Maximal energy isolated vortices in a uniform shear flow

Nonlinear Analysis 38 (1999) 123 – 135

Maximal energy isolated vortices in a uniform shear ow Wu Yong-Hui, Mu Mu ∗ LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, People’s Republic of China Received 1 November 1998; accepted 9 November 1998

Keywords: Isolated vortex; Background shear ow; Rearrangement theory

1. Introduction Coherent vortices are one of the most important atmospheric patterns in which most of the energy is concentrated upon a bounded domain. Usually, they have long life span and can coexist with the surrounding background elds. Many previous works are related to the isolated vortices without the external shear ows. However, almost all the systems of geophysical interest are associated with a mean background ow, and coherent vortices are often observed in natural shear ow. One of the known examples is the Great Red Spot of Jupiter, which shows that it is nearly a two-dimensional vortex embedded in a zonal ow with a shear approximately twice the environmental vorticity. In most of the cases they are anticyclones situated in the region of anticyclonic shear of the zonal ow, and their longitudinal extension (in the direction of the external ow) is typically about twice as long as the latitudinal extension. On the other hand, laboratory experiments [10] and numerical simulations [7,11] show that vortices embedded in a shearing zonal ow interact quite dierently when compared to those without the external shear ow. If the zonal ow’s shear and the vortex’s strength are of the same order and opposite sign, then the vortex is destroyed into thin spiral fragements. So, it is rarely observed in nature. If the signs of the zonal ow’s shear and the vortex’s strength are the same, the vortex redistributes its vorticity so that the maximum value is at the center, and its shape is determined by the ratio of its vorticity to the shear of the surrounding zonal ow. The dynamics depends crucially on the exchange between ∗

Corresponding author.

0362-546X/99/$ - see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 1 0 2 - 9

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the self-energy of the vortices and the interaction energy of the zonal ow with the vortices (see [7,5,8]). The addition of zonal ow complicates the vortex dynamics not only because the zonal ow advects and shears the vortices, but also because the vortices react back upon and change the zonal ow. Theoretically, to illustrate the above phenomena, we restrict ourselves to the incompressible Euler ow with zonal ows of uniform shear. From the above observations and experiments one expects that such an isolated stationary vortex should have its maximum at the center and be elongated in the direction of the external shear ow. Nycander [9] proved the existence and its stability of the needed isolated stationary vortex by means of the rearrangement theory under the condition that the vorticity anomaly has both positive upper and lower bounds as the shear ow strength S ¿ 0. However, just as [9] pointed out, the assumption that the vorticity anomaly has a positive lower bound is quite unreasonable since it may lead to the discontinuity of the vorticity anomaly at the boundary of its support, and then excite complex phenomena such as shock waves, etc. This paper is motivated by the conjecture of [9] that the maximal energy vortex can be obtained without the assumption that the vortex has a positive lower bound. We shall remove this restriction to obtain the existence of the desired maximal energy vortex. Meanwhile, we also get rid of the assumption that the supports of the maximal energy sequences are con ned to a bounded domain D of R2 which was needed by [9] for the proof of the compactness of the maximal sequence.

2. Problems and rearrangement theory First, we introduce some notations which will be used later. Denote Lp (R2 ) the usual Lebesque space with the norm Z kqkp =

R2

|q(x)|p d x

1=p ;

∀q ∈ Lp (R2 ):

Denote D(R2 ) the function space which contains all the in nitely dierentiable functions, and D(R2 )0 the dual space of D(R2 ). For a given function q(x) ∈ D(R2 ), denote supp(q) = {x ∈ R2 | q(x) 6= 0}; the support of function q (see [1]). In the general case, we say that a distribution T ∈ D(R2 )0 vanishing in an open set U of if T () = 0 for every  ∈ D( ) with support contained in U . The support of T , denoted by supp(T ), is de ned as the smallest closed set F of such that T vanishes in \F (see [12]). We also denote meas(B) the measure of the set B in R2 , and C and Ci the dierent constants which depend on the known data. The motion of the ow can be formulated as @! + J ( ; !) = 0; @t

(2.1)

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

125

where the potential vorticity ! = ∇2 . Eq. (2.1) shows that the potential vorticity is frozen into the incompressible ow given by the stream function . With the background ow (or external ow) V = −Sx2 e1 ; where S is the strength of the shear, e1 is the unit vector in the x1 -(eastward) direction, the motion is reformulated as @q + J ( ; q) = 0; @t

(2.2)

where q = ∇2  is the Rvorticity anomaly, = Sx22 =2 +  is the stream function of the total ow V + v;  = R2 q(y) ln|x − y| dy; v = e3 × ∇, and e3 is the unit vector in the vertical direction. The Poisson bracket (or Jacobian) is de ned as J (f; g) = fx1 gx2 − fx2 gx1 : In this way, for a given vorticity anomaly q, the stream function  is uniquely determined. System (2.2) possesses the following conservation laws: Z Z 1 1 1 2 (Sx + )q d x = − Sx2 q d x + k∇vk22 E(q) = − 2 R2 2 2 R2 2 2 Z ZZ 1 1 2 q(x)q(r)ln(|x − r|) d x dr; (2.3) =− Sx q d x − 2 R2 2 4 Z CF (q) =

F(q) d x;

(2.4)

where d x = d x1 d x2 ; dr = dy1 dy2 are the area elements in the plane. F(s) is any continuous function of s, and CF the Casimir invariant. E(q) is usually called the energy of the ow. It contains two parts, one is the self-energy 12 k∇vk22 , the other is R the interaction energy of the external ow with the vortex − 12 R2 Sx22 q d x. For a given potential vorticity q(x1 ; x2 ), denote q = {h(x) | CF (h) = CF (q)

for all continuous functions F};

the function set in which any function keeps all the Casimirs CF of q, and we call it the rearrangement set of function q. The purpose of this paper is to nd the maximal energy solution in the set q , i.e. to nd some q˜ ∈ q such that E(q) ˜ = max E(q1 ): q1 ∈q

From the introduction we know that the most favorable conditions to nd the maximal energy solution is that the vorticity anomaly has the same sign as that of the shear

ow. Hence, we assume that 1. q has the same sign as that of S; 2. the support of q is compact in R2 ; 3. 0 ≤ q(x1 ; x2 ) ≤ qmax in R2 , where qmax is a constant.

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It is worthwhile noticing that we remove the condition 0 ¡ qmin ≤ q, where qmin is a constant, which has been used by Nycander (1995) in his proof of the existence of the maximal energy solution. The restriction q ≥ qmin ¿ 0 is somewhat quite unreasonable, since it implies discontinuity on the boundary of the supp(q), which may excite a very complex phenomenon governed by the Euler equations, such as the shock waves, etc. To get the isolated maximal energy vortex, we rst prove Rthat E(q) is upper bounded in the function set q . Note that E = S1 + S2 , where S1 = − 12 Sx22 q d x is the interaction R energy of the vorticity with the zonal shear ow, and S2 = − 12 q d x = 12 k∇vk22 is the self-energy of the vorticity anomaly. Since S1 ≤ 0 from condition 1, we need only obtain that S2 is upper bounded. To do so, we make use of the rearrangement theory. For a given function q(x1 ; x2 ) ≥ 0, denote q∗ the symmetric decreasing rearrangement (SDR) of function q such that q∗ (x1 ; x2 ) is even and nonincreasing in both variables for xi ¿ 0; i = 1; 2. We have Lemma 1. Assume that p(x) ≥ 0; q(x) ≥ 0 in R2 ; and p; q ∈ L2 (R2 ). Let q∗ (x) be the SDR of q; we have Z Z p(x)q(x) d x ≤ p∗ (x)q∗ (x) d x: Denote Tq(x) = −

R R2

ln|x − r1 |q(r1 ) dr1 ; we have

Lemma 2. Assume q ≥ 0 a.e. in R2 . Then Z Z q(x)Tq(x) d x ≤ q∗ (x)Tq∗ (x) dr: The proofs of the above two lemmas can be found in [4], and a similar proof of Lemma 2 is given by [2]. Lemma 3. Let 1 ¡ p ¡ ∞ and let q ∈ Lp (R2 ) have bounded support. Then Tq ∈ 2;p 2 Wloc (R ) and −∇2 Tq = q almost everywhere in R2 . Moreover, if 0 ¡ a ¡ ∞; q(x) in p 2 L (R ) for some p ¿ 2. Then |DTq(x)| ≤

p−1 (2)1=p a(p−2)=(p−1) kqkp ; p−2

where q ∈ Lp (R2 ); vanishes outside of a set with an area of a2 ; and x ∈ R2 . Proof. The rst part is just a special case of Lemmas 4.1 and 4.2 of [3]. Though the second part is similar to Lemma 7.12 in [3], the domain we deal with is unbounded, and the proof is dierent, so we give the proof here. Let G(x; y) = (1=2) ln|x − y|, obviously, ∇x G(x; y) =

1 x−y : 2 |x − y|2

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

127

Using Lemma 1, we obtain Z Z 1 1 |q(y)| dy |DTq(x)| = − ∇x G(x; y) · q(y) dy ≤ 2 |x − y| Z ≤ ≤

∞ 0

1 ∗ |q (r)|r dr ≤ r

Z

0

0

∗

1=p Z

p

|q (r)| r dr

0

0

(r)

−p0 =p

1=p0

p−1 (2)1=p a(p−2)=(p−1) kqkp : p−2

(2.5)

This ends the proof of our lemma. By virtue of Lemma 3 we can obtain an upper bound of the total energy E(q) Z Z 1 1 q d x = − Tq ∇2 Tq d x E(q) ≤ S2 = − 2 2 Z
2 1 |∇Tq|2 d x ≤ p − 1=p − 2(2)1=p a(p−2)=(p−1) kqkp a2 : = 2 Therefore, E(q) is upper bounded in the function set q . In the special case of q(x) ≥ 0, we can get another estimate of E(q), in which the parameter C does not depend upon a, i.e. the area of the support of q. Lemma 4. Assume q(x) ≥ 0; and q(x) ∈ Lp (R2 ) with compact support. Then Tq(x) ≤ Ckq(x)kp : Proof. Denote qR∗ the radial symmetric nonincreasing rearrangement function of q. Since Tq(x) = − R2 ln|x − y|q(y) dy, it follows from Lemma 1 that Z Z 1 q(y) dy − ln ln|x − y|q(y) dy Tq(x) = |x − y| |x−y|¡1 |x−y| ≥ 1 Z Z 1 1 ∗ ln ln q(y) dy ≤ q (y) dy ≤ |x − y| |y| |x−y|¡1 |y|¡1 Z = 2

1 0

1 r ln q∗ (r) dr ≤ 2 r

1−1=p

≤ (2)

"Z 0

1



1 r ln r

"Z 0

1

∗

p

#1=p "Z

q (r) r dr

0

1



1 r ln r

#1=p0

p 0 dr

#1=p0

p0 dr

kqkp :

By Lemma 4 for any q˜ ∈ q we have that E(q) ˜ ≤ S2 ≤ Ckqkp kqk1 and we obtained another upper bound of E(q) ˜ for q˜ ∈ q .

(2.6)

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W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

Now, we are in the position to get the maximal energy solution in q . We show that maxq∈ ˜ q E(q) is reachable on every isovortical surface q . Let   q(x ˜ 1 ; x2 ) = q(−x ˜ ˜ 1 ; −x2 );   1 ; x2 ) = q(x   X 0 0 ; x ) ≥ q(x ˜ ; x ); for 0 ≤ x ≤ x ; q(x ˜ = q ˜ ∈ SD 1 2 1 1 2 1 q     q and q(x ˜ 1 ; x20 ); for 0 ≤ x2 ≤ x20 ; ˜ 1 ; x2 ) ≥ q(x be the subset of the rearrangement function set q , in which all the functions are symmetric nonincreasing in both variables for x ¿ 0. Generally, it contains more than one element, for example, let q = exp(−x12 − x24 ). Noticing that q itself is symmetric decreasing in both x1 and x2 for xi ¿ 0 (i = 1; 2), we have q ∈ SD q . We can also make a spherical rearrangement (or Schwarz symmetrization) q∗ of q (see [6]), i.e. q∗ is a radial, nonincreasing in |x| function such that for any  ¿ 0, meas{q∗ ≥ } = meas{q ≥ }: It is obvious that q∗ ∈ SD q . However, since q itself is not a radial symmetric function, ∗ therefore q 6≡ q . Hence, SD q contains more than one element. By Lemmas 1 and 2, we have sup E(p) = sup E(p):

p∈q

p∈SD q

In the following, we shall prove that the maximal energy solution is obtained in the subset SD q of q . Lemma 5. For ∀f(x) ∈ SD q ; we have f(x1 ; x2 ) ≤ C(1 + |x1 |)−1=p1 (1 + |x2 |)−1=p1 ; where p1 ¿ 0 is any parameter; C is a constant depending upon qmax ; p1 ; and the areas of the support of q. Proof. For a.e. x = (x1 ; x2 ) ∈ R2 , we have Z |x1 | Z |x2 | Z Z |f(x1 ; x2 )|p1 d x ≥ |f(x1 ; x2 )|p1 d x R2

−|x1 |

−|x2 |

≥ 4|x1 x2 | |f(x1 ; x2 )|p1

(2.7)

and |f(x1 ; x2 )| ≤ |4x1 x2 |−1=p1 kfkp1 ≤ 4−1=p1 qmax A1=p1 |x1 x2 |−1=p1 ;

(2.8)

where A = meas(supp(q)). On other hand, since 0 ¡ q ≤ qmax , and q has compact support, we have |f(x)| ≤ qmax for x ∈ R2 . This together with (2.8) implies |f(x1 ; x2 )| ≤ C(1 + |x1 |)−1=p1 (1 + |x2 |)−1=p1 : Lemma 6. (Theorem 2.22 of Adams [1]). Assume ⊂ Rn ; n ∈ N . 1 ≤ p ¡ ∞; K ⊂ Lp ( ); and there exists a subset sequence { j } of such that

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

129

1. ∀j ; j ⊂ j+1 ; and j → ; in Lp ( j ); 2. ∀j ; the restriction of K on j is precompact R 3. ∀ ¿ 0; there exists j such that ∀u ∈ K; − j |u(x)|p d x ¡ . Then K is precompact in Lp ( ). Now, for any maximizing sequence {qi } ⊂ SD q , we prove that for ∀p ¿ 1; {qi } is p 2 precompact in L (R ). It follows from Lemma 3 that {∇Tqi } are uniform bounded 1;2p 2 (R ), and by Sobolev embedding theorem, it is precompact in Lploc (R2 ), and in Wloc 2 qi = ∇ Tqi (i = 1; 2; : : :) is upper bounded and has compact support. Hence {qi } is precompact on every Lp (DH ), where DH = [ − H; H ]2 , for H = 1; 2; : : : . The remainder is to prove that condition 3 is satis ed. In fact, choosing p1 ¡ 1, we have Z Z Z 2 −1=p1 q (x) d x ≤ C (1 + |x1 |) d x1 (1 + |x2 |)−1=p1 d x2 |x1 |¿A;|x2 |¿A i |x1 |¿A |x2 |¿A ≤ 4Cp12 (1 + A)2−2=p1 =(1 − p1 )2 : Let

A ¿ (4p12 C=((1

2

p1 =(2−2p1 )

− p1 ) )) Z Z 2 qi (x) d x ≤ : |x1 |¿A |x2 |¿A

(2.9)

, then we have (2.10)

By Lemma 6 {qi } are precompact in Lp (R2 ), which implies that there exists q˜ ∈ Lp ∩L∞ such that qi → q˜ strongly in Lp (R2 ): Moreover, similar to the proof of [9], we have that q˜ ∈ SD q ⊂ q . This ends our proof. In the following, we prove that the maximal energy vortex is actually isolated. To do so, we need only prove that supp(q) ˜ is contained in a bounded set in R2 . Here, we follow the idea of [9] but without the assumption that q ¿ qmin ¿ 0 and the assumption that most of the vorticity is concentrated in a small region near the origin. The conclusion is obtained by the method of contradiction. Assume that supp(q) ˜ is not bounded in R2 . Without loss of generality, we assume that the set {(x1 ; 0) | x1 ∈ R} ⊂ supp(q). ˜ First, we can nd L ¿ max{4; exp(e)} large enough such that Z Z 1 q(x) ˜ dx ≥ q(x) ˜ dx (2.11) 2 R2 [−L; L]2 and ∀x ∈ [−L; L]2 , de ne Z q(y)ln|x ˜ − y| dy: 1 (x) ≡ R2 =[−L; L]2

(2.12)

We have 1 (x) ≤ CL1−1=p1 ln L:

(2.13)

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W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

It follows from Lemma 5 that for 0 ¡ p1 ¡ 1, there exists C ¿ 0 such that q(y) ˜ ≤ C|y1 y2 |−1=p1 . Then for x ∈ [−L; L]2 ; y ∈ R2 =[−L; L]2 we have |x| ≤ |y|; and Z q(y)ln(2|y|) ˜ dy 1 (x) ≤ R2 =[−L; L]2

Z

≤ C(ln 2)

R2 =[−L; L]2

Z ≤ 4C(ln 2)

L

Z +C ≤C

∞

R2 =[−L; L]2

|y1 y2 |−1=p1 dy + C

y1−1=p1

≤ CL

R2 =[−L; L]2

|y1 y2 |−1=p1 ln|y| dy

2 dy1

|y1 y2 |−1=p1 ln|y1 |ln|y2 | dy1 dy2

p1 L1−(1=p1 ) + 4C 1 − p1 1−1=p1

Z

Z L

∞

y11−1=p1 ln y1 dy1

2

ln L;

(2.14)

where we have to make use of ln|y1 | + ln|y2 | ≤ ln|y1 | · ln|y2 | for |yi | ¿ exp(e); i = 1; 2. ˜ we Secondly, let r1 = (x01 ; x02 ) = (L=2; 2A=L). Since q˜ ∈ SD q and A = meas(supp(q)), have r1 ∈= supp(q). ˜ Now, we move the in nitesimal circulation dQ from rL = (L; 0) to the point r1 . The energy change due to this rearrangement is given by   2SA2 + (r ) − (r ) ; (2.15) dE = −dQ 1 L L2 ˜ If we can prove that where ∇2  = q. 2SA2 ¡ (rL ) − (r0 ); L2

(2.16)

then dE is positive, which contradicts the assumption that q˜ is the maximal energy solution. Hence the support of q˜ cannot touch the boundary of [ − L; L]2 . By de nition we have Z Z q(y) ˜ ln|rL − y| dy − q(y) ˜ ln|r1 − y| dy (rL ) − (r0 ) = R2 =[−L; L]2

R2 =[−L; L]2

Z +

[−L; L]2

q(y) ˜ ln

= I1 − I2 + I3 ; where

Z I1 =

R2 =[−L; L]2

q(y) ˜ ln|rL − y| dy;

(L − y1 )2 + y22 dy (L=2 − y1 )2 + (y2 − 2A=L)2 (2.17)

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

131

Z I2 =

R2 =[−L; L]2

q(y) ˜ ln|r1 − y| dy;

Z I3 =

[−L; L]2

q(y) ˜ ln

(L − y1 )2 + y22 dy: (L=2 − y1 )2 + (y2 − 2A=L)2

From inequality (2.13), we have I1 − I2 ≥ − CL1−1=p1 ln L:

(2.18)

Moreover, we can divide I3 as Z I3 =

[−L; L]2

q(y) ˜ ln

(L − y1 )2 + y22 dy (L=2 − y1 )2 + (y2 − 2A=L)2

q(y) ˜ ln

(L − y1 )2 + y22 dy (L=2 − y1 )2 + y22

Z =

[−L; L]2

Z +

[−L; L]2

q(y) ˜ ln

(L=2 − y1 )2 + y22 dy = I31 + I32 (L=2 − y1 )2 + (y2 − 2A=L)2

(2.19)

and Z I31 =

[−L; L]2

Z =

Z

L

dy2

−L

Z +

q(y) ˜ ln L

0

q(y) ˜ ln

Z

L −L

dy2

(L − y1 )2 + y22 dy (L=2 − y1 )2 + y22

0

−L

(L − y1 )2 + y22 dy (L=2 − y1 )2 + y22

q(y) ˜ ln

(L − y1 )2 + y22 dy = I311 + I312 : (L=2 − y1 )2 + y22

(2.20)

Notice that Z

L 0

q(y) ˜ ln Z

=

0

Z =

L=2

L=2

0

Z +

0

(L − y1 )2 + y22 dy1 (L=2 − y1 )2 + y22

q(y) ˜ ln

(L − y1 )2 + y22 dy1 + (L=2 − y1 )2 + y22

q(y ˜ 1 ; y2 ) ln L=2

Z

L

L=2

q(y) ˜ ln

(L − y1 )2 + y22 dy1 (L=2 − y1 )2 + y22

(L − y1 )2 + y22 dy1 (L=2 − y1 )2 + y22

q(L=2 ˜ + y1 ; y2 ) ln

(L=2 − y1 )2 + y22 dy1 : y12 + y22

(2.21)

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W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

Since q˜ ∈ SD ˜ 1 ; y2 ) ≥ q(y ˜ 1 + L=2; y2 ) for y1 ≥ 0. It follows that q , we have q(y Z L=2 Z L (L − y1 )2 + y22 (L − y1 )2 + y22 q(y) ˜ ln dy ≥ q(y ˜ + L=2; y ) ln dy1 ≥ 0: 1 1 2 (L=2 − y1 )2 + y22 y12 + y22 0 0 Hence, I311 ≥ 0: It is easy to verify that for (y1 ; y2 ) ∈ [−L; 0] × [−L; L] ln

20 (L − y1 )2 + y22 ≥ ln ; 13 (L=2 − y1 )2 + y22

we have Z I312 =

Z

L

dy2

−L

≥ ln

20 13

Z

0

−L L

Z

−L

q(y) ˜ ln 0

−L

(L − y1 )2 + y22 dy (L=2 − y1 )2 + y22

q(y ˜ 1 ; y2 ) dy ≥

1 20 ln kqkL1 (R2 ) ; 4 13

(2.22)

which yields that I31 ≥

1 20 ln kqkL1 (R2 ) : 4 13

(2.23)

If y2 ¡ A=L; it follows that y22 ≤ (y2 − 2A=L)2 , and we have Z I32 =

[−L; L]2

Z ≥−

L

−L

(L=2 − y1 )2 + y2 dy2 dy1 (L=2 − y1 )2 + (y2 − 2A=L)2

q(y ˜ 1 ; y2 ) ln Z

dy1

A=L

−L

q(y ˜ 1 ; y2 )

(L=2 − y1 )2 + (y2 − 2A=L)2 dy2 : (L=2 − y1 )2 + y2

(2.24)

Hence, − I32 ≤ I321 + I322 ; where

(2.25)

Z I321 =

and

y2 ¡A=L;|y1 −L=2| ≥ 1=2

q(y ˜ 1 ; y2 )

Z I322 =

y2 ¡A=L;|y1 −L=2|≤1=2

q(y ˜ 1 ; y2 )

(L=2 − y1 )2 + (y2 − 2A=L)2 dy2 (L=2 − y1 )2 + y2

(L=2 − y1 )2 + (y2 − 2A=L)2 dy2 dy1 : (L=2 − y1 )2 + y2

Since ln(1 + x) ¡ x for x ¿ 0, we have ln

(L=2 − y1 )2 + (y2 − 2A=L)2 ≤ 4[(y2 − 2A=L)2 − y22 ]; (L=2 − y1 )2 + y2

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

133

if y2 ¡ A=L and |y1 − L=2| ≥ 1=2. For 0 ¡ p1 ¡ 1 Z 8A[ − 2y2 + 2A=L]=Lq(y ˜ 1 ; y2 ) dy2 dy1 I321 ≤ y2 ¡A=L;|y1 −L=2| ≥ 1=2

≤ 16A3 qmax =L2 −

16A L

Z y2 ¡A=L;|y1 −L=2| ≥ 1=2

y2 q(y ˜ 1 ; y2 ) dy2 dy1

Z 16A y2 q(y ˜ 1 ; y2 ) dy2 dy1 L y2 ¿0;|y1 −L=2| ≥ 1=2 Z 3 2 16A ≤ 16A qmax =L − Cy2 (1+y1 )−1=p1 (1+y2 )−1=p1 dy2 dy1 L y2 ¿0;|y1 −L=2| ≥ 1=2 ≤ 16A3 qmax =L2 +

32CA ≤ 16A qmax =L + L 3

2

≤ 16A3 qmax =L2 +

Z 0

L

(1 + y2 )

y2 ¡A=L;|y1 −L=2|≤1=2

q(y ˜ 1 ; y2 ) ln

|y2 |¡A=L;|y1 |≤1=2

q(y ˜ 1 + L=2; y2 ) ln

Z +

y2 ¡−A=L;|y1 |≤1=2

|y2 |¡A=L;|y1 |≤1=2

q(y ˜ 1 + L=2; y2 ) ln

Z +

0

L

(1 + y1 )−1=p1 dy1

y2 ¿A=L;|y1 |≤1=2

y12 + (y2 − 2A=L)2 dy2 dy1 y12 + y2

y12 + (y2 − 2A=L)2 dy2 dy1 y12 + y2

q(y ˜ 1 + L=2; y2 ) ln

y12 + (y2 + 2A=L)2 dy2 dy1 y12 + y2

= J1 + J2 :

(2.27)

For y2 ¿ A=L, we have y12 + 9y22 y12 + (y2 + 2A=L)2 ≤ ≤ 9; y12 + y22 y12 + y22 and

Z J2 =

y2 ¿A=L;|y1 |≤1=2

Z ≤ 2C ln 3

(2.26)

y12 + (y2 − 2A=L)2 dy2 dy1 y12 + y2

q(y ˜ 1 + L=2; y2 ) ln

Z =

dy2

(L=2 − y1 )2 + (y2 − 2A=L)2 dy2 dy1 (L=2 − y1 )2 + y2

Z =

Z

32CA [(1 + L)2−1=p1 − 1]: L(2 − 1=p1 )(1=p1 − 1)

Z I322 =

1−1=p1

L

A=L

q(y ˜ 1 + L=2; y2 ) ln

(1 + y2 )−1=p1 dy2

Z

y12 + (y2 + 2A=L)2 dy2 dy1 y12 + y22

1=2

−1=2

(1 + L=2 + y1 )−1=p1 dy2

134

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

≤

2C ln 3 (L=4)1=p1 (1=p1

− 1)

:

(2.28)

Choosing L large enough such that (A=L)2 + 1=4 ¡ 1, we get Z y2 + (y2 − 2A=L)2 q(y ˜ 1 + L=2; y2 ) ln 1 dy2 dy1 J1 = y12 + y2 |y2 |¡A=L;|y1 |≤1=2 Z 1 ≤ q(y ˜ 1 + L=2; y2 ) ln 2 dy2 dy1 y + y22 |y2 |¡A=L;|y1 |≤1=2 1 ≤ 2C(L=4)−1=p1

(2.29)

and I322 ≤ C1 L−1=p1 ;

(2.30)

where C1 ¿ 0 is a constant depending on C and p1 only. Hence, I32 ≤ C max{L2−2=p1 ; L−2 ; L−1=p1 }:

(2.31)

Combining all the above estimates, we have 20 kqkL1 (R2 ) − C2 max{L2−2=p1 ; L−2 ; L1−1=p1 ln L}: 13 It follows that for L large enough (rL ) − (r1 ) ¿

(rL ) − (r1 ) ¿

(2.32)

20 kqkL1 (R2 ) − C2 max{L2−2=p1 ; L−2 ; L1−1=p1 ln L} 13

2SA2 ; (2.33) L2 this implies that dE ¿ 0, which contradicts the fact that q˜ is the maximal energy vortex on the isovortical surface. This ends our proof. ¿

3. Discussions We obtained the existence of the isolated vortex on every isovortical surface under the condition that the vorticity anomaly everywhere has the same sign as the external shear ow. Moreover, the isolated vortex attains the maximal energy on every isovortical surface and therefore is formally stable. We remove the condition that the vorticity anomaly has a positive lower bound needed by Nycander [9], i.e. 0 ¡ qmin ≤ q in the case of S ¿ 0, which is a very strong restriction to the vorticity anomaly. It is easy to see that the more concentrated upon the center, the greater is S2 ; while the more narrow it is along the x2 direction, the greater is S1 . However, since the ow is incompressible, we have to balance the mass between two directions. If S ¿ 0, then the vorticity will be elongated along the shear ow direction, and the ratio of the along shear length to the vertical direction depends upon the strength of the shear ow.

W. Yong-Hui, M. Mu / Nonlinear Analysis 38 (1999) 123 – 135

135

Acknowledgements The author would like to thank G.R. Burton for his useful suggestion. This work is supported by the Natural Science Foundation of China (No. 49455007 and 49775262). References [1] Adams, Soblev Spaces, Springer, Berlin, 1963. [2] G.R. Burton, Steady symmetric vortex pairs and rearrangements, Proc. Roy. Soc. Ed. 108A (1988) 269. [3] D. Gilbarg, N.S. Trudinger, Elliptic Partial Dierential Equations of Second Order, Springer, Berlin, 1977. [4] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, London, 1934. [5] S. Kida, Motion of an elliptic vortex in a uniform shear ow, J. Phys. Soc. Jpn. 50 (1981) 3517–3520. [6] E.H. Lieb, Existence and uniqueness of the minimizing sol. of Choquard’s nonlinear eq., Stud. Appl. Math. 57 (1977) 93 –105. [7] P.S. Marcus, Vortex dynamics in a shearing zonal ow, J. Fluid Mech. 215 (1990) 393. [8] D.W. Moore, P.G. Saman, Structure of a line vortex in an imposed strain, in: J.H. Olsen, A. Goldburg, M. Rogers (Eds.), Aircraft Wake Turbulence and its Detection, Plenum Press, New York, 1971, pp. 339 –354. [9] J. Nycander, Existence and stability of stationary vortices in a uniform shear ow, J. Fluid Mech. 287 (1995) 119. [10] J. Sommeria, S.D. Meyers, H.L. Swinney, Laboratory simulation of Jupiter’s great red spot, Nature 331 (1988) 689. [11] S. Toh, K. Ohkitani, M. Yamada, Enstrophy and momentum uxes in two-dimensional shear ow turbulence, Physica D 51 (1991) 569. [12] K. Yosida, Functional Analysis, 6th ed., Springer, Berlin, 1980, pp. 501.