dongho chae and oleg yu imanuvilov- existence of axisymmetric weak solutions of the 3-d euler...

8/3/2019 Dongho Chae and Oleg Yu Imanuvilov- Existence of Axisymmetric Weak Solutions of the 3-D Euler Equations for Ne

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Electronic Journal of Differential Equations, Vol. 1998(1998), No. 26, pp. 117.ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.eduftp (login: ftp) 147.26.103.110 or 129.120.3.113

EXISTENCE OF AXISYMMETRIC WEAK

SOLUTIONS OF THE 3-D EULER EQUATIONS

FOR NEAR-VORTEX-SHEET INITIAL DATA

Dongho Chae & Oleg Yu Imanuvilov

Abstract. We study the initial value problem for the 3-D Euler equation when thefluid is inviscid and incompressible, and flows with axisymmetry and without swirl.On the initial vorticity 0, we assumed that 0/r belongs to L(log L(R3)) with > 1/2, where r is the distance to an axis of symmetry. To prove the existence ofweak global solutions, we prove first a new a priori estimate for the solution.

Introduction

We consider the Euler equations for homogeneous inviscid incompressible fluidflow in R3

v

t+ (v )v = p , div v = 0 in R+ R3 , (1)

v(0, ) = v0 , (2)

where v(t, x) = (v1(t, x), v2(t, x), v3(t, x)) is the velocity of the fluid flow and p(t, x)is the pressure. The problem of finite-time breakdown of smooth solutions to (1)-(2) for smooth initial data is a longstanding open problem in mathematical fluidmechanics. (See [6,13,14] for a detailed discussion of this problem.) The situationis similar even for the case of axisymmetry (see e.g.[11], [4]). In the case of axisym-metry without swirl velocity (-component of velocity), however, we have a globalunique smooth solution for smooth initial data [14,17]. In this case a crucial role

is played by the fact that (t, x)/r (where = curl v, r =

x21 + x22) is preserved

along the flow, and the problem looks similar to that of the 2-D Euler equations.This apparent similarity between the axisymmetric 3-D flow without swirl and

the 2-D flow for smooth initial data breaks down for nonsmooth initial data. Inparticular, Delort [8] found the very interesting phenomenon that for a sequence ofapproximate solutions to the axisymmetric 3-D Euler equations with nonnegativevortex-sheet initial data, either the sequence converges strongly in L2loc([0, )R3),or the weak limit of the sequence is not a weak solution of the equations. Thisis in contrast with Delorts proof of the existence of weak solutions for the 2-DEuler equations with the single-signed vortex-sheet initial data, where we have weak

1991 Mathematics Subject Classifications: 35Q35, 76C05.Key words and phrases: Euler equations, axisymmetry, weak solution.c1998 Southwest Texas State University and University of North Texas.

Submitted October 9, 1998. Published October 15, 1998.Partially supported by GARC-KOSEF, BSRI-MOE, KOSEF(K950701), KIAS-M97003,and the SNU Research Fund.

1


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2 Dongho Chae & Oleg Yu Imanuvilov EJDE1998/26

convergence for the approximate solution sequence. Due to the subtle concentrationcancellation type of phenomena in the nonlinear term, the weak limit itself becomesa weak solution [7,10,15]. We refer to [13, Section 4.3] for an illuminating discussionon the differences between the the quasi 2-D Euler equations and the pure 2-DEuler equations for weak initial data.

In this paper we prove existence of weak solutions to (1)-(2) for the axisymmetricinitial data without swirl in which the vorticity satisfies

0r

1 + log+ 0r

L1(R3), > 12

,

where log+ t = max{0, log t}. The idea of proof is as follows. We divide R3 intotwo parts: the region near the axis of symmetry, and the region away from theaxis. For the latter region, using the 2-D structure of the equations expressed incylindrical coordinate system, we obtain strong compactness for the approximatesolution sequence using arguments previously used in the 2-D problem in [3]. Forthe region near axis, we could not adapt the previous 2-D arguments. See the next

section for explicit comparison between the nonlinear terms in the pure 2-D Eulercase and our case. Here we use a new a priori estimate for the axisymmetric flow,combined with Delorts argument in [8] to overcome these difficulties.

To the authors knowledge this a priori estimate (See Lemma 2.1) is completelynew for the 3-D Euler equations with axisymmetry. On the other hand, the resultsobtained in this paper improve substantially the results in [5], where the authorsproved existence of weak solutions for

0r

L1(R3) Lp(R3), p > 65

.

It would be very interesting to study (1)-(2) with initial data in L1(R3).

1. Preliminaries

By a weak solution of the Euler equations with an initial data v0, we mean thevector field v L([0, T]; (L2loc(R3))3) with div v = 0 such that

T0

R3

[v t + v v : ] dxdt +R3

v0 (0, x) dx = 0 ,

for all C([0, T]; [C0 (R3)]3) with div 0 and (T, x) 0 Here we haveused the notation v v : =

3i,j=1 vivj(i)xj .

We are concerned with the axisymmetric solutions to the Euler equations. Byan axisymmetric solution of equations (1)-(2) we mean a solution of the form

v(t, x) = vr(r, x3, t)er + v(r, x3, t)e + v3(r, x3, t)e3

in the cylindrical coordinate system, using the canonical basis

er = (x1r

,x2r

, 0), e = (x2r

, x1r

, 0), e3 = (0, 0, 1), r =

x21 + x22 .


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EJDE1998/26 Existence of axisymmetric weak solutions 3

For such flows the first equation in (1) can be written as

DvrDt

(v)2

r= p

r, (3)

D

Dt (rv) = 0 , (4)Dv3

t= p

x3, (5)

for each component of velocity in the cylindrical coordinate system, where

D

Dt=

t+ vr

r+ v3

x3.

On the other hand, the second equation of (1) becomes

r (rvr) +

x3 (rv3) = 0 . (6)

We observe that -component of the vorticity equation is written as

D

Dt

r

=

1

r4

x3(rv)

2 , (7)

where

=vrx3

v3r

(8)

is the component of the vorticity vector . If we assume that the initial velocity

v0 Vm = {v [ Hm(R3)]3 : div v = 0}with m 4 is axisymmetric, then due to the symmetry properties of the Eulerequations, and by the existence of local unique classical solutions [12], the solutionremains axisymmetric during its existence. Here we used the standard Sobolevspace

Hm(R3) = {u L2(R3) : Du L2(R3), || m} .Furthermore, if v0 has no swirl component, i.e. v0,=0, then (4) and (7) implythat

D

Dt r

= 0 t > 0 . (9)

We observe that in this case the vorticity becomes (t, x) = (t,r,x3)e. Thus,we have, in particular,

|(t, x)| = |(t,r,x3)| ,where | | denotes the Euclidean norm in R3 in the left hand side, and the absolutevalue in the right hand side of the equation. In [17] Saint-Raymond proved existenceof a global unique smooth solution for smooth v0 without swirl.

Below we show explicitly the difference between the nonlinear terms for the 2-DEuler equations and those for 3-D Euler equations with axisymmetry and without


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swirl. In the weak formulation of the 2-D Euler equations, if we use a test functionof the form = ( x2 ,

x1

) in order to satisfy div = 0, thenT0

R2

[v v : ] dxdt =T0

R2

(v21 v22)

2

x1x2 v1v2

2

x21

2

x22

dxdt.

On the other hand, in the axisymmetric 3-D Euler equation without swirl, if weuse as a test function (t, x) = r(t,r,x3)er + 3(t,r,x3)e3 with

r =1

r

x3, 3 = 1

r

r

to satisfy (rr)r

+ (r3)x3

= 0, thenT0

R3

[v v : ] dxdt = 2T0

RR+

(v2r v23)

2

rx3 vrv3

2

r2

2

x23

+vrv3

r

r v

2r

r

x3

drdx3 dt .

Here we have extra two nonlinear terms compared to the 2-D case, which haveapparent singularities on the axis of symmetry.

Before closing this section, we provide a brief introduction to the Orlicz spaces.For more details see [1,9], and for applications to the 2-D Euler equations, see [3,16].By an N-function we mean a real valued function A(t), t 0 which is continuous,increasing, convex, and satisfies

limt0

A(t)

t= 0, lim

t

A(t)

t= + .

We say that A(t) satisfies 2-condition near infinity if there exist k > 0, t0 0such that

A(2t)

kA(t)

t

t0 .

We denote A(t) B(t) if for every k > 0

limt

A(kt)

B(t)= .

Let be a domain in Rn. Then the Orlicz class KA() is defined as the set allfunctions u such that

A(|u(x)|) dx < . On the other hand, the Orlicz spaceLA() is defined as the linear hull of the Orlicz class KA(). The set LA() is aBanach space equipped with the Luxembourg norm

uA = inf

k :

A(u

k) dx 1 .

In general KA() LA(), but in case the domain is bounded inRn

, and theN-function A satisfies the 2-condition near infinity we have KA() = LA() (see[1]). For example Lp(), 1 < p < is an Orlicz space with N-functions given byA(t) = tp.

Recall that for a bounded domain we have the continuous imbedding, [1],

LA() LB() if A(t) B(t).Also recall the following duality relations [3, Lemma 4]. (Below X denotes thedual of X)


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Lemma 1.1. Let be a bounded domain in Rn, and > 0. Let A(), B() beN-functions given by A(t) = t(log+ t), B(t) = exp(tq/) 1, where t 0. Then,we have

LB() = LA() .

By the Orlicz-Sobolev space WmLA() we mean a subspace of the Orlicz space

LA() consisting of functions u such that the distributional derivatives Du arecontained in LA() for all multi-index with || m, equipped with a Banachspace norm

um,A = max||m

DuA .

The following lemma corresponds to a special case of the general result by Don-aldson and Trudinger [9].

Lemma 1.2. Let R2 be a bounded domain, and B(t) = exp(t2) 1, then wehave a continuous imbedding

H10 () LB().

Moreover, for any N-function A(t) with A(t) B(t) we have a compact imbedding

H10 () LA().

Combining dual of the compact imbedding in Lemma 1.2, and Lemma 1.1 wehave

Corollary 1.1. Let R2 be a bounded domain and A(t) = t(log+ t) with > 12 . Then we have the compact imbedding

LA() H1().

2. Main Results

Our main result is as follows:

Theorem 2.1. Suppose > 12 is given. Let v0 V0 be an axisymmetric initialdata with v0, 0, and |0r |

1 + (log+ |0r |)

L1(R3). Then there exists a weaksolution of problem (1)-(2). Moreover, the solution satisfies

v(t, )V0 v0V0 ,

and

R3

(t, )r 1 + log+ (t, )r

dx R3

0r 1 + log+ 0r

dx

for almost everyt [0, ).In this section our aim is to prove the above theorem. Below we denote

Q = [0, T] R3, G = {(r, x3) R2 | r > 0, x3 R}.

We start from establishment of the following a priori estimate.


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Lemma 2.1. Let v(t, x) C([0, T]; [C1(R3)H1(R3)]3)C([0, T]; V0) be theclassical solution of the Euler equations for the axisymmetric initial data v0 with-out the swirl component, and with the vorticity satisfying 0r L1(R3). Then the following estimate holds:

T0

R3

11 + x23

vrr

2dxdt Cv02V0 + 0r L1(R3) . (10)

Proof. The velocity conservation law for the Euler equations implies the estimate

rvrL(0,T;L2(G)) +

rv3L(0,T;L2(G)) Cv0V0 . (11)

Moreover, (9) immediately yields the estimate for L1-norm of vorticity

(t, )L1(G) 0L1(G).

We set (x3) =x3 1/(1 + 2) d. Multiplying (9) by 2r(x3) scalarly in

L2(0, T; L2(G)) and integrating by parts, we obtain

0 =

R3

r

dx

T

0

T0

G

2v3 drdx3 dt

=

R3

r

dx

T

0

+

T0

G

2v3

v3r

vrx3

drdx3 dt

=

R3

r

dx

T

0

T0

+

v23(t, 0, x3) dx3 dt

+T0

G 2

v3vr +

vrv3x3

dr dx3 dt ,

(12)

where we used the regularity assumption of solution v, and the integration by partsused above can be justified easily. Indeed,

T0

G

2v3

v3r

vrx3

drdx3 dt

= limrk+

2

T0

+

rk0

v3v3r

dr dx3 dt

lim

bk+2 T

0bkbk

0

v3vr

x3dr dx3 dt

= T0

+

v23(t, 0, x3) dxdt + limrk+

T0

+

v23(t, rk, x3) dx3 dt

limbk+

T0

0

2v3vrdrdt

bk

bk

+ limbk+

T0

bkbk

0

2

v3vr +

vrv3x3

dr dx3 dt .


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for all sequence rk +. Since v C([0, T]; (C1(R3))3),(0,T)R3

|v|2 dxdt =2+0

T0

+

|v|2 dx3 dt

r dr

=2

T0

+0

|v|2r dr dt dx3 < ,and limx3

(x3) = 0 one can find a sequence rk + and bk + suchthatT

0

v23(t, rk, x3) dx3 dt 0, limbk+

T0

0

2v3vr dtdr

bk

bk

0 .

From (6) we havev3x3

= vrr

vrr

. (13)

Therefore (12) and (13) imply

0 =

R3

r

dx

T

0

T0

+

v23(t, 0, x3) dx3 dt

+

T0

G

2

v3vr (vr)

2

r vr vr

r

drdx3 dt . (14)

Since, by assumption, v(t, x) is a smooth and axisymmetric vector field

vr(t, 0, x3) = 0 t R1+, x3 R1.

Thus integration by parts in (14), which can be justified similarly to the above,implies

T0

+

v23(t, 0, x3) dx3 dt +

T0

G

2(x3)(vr)

2

rdr dx3 dt

=

T0

G

2v3vr drdx3 dt +

R3

r

dx

T

0

.

Since (x3) > 0, |(x3)| < C for all x3 R1 we obtain the inequality

2 T

0G

(vr)2

r drdx3dt 2 T

0G

|v3vr| drdx3 dt + C0r L1(R3)

2T

0

G

(vr)

2

rdrdx3 dt

12T

0

G

rv23()2

drdx3 dt

12

+C0

r

L1(R3)

.

Hence by the Cauchy-Bunyakovskii inequality we have

T0

G

|| (vr)2

rdrdx3 dt C

T0

R3

v23||2

dxdt +

0,r

L1(R3)

. (15)


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Since supx3R |(x3)|2/|(x3)| C, inequalities (11) and (15) imply the estimate(10).

Now, let v0 be an axisymmetric initial datum without the swirl component suchthat

v0

v0 in V0, v0

(C(R3))3 ,

0,

r 0,

rin L1(R3) . (16)

Such an approximation v0 for any axisymmetric function, v0 V0 without swirlwas constructed in [17] for example. In [17] also, it was proved that in this casethere exists a unique solution of the problem (1)-(2), v(t, ) C([0, T]; [C2(R3)]3)L2(0, T; H1(R3)). Without loss of generality, passing to a subsequence if it is nec-essary, we may assume that

v v weakly in [L2((0, T) R3)]3 . (17)

We have

Lemma 2.2. Let{v(x, t)}(0,1) be a sequence of smooth solutions of (1)-(2) as-sociated with the initial datum {v

0} with axisymmetry and without swirl, and sat-isfying (16) and (17). Then, for each C([0, T]; C0(R3)), we have

Q

[(vr)2 (v3)2]dxdt

Q

[(vr)2 (v3)2]dxdt as +0 . (18)

Remark. The above lemma is very similar to Delorts in [8], where he proved itin particular under the assumptions on the sequence {v(x, t)} that

(x, t) 0 almost everywhere in (0, T) R3, and{} is uniformly bounded in L(0, ; L1(R+ R, (1 + r2) dr dx3)) .

In our case, however, we only need to assume 0r L1(R3), and {v} is the associ-ated sequence of approximated solutions.

Proof of Lemma 2.2. We follow Delorts arguments. Denote

(1f)(x) = 14

R3

f(y)

|x y| dy .

Relation v = 1curl implies

v1 = 13

2 ,

v2 =

131 ,

v3 = 1(2

1 12) .

Let C0 (R3), in 0, 1 for all (t, x) supp . Set

v1 = 13(

2) + w

1 ,

v2 = 13(2) + w2 ,v3 =

1(2(1) 1(2)) + w3 ,


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where

w1 = [, 1

x3]2 ,

w2 =

[, 1

x3

]1 ,

w3 = ([, 1

x2]1 [, 1

x1]2) ,

where [A, B] = AB BA is the commutator of operators A, B, and i = /xi .Note that wi are uniformly bounded in L

(0, T; H1loc(R3)) H1(0, T; H4loc (R3))

for each i = 1, 2, 3. Really let us prove this claim for example for w1. Denotez =

1 x3

(2). Then z =x3

(2), and

(v1) = 3(2) 32 + 2

3i=1

xi(v1

xi) v1 .

Denote u = v1 z. Then

u = (3v1)

x3+

2

x23v1 +

(3v3)

x1

2

x1x3v3 + 2

3i=1

xi(v1

xi) v1

and

u = 1

(3v

1)

x3+

(3v3)

x1+ 2

3i=1

xi(v1

xi)

+ 12

x23

v1 2

x1

x3

v3 v1 ,where the first component of u is bounded in L

(0, T; H1(R3)), and the secondone is bounded in L(0, T; H1loc(R

3)) due to compactness of supp in [0, T] R3.Since the function also has a compact support in [0, T] R3, the first part of ourstatement is proved. To prove the uniform boundness of

w1t in L

2(0, T; H4loc (R3))

we first recall that for any smooth solution v of the 3-D Euler equations with initialdata v0, we have in general

v(t1) v(t2)H3(Br) C(r)v02V0 |t1 t2|

for all t1, t2 with 0 < t1 t2 < T, where Br is a ball with the center 0 and radiusr (see e.g. [5]). This estimate implies immediately thatvt

L(0,T;H3(Br))

C(r), (19)

where C is independent of . Taking the time derivative of u we havewtL2(0,T;H4(Br))

C(r)vt

L2(0,T;H3(Br))

+ vL2(0,T;V0)

C(r).


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Hence to prove (18) we need only to pass to the limit in the following equation.

A =

Q

[(13(2))

2 + (13(1))

2 (12(1))2 (11(2))2

+ 2(12(1))(

11(2))]dxdt.

After simplifications we have:

A = (2, 2(21 23)(2))L2(Q) + (1, 2(22 23)(1))L2(Q)

2(1, 212(2))L2(Q) + A0 = A1 + A0,

where

A0 = (2, [

13, ](13(

2)))L2(Q)+(

1, [

13, ](13(

1)))L2(Q)

(1, [12, ](12(1)))L2(Q) (2, [11, ](11(2)))L2(Q)+ 2(1, [

12, ](11(

2)))L2(Q).

Since each sequence [1j , ](1k(

)) belongs to a compact set in H

1loc(R

3),we obtain

A0 A0 as 0for a subsequence. In [8] Delort proved that the term A1 can be rewritten as follows

A1 =

T0

G

G

K(t,r,x3, r, x3)(

)(t,r,x3)(

)(t, r

, x3) dr dx3 dr dx3 dt ,

where the function K(t,r,x3, r, x3) satisfies

K

C on{

(r, x3, r, x

3

)R+

R

R+

R; (r, x3)

= (r, x

3

)}

and K is locally bounded on R+ RR+ R.Let () C0 (R1), () 0 for any R1 and 1 in some neighborhood

of 0. Set

A1 = I,1 + I

,2 =

T0

G

G

K(t,r,x3, r, x3)

1

r

1

r

1 |r r| + |x3 x3|

()(t,r,x3)(

)(t, r

, x3) dr dx3drdx3dt

+T

0 G G K(t,r,x3, r, x3)

r

1 r

1 |r r| + |x3 x3|

+

r

1

|r r| + |x3 x3|

+

|r r| + |x3 x3|

()(t,r,x3)(

)(t, r

, x3) dr dx3 dr dx3dt. (20)

Our aim is to prove that for any > 0 there exist 0 > 0 and 0 > 0 such that

|I,2 | (0, 0), (0, 0). (21)


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We start from the following estimate

|I,2 | C T

0

|r|c

+

||dx3drdt0

r

L1(R3)

+ 1

+T0(GG){|rr|+|x3x3| 0 there exists 0 > 0, 0 > 0 such that

|B,| 4a

(0, 0), (0, 0) , (26)

where a = C(0r

L1(R3)

+ 1). On other hand, from

(GG){|rr|+|x3x3|c}

|()(t,r,x3)()(t, r, x3)| dr dx3 dr dx3 dt


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after the change of variables we obtain

T0

G

|()(t, r, x3)|({|r|+|x3|c}

|()(t, r + r, x3 + x3)|drdx3) dr dx3dt

C0

rL1(R3)

T0 ( sup(r,x3)G

{|r|+|x3|c} |(

)(t, r + r

, x3 + x

3)| dr dx3) dt

C0r

L1(R3)

T0

( sup(r,x3)G

{|r|+|x3|c}

|(t, r + r, x3 + x3)| dr dx3) dt

C0r

L1(R3)

T0

sup(r,x3)G

{(r,x3)G||rr|+|x3x3|c}

0,

r(X1 (t, x))

dxdt.(27)

Set ({x R3||r r| + |x3 x3| c}) = (). Then

(GG){|rr

|+|x3x

3|c}

|()(t,r,x3)||()(t, r, x3)| dr dx3 dr dx3 dt

CT0r

L1(R3)

supB

(B)()

B

0r dx . (28)

Since () 0 as 0 for any > 0, one can find 0 > 0 and 0 > 0 such thatright hand side of (28) is less than or equal to 4 for all (0, 0) and (0, 0).Then, taking into account (28) , we obtain (21).

On the other hand, we have

K(t,r,x3, rx3)

1

r

1

r

1

|r r| + |x3 x3|

C([0, T] G G).

HenceI,1 I1 as 0. (29)

Thus by (21) and (29),A1 A1.

The proof of the lemma is complete.

Let us introduce a class of axisymmetric vector fields without a swirl component,L2a(R

3) = {v (L2(R3))3 | v = v(r, x3), v = 0}. For a given N-function A(t),following [3], we introduce

QA(R3) = {v L2a(R3) W1LA(R3) | div v = 0, curl v LA(R3)}

equipped with the Banach space norm vQA(R3) = (v2L2(R3)+curl v2L2A(R3)

)1/2.

Here the derivatives are in the distribution sense. We can extend our definition toQA() for any axisymmetric domain in R3.

Now, we establish the following compactness lemma, which is an axisymmetricanalogue of Lemma 6. of [3].


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Lemma 2.3. LetA(t) be an N-function satisfying the 2condition, and satisfiesA(t) t(log+ t) 12 . Then for any bounded sequence {v} in QA(R3) there exists asubsequence, denoted by the same notation, {v} and v QA(R3) such that

lim0

R

3

|v

|2 dx =

R

3

|v

|2 dx

for any given axisymmetric test function C0 (R3) with supp {(r, x3) R2 | r > 0}.

Proof. Let {v} be a uniformly bounded sequence in QA(R3). Then, there existsa subsequence, denoted by {v}, and v in QA(R3) such that

v v weakly in L2(R3) . (30)

For such v() we introduce stream functions () = ()(r, x3) such that

v

()

r = 1

r

()

x3 , v

()

3 =

1

r

()

r .

Let a function C0 (R3) and a bounded domain W with supp W G begiven. Then, by integration by part we obtain

R3

|v()|2 dx =R3

((v()r )2 + (v

()3 )

2)dx

= 2

R3

v()r

1

r

()

x3+ v

()3

1

r

()

r

rdrdx3

= 2

R3

x3v()r

() r

v()3

()

+v()rx3

() v()3

r()

drdx3

= 2

G

x3v()r

() r

v()3

()

drdx3

+ 2

G

()

()drdx3 = {1}() + {2}() . (31)

Since

L2(W) C(W)

r

L

2(W)

= C(W)vL2(W) C,

we obtain by Rellichs compact imbedding lemma that

1 1 strongly in L2(W) 1 C0 (W)

after choosing a subsequence. This, combined with (30), provides easily that {1} {1} in (31) as 0.

To prove {2} {2} we observe that

, (32)


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andL

t(log+ t)12(W) CLA(W) C2 , (33)

where B(t) = exp(t2) 1. Since A(t) = t(log+ t) t(log+ t) 12 by hypothesis,applying Corollary 1.1, we find that there exists a subsequence {} and inH1

(W) LA(W) such that in H1(W) . (34)

We decompose our estimateG

( )drdx3

W

( )drdx3 +

W

( )drdx3

= J1 + J2 .

From (32) and (34) we obtain

J1

C

H1(W)

H1(W)

0

after choosing a subsequence, if necessary. On the other hand, the convergenceJ2 0 for another subsequence, if necessary, follows from (32). This completesthe proof of the lemma.

Using Lemma 2.3 we establish the following

Lemma 2.4. Suppose a sequence {v} and v be given as in (1.7), Lemma 2.2. Let(r) C(R+), (r) 0, (r) = 1, r [1, ] and (r) = 0 for r < 12 . Then forany > 0 and C([0, T]; C0 (R3)) we have

Q

r

|v v|2dxdt 0 as +0 , (35)

after choosing a subsequence.

Proof. Let W be any given bounded domain in G whose closure does not intersectwith the axis of symmetry. By conservation of L2(R3) norm of velocity we have

v(t, )2L2(W) C(W)v(t, )2L2(R3) = C(W)v02L2(R3) C(W, v0) . (36)

On the other hand, the conservation of(t,x)

r along the flow, (9), implies

(t, )LA(W) C(W)

r(t, )

LA(R

3)

= C(W)

0r

LA(R

3)

C(W, v0), (37)

where A = A(t) = t(log+ t). Combining (36) and (37), we find that

supt[0,T]

v(t, )QA(W) C. (38)

From the estimate (38), combined with (19), together with Lemma 2.3, we deduceby using the standard compactness lemma that there is a subsequence {v(t,r,x3)}such that

v v strongly in L2([0, T] W) .


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Now (35) follows from this immediately. The lemma is proved.

Proof of Theorem 1.1 To prove the theorem we have only to show that

I =

Q

vi vjdxdt I =

Q

vivjdxdt, (39)

for all i, j {1, 2, 3}, and C([0, T]; C0 (R3)). Let () C(R1+), 0 1, () = 1 for all [1, +) and () = 0 for [0, 12 ]. For any > 0 we set

I = I,1 + I,2 =

Q

r

vi v

jdx +

Q

1

r

vi v

j d x .

By Lemma 2.4

I,1 Q

r

vivjdx as 0 . (40)

Hence the statement of theorem will be proved, if we show that for any > 0 there

exists 0 > 0 such that for all (0, 0) one can find 0() > 0 that|I,2 | (0, 0) . (41)

Indeed, in case either i or j equals 1 or 2, by Lemma 2.1 we have

Q

1

r

vi v

j dx

2T0

+

0

r|vrvj| dr dx3 dt

CT0

+

0

|vrvj | drdx3 dt (42)

C T0

+

+0

1

1 + x23|vr |2

r drdx3 dt

12 v0V0C

v02V0 +

0r

L1(R3)

+ 1

12

(v0V0 + 1) .

Hence taking parameter 0 = 1 and parameter 0 sufficiently small, we obtain (41).Let us consider the case i = j = 3. Then

|I,2 | CQ

1 r

(v3)2 dxdt. (43)

Set (r) = 1 (r). Let 1 > 0 be such thatQ

r

((vr)

2 (v3)2) dxdt 4C (0, 1). (44)

In the above we also proved that for each > 0 there exists 2 > 0 such thatQ

(vr)2r

dxdt

4C (0, 2), (0, 1). (45)


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Let 0 = min{1, 2}.Note that by Lemma 2.2 for (0, 0)

Q

[(vr)2 (v3)2]

r

dxdt

Q

[(vr)2 (v3)2]

r

dxdt,

as +0.Thus for every > 0 and (0, 0) one can find 0() thatQ

[(vr)2 (v3)2 (vr)2 + (v3)2]

r

dxdt

4C

(0, 0()). (46)

Inequalities (43)-(46) imply (41). Since now (36) is proved for an arbitrary i, j {1, 2, 3}, the proof of the existence part of Theorem 2.1 is complete. The inequalitiesfor the energy and the vorticity follow immediately by the energy conservation forvelocity and the conservation of

rfor the smooth approximate solutions, and

taking limit for suitable subsequence. This completes the proof of the theorem.

Acknowledgment. The authors would like to express gratitude to the anonymousreferee for a very constructive criticism with helpful suggestions.

References

[1] R. A. Adams, Sobolev spaces, Academic Press. (1975).

[2] T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-DEuler equations, Comm. Math. Phys. 94 (1984), 6166.

[3] D. Chae, Weak solutions of 2-D incompressible Euler equations, Nonlinear Analysis-TMA 23(1994), 329-638.

[4] D. Chae and N. Kim, On the breakdown of axisymmetric smooth solutions for the 3-D Eulerequations, Comm. Math. Phys. 178 (1996), 391398.

[5] D. Chae and N. Kim, Axisymmetric weak solutions of the 3-D Euler equations for incom-pressible Fluid Flows, Nonlinear Analysis-TMA 29 (1997), no. 12, 1393-1404.

[6] P. Constantin, Geometric statistics in turbulence, SIAM Review 36 (1994), 173.

[7] J. M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc.(1991), no. 4, 553586.

[8] J. M. Delort, Une remarque sur le probleme des nappes de tourbil lon axisymetriques surR3,J. Funct. Anal. 108 (1992), 274295.

[9] T. K. Donaldson and N.S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Func.Anal. 8 (1971), 5275.

[10] L. C. Evans and S. Muller, Hardy spaces and the two dimensional Euler equations withnonnegative vorticity, J. Amer. Math. Soc. 7 (1994), 199219.

[11] R. Grauer and T. Sideris, Numerical computation of 3-D incompressible ideal fluids with swirl,Phys. Rev. Lett. 67 (1991), 35113514.

[12] T. Kato, Nonstationary flows of viscous and ideal fluid in R3, J. Funct. Anal. 9 (1972),

296305.[13] P. L. Lions, Mathematical topics in fluid mechanics, Vol 1, Incompressible models, Oxford

Univ. Press (1996).

[14] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. PureAppl. Math. 39 (1986), S 187-220.

[15] A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana U.Math. J. 42 (1993), 921939.

[16] A. B. Morgulis, On existence of two-dimensional nonstationary flows of an ideal incompress-ible liquid admitting a curl nonsummable to any power greater than 1, Siberian Math. J. 33(1992), 934937.


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[17] X. Saint-Raymond, Remarks on axisymmetric solutions of the incompressible Euler system,Comm. P.D.E. 19 (1994), 321334.

Dongho Chae

Department of Mathematics, Seoul National University,

Seoul 151-742, Korea

E-mail address: [email protected]

Oleg Yu Imanuvilov

Korean Institute for Advanced Study,

207-43 Chungryangri-dong Dongdaemoon-ku, Seoul, Korea

E-mail address: [email protected]

dongho chae and oleg yu imanuvilov- existence of axisymmetric weak solutions of the 3-d euler...

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