Cω0 2 exp C ω p dt . sup ω2 + C 0≤t≤T

0

0

660

D. Chae, J. Lee

For the proof of Theorem 3 and Theorem 4, we will use the standard continuation principle for the local strong solution. Proof of Theorem 3. Proof of (i): First note that there exists maximal time T0 such that there is a unique classical solution u ∈ C((0, T0 ); Ls (R3 )), s > 3 C and u(τ )s ≥ s−3 with constant C, which is independent of T0 (T0 −τ )

2s

and s.( See [9].) We provide a priori estimate for the smooth axisymmetric solution. We write the er and eθ components of the vorticity equation as follows. ˜ r 1 1 Dω 2 2 − ∂r + ∂r + ∂3 ω r + 2 ω r − (ω r ∂r + ω3 ∂3 )ur = 0. (10) Dt r r ˜ θ 1 Dω 2 2 − ∂r + ∂r + ∂3 ω θ Dt r 1 1 1 + 2 ω θ + uθ ω r − {(ω r ∂r + ω3 ∂3 ) uθ + ω θ ur } = 0. r r r

(11)

Multiplying the both sides of (10) and (11) by ω r and ω θ respectively, and integrating over (0, ∞) × (−∞, ∞), we get the following equalities. 1d r 2 θ 2 ˜ r |2 + |∇ω ˜ θ |2 drdx3 (ω ) + (ω ) drdx3 + |∇ω 2 dt (ω θ )2 (ω r )2 + drdx + drdx3 3 2 r r2 1 (∂r ω r )ω r drdx3 − = [(ur ∂r + u3 ∂3 )ω r ]ω r drdx3 r 1 (∂r ω θ )ω θ drdx3 + [(ω r ∂r + ω3 ∂3 )ur ]ω r drdx3 + r − [(ur ∂r + u3 ∂3 )ω θ ]ω θ drdx3 + [(ω r ∂r + ω3 ∂3 )uθ ]ω θ drdx3 1 r θ θ 1 θ r θ ω ω u drdx3 + ω u ω drdx3 − r r r u 1 1 ∂r ω r ω r drdx3 − (ω r )2 drdx3 = r 2 r 1 r r r ∂r ω θ ω θ drdx3 + ω ∂r u ω drdx3 + r r θ u 1 u (ω θ )2 drdx3 − ∂3 ur ω r drdx3 + 2 r r


−

r r

∂r u ∂3 u ω drdx3 +

−

θ

uθ ∂3 uθ ω θ drdx3 r

ω r ∂r uθ ω θ drdx3

1 θ r θ ∂r u ∂3 u ω drdx3 − u ω ω drdx3 r r u 1 1 ∂r ω r ω r drdx3 − (ω r )2 drdx3 = r 2 r + ω r ∂r ur ω r drdx3 r u 1 1 θ θ ∂r ω ω drdx3 + (ω θ )2 drdx3 + r 2 r θ u − ∂3 ur ω r drdx3 r θ u r θ ω ω drdx3 − −2 ∂r uθ ∂3 ur ω r drdx3 r := I1 + I2 + I3 + I4 + I5 + I6 + I7 + I8 . −

θ

θ θ

661

(12)

We will estimate I1 ,...,I8 as follows. First, it is easily seen from Young’s inequality that (ω r )2 1 1 (∂r ω r )2 drdx3 , |I1 | ≤ drdx3 + 2 r2 2 and

(ω θ )2 1 1 (∂r ω θ )2 drdx3 . drdx3 + |I4 | ≤ 2 r2 2 By means of the Hölder inequality, the Young inequality and the GagliardoNirenberg inequality, we get 12 1 2 (ω r )2 r 2 r 2 (u drdx ) (ω ) drdx |I2 | ≤ 3 3 r2 (ω r )2 ≤) drdx3 + C ur 2L˜ γ ω r 2 2γ ˜ γ−2 r2 L 2(γ−2) r 2 4 (ω ) γ r 2 r r γ ˜ ≤) drdx + C u ω ∇ω 3 ˜γ ˜2 ˜2 L L L r2 2γ (ω r )2 r γ−2 ˜ r 2˜ 2 , ≤) drdx + C u ω r 2L˜ 2 + )∇ω 3 ˜ L Lγ r2

r r r u (∂r ω )ω drdx3 |I3 | = 2

662

D. Chae, J. Lee

2γ

˜ r 2˜ 2 + C ur γ−2 ω r 2˜ 2 , (∂r ω r )2 drdx3 + )∇ω ˜γ L L L

≤) and

(ω θ )2 (ur )2 (ω θ )2 drdx3 |I5 | ≤ ) drdx3 + C r2 2γ (ω θ )2 θ 2 r γ−2 θ 2 ˜ ≤) drdx + ) ∇ω + C u 3 ˜2 ˜2 . ˜ γ ω L L L r2

˜ r ˜ p ≤ Cω θ ˜ p , which follows from On the other hand, we note that ∇u L L Corollary 2 and Lemma 2. By use of the above fact, the Young inequality, the Hölder inequality, and the Gagliardo-Nirenberg inequality again, we are lead to (ω r )2 θ r 2 |I6 | ≤ C u ∂3 u L˜ 2 + ) drdx3 r2 (ω r )2 ≤ C uθ 2L˜ γ ∂3 ur 2γ + ) drdx3 ˜ γ−2 r2 L (ω r )2 θ 2 θ ≤ C u L˜ γ ω 2γ + ) drdx3 ˜ γ−2 r2 L 2(γ−2) 4 (ω r )2 γ θ 2 θ θ γ ˜ ≤ C u L˜ γ ω L˜ 2 ∇ω L˜ 2 + ) drdx3 r2 2γ (ω r )2 θ γ−2 θ 2 θ 2 ˜ ≤ C u L˜ γ ω L˜ 2 + )∇ω L˜ 2 + ) drdx3 . r2 Similarly to the above, we estimate θ

2γ γ−2 ˜γ L

|I7 | ≤ C u

ω r 2L˜ 2

˜ r 2˜ 2 + ) + )∇ω L

(ω θ )2 drdx3 . r2

Since it is not easy to estimate I8 directly, we integrate by parts first. ∂r ∂3 uθ ur ω r drdx3 + ∂r uθ ur ∂3 ω r drdx3 I8 = = ∂r ω r ur ω r drdx3 − uθ ∂r ur ∂3 ω r drdx3 − uθ ur ∂r ∂3 ω r drdx3 r r r = 2 ∂r ω u ω drdx3 − uθ ∂r ur ∂3 ω r drdx3 + uθ ∂3 ur ∂r ω r drdx3 := I81 + I82 + I83 .


663

As previously, thanks to the Young inequality, the Hölder inequality, and the Gagliardo-Nirenberg inequality, we obtain 1 r 2 (ur )2 (ω r )2 drdx3 (∂r ω ) drdx3 + C |I8 | ≤ ) 2γ ˜ r 2˜ 2 + C ur γ−2 ω r 2˜ 2 , ≤) (∂r ω r )2 drdx3 + )∇ω ˜γ L L L

|I82 |

≤)

≤) and

|I83 |

≤)

r 2

(∂3 ω ) drdx3 + C

(uθ )2 (∂r ur )2 drdx3 2γ

˜ θ 2˜ 2 + C uθ γ−2 ω θ 2˜ 2 , (∂3 ω r )2 drdx3 + )∇ω ˜γ L L L 2γ

˜ θ 2˜ 2 + C uθ γ−2 ω θ 2˜ 2 . (∂r ω r )2 drdx3 + )∇ω ˜γ L L L

Choosing ) sufficiently small, we have d ˜ r |2 + |∇ω ˜ θ |2 drdx3 (ω r )2 + (ω θ )2 drdx3 + C |∇ω dt (ω θ )2 (ω r )2 +C + 2 drdx3 r2 r 2γ

2γ

≤ C(ur L˜γ−2 + uθ L˜γ−2 )(ω θ 2L˜ 2 + ω r 2L˜ 2 ). γ γ

(13)

Using Gronwall’s inequality, we get the following a priori estimates. T ˜ r 2˜ 2 + ∇ω ˜ θ 2˜ 2 dt sup (ω r (t)2L˜ 2 + ω θ (t)2L˜ 2 ) + C ∇ω L L 0

t∈[0,T0 )

θ 2 ω r 2 + ω +C r ˜ 2 r ˜ 2 dt 0 L L r 2 θ 2 ≤ C(ω0 L˜ 2 + ω0 L˜ 2 ) 1 + exp C

T

(14) T

r

2γ γ−2 ˜γ L

u

0

θ

2γ γ−2 ˜γ L

+ u

dt

.

Applying Theorem 2 and above a priori estimates, we get ∞ 1 ∞ ∞ θ 2 θ 2 |ω | dx = |ω | rdrdx3 + |ω θ |2 rdrdx3 R3

−∞ 0 ∞ 1

≤

−∞

0

|ω θ |2 drdx3 +

−∞ 1 ∞ ∞

−∞

1

|ω θ |2 r7 drdx3 .

In the above inequality, right hand side is bounded by some constant by Theorem 2 and (15). By Remark 4 below the end of the proof of the Theorem

664

D. Chae, J. Lee

1 and the Sobolev embedding theorem, we find that uL∞ ((0,T0 );L6 (R3 )) is bounded by some constant. Thus we can continue our local smooth solution until T by the standard continuation argument. Proof of (ii) : The θ-component of the axisymmetric Navier-Stokes equations is as follows. 1 1 1 ∂t uθ + (ur ∂r + u3 ∂3 )uθ − (∂r2 + ∂32 + ∂r )uθ + 2 uθ + uθ ur = 0. (15) r r r Multiplying the both sides of (15) and (8) by 2|uθ |2 uθ and ω θ , and integrating over (−∞, ∞) × (0, ∞), we get 1d 1d θ 4 |u | drdx3 + |ω θ |2 drdx3 2 dt 2 dt 3 θ 2 2 ˜ ˜ θ |2 drdx3 |∇(u ) | drdx3 + + |∇ω 2 |uθ |4 (ω θ )2 +2 drdx + drdx3 3 r2 r2 r 1 5 θ 2 θ 2 θ 4 u |u | drdx3 ≤ ∂r (u ) (u ) drdx3 + r 4 r r 1 u (ω θ )2 drdx3 ∂r ω θ ω θ drdx3 + 1 + r r 2 ωθ + |∂3 (uθ )2 | drdx3 := I1 + I2 + I3 + I4 + I5 . r Using Young’s inequality, we estimate θ 2 2 (u ) 1 ˜ θ )2 2˜ 2 , + 1 ∇(u |I1 | ≤ L 2 r L˜ 2 2 θ 2 ω 1 ˜ θ )2˜ 2 , + 1 ∇(ω |I3 | ≤ L 2 r L˜ 2 2 and

θ 2 ω 1 ˜ θ 2 2 1 . |I5 | ≤ ∇(u ) L˜ 2 + 2 2 r L˜ 2

On the other hand, by virtue of the Gagliardo inequality and Hölder inequality, we have γ

˜ θ )2 2˜ 2 |I2 | ≤ Cω θ L˜γ−1 (uθ )2 2L˜ 2 + )∇(u γ L and

γ

˜ θ 2˜ 2 . ω θ 2L˜ 2 + )∇ω |I4 | ≤ Cω θ L˜γ−1 γ L


665

Choosing ) sufficiently small and using Gronwall’s inequality, we obtain that T0 γ θ 4 θ 2 θ 4 θ 2 sup (u L˜ 4 +ω L˜ 2 ) ≤ C(u0 L˜ 4 +ω0 L˜ 2 ) exp(C ω θ L˜γ−1 dt). γ 0≤t≤T0

0

By the same argument below (15) in the proof of (i), we obtain (ii).

Proof of Corollary 1. For the axisymmetric solution for initial data without swirl, uθ ≡ 0, and thus ω3 ≡ 0. In view of (ii) of Theorem 3, it suffices to prove T |ω θ |2 drdx3 dt < ∞. (16) 0

θ

The main difficulty of the global existence is that we do not know if ωr2 ∈ θ L2 (0, T ; L2 (R3 ))( See [13].). Choose ωr as a test function of the vorticity equation and integrate over R3 . Since ω r , ω3 and uθ vanish in this case, we get the following inequality by integrating by parts. (ω θ )2 1d 1 1 ˜ θ |2 drdx3 |ω θ |2 drdx3 + |∇ω drdx + 3 2 dt 2 r2 2 ≤C (ω θ )2 drdx3 . The Gronwall inequality gives us that ˜ 2 (R3 )). ω θ ∈ L∞ (0, T ; L

Thus, (16) is now proved.

Proof of Theorem 4. (i) Let T0 be the maximal time as in the proof of Theorem θ 3 θ 3. We wish to multiply the both sides of (8) and (15) by ωr2 and (ur4) , respectively, and integrate over R3 . Indicated in the proof of Corollary 1 ( See also [13].), however, we do not know if they belong to L2 (0, T0 ; L2 (R3 ). θ To justify the procedure, we multiply the both sides of (8) and (15) by rω2− θ 3

) and (u with sufficiently small ) > 0, respectively and integrate over R3 . r 4− Then we have the following by the integration by parts.

θ 4 u 1d dx 1− 2 1− 4 4 dt 3 3 r r R R θ 2 θ 2 2 ω u 1 ˜ ˜ dx + ∇ ∇ 1− dx 1− 2 3 3 2 4 r r R R

1d 2 dt +

ωθ

2

dx +

666

D. Chae, J. Lee

θ 2 ) (uθ )4 ω ) ) 3 − ) 1− dx + dx 4 R3 r2− 2 4 2 R3 r6− r θ 2 θ 2 θ u ω u ω ≤) ∂ dx + dx 3 r1− 2 r1− 2 r1− 2 R3 r R3 r θ 4 u u +(2 − )) dx := I1 + I2 + I3 . 1− r 4 R3 r

Note that boundary terms vanish (see [13] and Lemma 4). By virtue of the Hölder inequality, the Gagliardo-Nirenberg inequality, and the Young inequality, we obtain r θ 2 u ω |I1 | ≤ ) r r1− 2 2γ γ γ−1 θ 2 r 2γ θ 2 u 2γ−3 ω ω 1 ˜ ∇ , ≤C + r1− 2 r γ 4 r1− 2 2 2 2 ωθ uθ ˜ |I2 | ≤ ∇ 1− 1− r 4 2 r 2 2 θ 2 2 2 θ ω u 1 ˜ ∇ ≤C , r1− 2 + 8 1− 4 r 2 2 and

r θ 2 2 u u |I3 | ≤ (2 − )) 1− r γ r 4 2γ r 2γ u 2γ−3 ≤C r γ

γ−1

uθ 2 2 1 uθ 2 2 ˜ 1− + ∇ . r 4 8 r1− 4 2

2

By the use of the Gronwall inequality, we have θ 2 θ 4 ω (t) u (t) 1 1 sup 1− + sup 2 t∈[0,T0 ] r 2 2 4 t∈[0,T0 ] r1− 4 4   2 θ 2 T0 θ 2 ω u  ˜ ˜ + ∇ (17) +C dt ∇ 1− 2 1− 4 r r 0 2 2  

2γ θ 2 uθ 2 2 T0 r 2γ−3 ω0 u 0 ≤ C  dt  1 + exp r1− 2 + r r1− 4 0 2 γ 2


667

If ) → 0, then we have the following by the Lebesgue dominated convergence theorem. θ 2 θ 4 ω (t) u (t) 1 1 sup sup + r 2 t∈[0,T0 ] r 4 t∈[0,T ] 2 4 0   T0 θ 2 2 2 θ  ˜ ω + ˜ u ∇ (18) +C dt ∇ r r 0 2 2  

2γ θ 2 uθ 2 2 T0 r 2γ−3 ω0 u 0 ≤ C  dt  1 + exp r + r r 2

0

2

γ

In order to complete the proof, we choose cut-off function η(r) such that 0 ≤ η ≤ 1 on r ≤ r20 and supp η ∈ {r ≤ r0 }. Multiplying the both sides of

θ 3 ωθ u 4 (8) and (15) by 1− , respectively, and integrate over R3 . η and η r 2 r 1− 4 Since the procedure is rather standard, we omit the details here. then we can deduce the following inequality. θ θ ω (t) 2 2 1 u (t) 4 1 sup η + sup η 2 t∈[0,T0 ] r 4 r t∈[0,T0 ] 2 4   T0 θ 2 2 2 θ ω u ˜ ˜ ∇ η2 η  dt +C + ∇ r r 0 2 2   2

2γ θ 2 2 uθ T0 r 2γ−3 ω0 2 u 0 ≤ C  dt η  1 + exp r η + r γ r 0 2 L (Γr0 ) 2 T0 T0 ruθ 44 dt + C ω θ 22 dt. (19) +C 0

0

The right hand side of (19) is controlled by the initial datum u0 2 , ruθ0 4 r and ur Lα (0,T ;Lγ (Γr )) , which is finite by hypothesis. Similarly to the proof 0 of Theorem 3, we consider the following estimate. ∞ r0 ∞ ∞ 2 θ 2 |ω | dx = |ω θ |2 rdrdx3 + |ω θ |2 rdrdx3 R3

−∞

≤C

0

∞

−∞

0

−∞

r0 2

(ω θ )2 drdx3 + C r

r0 2

∞

−∞

∞ r0 2

|ω θ |2 r7 drdx3 .

Thus it follows that u ∈ L∞ (0, T0 ; L6 (R3 )), which implies that we can continue our local smooth solution by the standard continuation argument.

668

D. Chae, J. Lee

Proof of (ii) is similar to that of (i). For simplicity, we do not present the cut-off function technique. It will be shown that the integrability of uθ is controlled by that of ur . First we multiply the both sides of the azimuthal component of the Navier-Stokes equations by |uθ |2 uθ and integrate over R3 , then we get 1d 3 θ 4 ˜ θ |2 |2 dx |u | dx + |∇|u 4 dt R3 4 R3 |uθ |4 |uθ |4 r u dx := I1 . + dx = − (20) 2 r R3 r R3 By means of the Young inequality, the Hölder inequality, and the GagliardoNirenberg inequality, it is established that θ 2 2 |u | |I1 | ≤ ) dx + C |uθ |4 (ur )2 dx r R3 R3 θ 2 2 |u | dx + Cur 2γ (|uθ |)2 22γ ≤) r γ−2 R3 θ 2 2 6 2(1− 3 ) |u | ˜ θ |2 γ ≤) dx + Cur 2γ |uθ |2 2 γ ∇|u 2 r R3 θ 2 2 2γ |u | ˜ θ |2 22 + Cur γγ−3 |uθ |2 22 . ≤) dx + )∇|u r R3 Thus from (20), we have the following inequality. θ 2 2 1d |u | θ 4 θ 2 2 ˜ |u | dx + C |∇|u | | dx + C dx 4 dt R3 r 3 3 R R 2γ

≤ Cur γγ−3 (uθ )2 22 . Using Gronwall’s inequality, we find that the following is immediate. T0 1 θ 4 ˜ θ |)2 |2 dxdt sup u 4 + C |∇(|u (21) 4 0≤t≤T0 3 0 R T0 T0 θ 2 2 2γ |u | θ 4 r γ−3 +C dxdt ≤ C(u0 4 ) exp u γ dt . r 0 R3 0 2γ T The right hand side of (22) is controlled by uθ0 4 and 0 ur γγ−3 dt. Multiplying the both sides of (8) by ω θ and integrating over R3 , we get θ 2 1d ω ˜ θ |2 dx + (ω θ )2 dx + |∇ω dx 2 dt R3 r 3 3 R R θ (ω θ )2 θ 2 ω dx := I2 + I3 . (∂3 (u ) ) dx + ur (22) =− r r R3 R3


669

As previously, by means of the Hölder inequality, the Young inequality, and the Gagliardo-Nirenberg inequality, it can be obtained that θ 2 ω dx + C (∂3 |uθ |2 )2 dx. |I2 | ≤ ) r R3 R3 Due to the hypothesis on the integrability of ur , it is derived that θ 2 ω dx + C (ur )2 (ω θ )2 dx |I3 | ≤ ) r R3 R3 θ 2 2γ ω ˜ θ 22 + C ur γγ−3 ω θ 22 . ≤) dx + )∇ω r R3 Choosing ) sufficiently small, the inequality (22) reduces to the following. θ 2 1d ω θ 2 θ 2 ˜ (ω ) dx + C |∇(ω )| dx + C dx 2 dt R3 r R3 R3 2γ ≤C (∂3 |uθ |2 )2 dx + Cur γγ−3 ω θ 22 . R3

For the completion of the proof we choose η as in the proof of (i). Multiplying the both sides of (15) and of (i). Multiplying the both sides of (15) and (8) by |uθ |uθ η 4 and ω θ η 4 , respectively, and proceeding similarly, we get T0 1 θ 2 2 ˜ θ η 2 )|2 dx sup ω η 2 + C |∇(ω 2 0≤t≤T0 3 R 0 T0 θ 2 2 ω η +C dxdt r R3 0 T0 θ 2 θ 2 2 θ 2 ˜ ∇(|u |η) 2 + ω 2 dt ≤ C ω0 η2 + 0 T0 2γ ur Lγ−3 dt . (23) × exp C γ (Γ ) r 0

0

The right hand side of (23) is controlled by the initial datum u0 H 1 , ruθ0 4 and the norm ur Lα (0,T ;Lγ (Γr0 )) , which is finite by hypothesis. Similarly to the case (i), we conclude that u is regular. Acknowledgements. We deeply thank to the anonymous referee for very careful reading of the paper, and many helpful and constructive criticism. This research is supported partially by KOSEF(K97-07-02-02-01-3) and BSRI-MOE.

670

D. Chae, J. Lee

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