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Stability investigation for steady solutions of the barotropic vorticity equation in R2
Commun Nonlinear Sci Numer Simulat 18 (2013) 541–546
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Stability investigation for steady solutions of the barotropic vorticity equation in R2 Fariba Bahrami a,⇑, Hadi Taghvafard a, Jonas Nycander b, Abbasali Mohammadi a a b
Faculty of Mathematical Sciences, University of Tabriz, 29 Bahman St., Tabriz 51665-163, Iran Department of Meteorology, Stockholm University, 106 91 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 25 April 2012 Received in revised form 20 July 2012 Accepted 30 July 2012 Available online 6 August 2012
a b s t r a c t In this paper, we investigate the p-norm stability for vortices of geophysical flows over a surface of variable height that are strict maximizers of the kinetic energy relative to all isovortical flows. In this note, stability means nonlinear stability in the p-norm on the space of vorticity. Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Nonlinear stability Vortices Rearrangements of a function Transport equation Barotropic vorticity equation Energy conserving solutions
1. Introduction The present study considers the p-norm stability for vortices of geophysical flows over a surface of variable height (e.g. a seamount in an ocean or a mountain in an atmosphere). The basic equation governing such flows is the two-dimensional barotropic vorticity equation
@q þ ½q; W ¼ 0; @t
ð1:1Þ
where W is the stream function, h is the height of the bottom topography relative to some reference value and q is given by
q ¼ DW þ h:
ð1:2Þ 2
Here, the symbol ½; denotes the Jacobian. The domain of the flow is R . The expression DW þ h in Eq. (1.2) is the quasi-geostrophic approximation of the potential vorticity. This approximation requires that h is small compared to the depth of the ocean. Much work has been devoted to finding stationary flow solutions of Eqs. (1.1) and (1.2). Such solutions must satisfy
q ¼ u W;
ð1:3Þ
where u is an arbitrary function. ⇑ Corresponding author. E-mail addresses: [email protected] (F. Bahrami), [email protected] (H. Taghvafard), [email protected] (J. Nycander), [email protected] (A. Mohammadi). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.07.024
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F. Bahrami et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 541–546
The simplest case is to assume that q ¼ const. Because of the nonuniform topography hðxÞ, where x 2 R2 , solving Eq. (1.2) then gives a nontrivial flow W in the form of a ‘Taylor column’: a stationary anticyclone above the seamount. This flow can be combined with a uniform external flow with arbitrary strength and direction, since such an external flow has zero potential vorticity [1]. Solutions of this kind have also been found for more complex model equations than (1.1), (1.2), allowing for large amplitude topography or baroclinicity [2]. More solutions can be obtained by assuming that u in Eq. (1.3) is a general linear function [3]. In this case it is possible to prove existence of stationary solutions by using well-known properties of linear eigenvalue problems. However, in most stationary solutions u is nonlinear. Existence of families of such solutions can be proved by using a variational principle. According to this principle, any potential voriticity field that maximizes or minimizes the energy in the set of rearrangements of a given function is a stationary solution of Eq. (1.1) [4]. In an application of this principle, a heuristic proof for the existence of both energy maximizing and energy minimizing families of localized vortex solutions of Eq. (1.1) was given by Nycander and LaCasce, who also studied the stability of these vortices numerically [5]. Existence of the energy maximizing vortices was proved rigorously by Nycander and Emamizadeh [6], and Emamizadeh and Bahrami [7,8] later considered the same problem in the half- and quarter-planes, respectively. A corresponding existence proof for energy minimizing vortices was given by Bahrami and Nycander [9]. For the existence of three-dimensional quasi-geostrophic vortices see [10–12]. Once the existence of a stationary flow has been proved, it is essential to study its stability. A fundamental stability theorem was proved by Arnol’d [14]. Essentially, he proved that if u in Eq. (1.3) is a monotonic decreasing function, the flow is nonlinearly stable. (He did not include a nonuniform topography h, but his theorem can easily be extended to include this.) He also proved a second therorem, which guarantees stability when u is a monotonic increasing function, but this is only valid in a bounded domain. A monotonic decreasing u corresponds to a potential vorticity field that minimizes the energy in a set of rearrangements, while an increasing u corresponds to a maximum energy flow. Physically, it is very natural that a minimum energy state is stable, and such cases are much more common than maximum energy states. A maximum energy flow cannot exist if the system studied allows waves that can transport away energy to infinity, for example gravity waves (for the shallow water equations) or Rossby waves (if the background potential vorticity is nonuniform). However, in the system (1.1), (1.2) studied here, there are no such waves, and maximum energy flows therefore exist, as proved by Nycander and Emamizadeh [6]. They have the form of an anticyclonic localized vortex attached to a seamount. These flows should be expected to be stable for the same reason that minimum energy flows are stable. Our purpose is to prove this. For a localized vortex attached to a seamount, an infinite domain is the natural setting, and we note that our proof is valid in an infinite domain, in contrast to Arnold’s second theorem [14]. Nevertheless, the unbounded domain causes major technical difficulties. To overcome these difficulties, we consider solutions q of equation ð1:1Þ staying in a bounded domain for all time. However, the corresponding stream functions W are defined in R2 . We provide sufficient conditions that under these assumptions the steady solutions of ð1:1Þ are stable. Here, stability means nonlinear stability in the p-norm on the space of vorticity. 2. Preliminary results Henceforth, we assume p 2 ð2; 1Þ and p0 is the conjugate exponent of p; that is, 1=p þ 1=p0 ¼ 1, and points in R2 are denoted by x ¼ ðx1 ; x2 Þ. We use the notation Br ðxÞ for the open ball at point x in R2 with radius r > 0. For A R2 ; jAj stands for the Lebesgue measure of A. For a measurable and non-negative function f on R2 , the strong support (or simply support) of f is denoted by suppðf Þ ¼ fxjf ðxÞ > 0g. If f and g are measurable and non-negative functions that vanish outside sets of finite measure in R2 , we say f is a rearrangement of g if
jfx 2 R2 : f ðxÞ P agj ¼ jfx 2 R2 : gðxÞ P agj; for every real a > 0. Let f 2 Lp ðR2 Þ be a non-negative function vanishing outside a bounded set. The set of all rearrangements with bounded support of f is denoted by F ðf Þ. For a bounded domain X R2 , the subset of F ðf Þ comprising functions vanishing outside of X is denoted by F X ðf Þ and the closed convex hull in Lp ðXÞ of F X ðf Þ is denoted by F X ðf Þw . For a non-negative f 2 Lp ðR2 Þ having bounded support, we set
Kf ðxÞ ¼
1 2
Z
log
R2
1 f ðyÞdy; jx yj
and g ¼ Kh where h 2 Lp ðR2 Þ is a non-negative function with bounded support. Now we can rewrite the vorticity Eq. (1.1) in the following form
qt þ r ðquÞ ¼ 0 in R2 ;
ð2:1Þ
?
u ¼ r ðKq þ gÞ; where r denotes the gradient in the space variable, and \ denotes a clockwise rotation through a right angle. Here q represents vorticity, and u represents velocity. The kinetic energy at time t is given by
F. Bahrami et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 541–546
Eðqðt; ÞÞ ¼
1 2
Z
qðt; ÞKqðt; Þ þ
R2
Z
543
gðÞqðt; Þ;
R2
whenever it is well defined. In case q is time independent, we write EðqÞ for q 2 Lp ðR2 Þ with bounded support. p 2 Suppose q0 2 Lp ðR2 Þ having bounded support, we say q 2 L1 loc ð0; 1; Lloc ðR ÞÞ is a solution of Eq. (2.1) with initial value q0 , if there exists a bounded set X R2 containing suppðhÞ and suppðqðt; ÞÞ for every t P 0; and also it satisfies the following equation in the distribution sense
Z
1
0
Z R2 ?
Z
wt q þ
R2
wð0; xÞq0 ðxÞdx þ
Z
1 0
Z R2
qu rw ¼ 0;
ð2:2Þ
u ¼ r ðKq þ gÞ; for all test functions w 2 C 1 ð½0; 1Þ R2 Þ with compact support in ½0; 1Þ R2 , we can denote this set of functions by Dð½0; 1Þ R2 Þ. Let f 2 Lploc ðR2 Þ, we say f is a steady solution of Eq. (2.1) if
Z
R2
fu rw ¼ 0;
for w 2 DðR2 Þ. In the sequel, we have some lemmas that are crucial to prove the main results. First, we begin by a result from Burton [15]. Lemma 2.1. Let U be a bounded domain in R2 and g 2 Ls ðUÞ for some s P 1, then we have
(i) kf ks ¼ kgks ; for f 2 F U ðgÞ, (ii) F U ðgÞw is weakly sequentially compact in Ls ðUÞ. The following lemma was proved by Nycander and Emamizadeh [6]. Lemma 2.2. There exists f0 2 Lp ðR2 Þ with bounded support such that it maximizes the energy functional E with respect to F ðf0 Þ, and if we set W ¼ Kf0 þ g, then W satisfies the following semilinear elliptic partial differential equation
DW ¼ u W þ h; almost everywhere in R2 for some increasing function u unknown a priori. The following lemma was proved by Emamizadeh [19]. Lemma 2.3. Let p > 2 and U be a bounded open subset of R2 . Then 0
K : Lp ðUÞ ! Lp ðUÞ is a compact linear operator; the compactness is in the sense that if ffn g is a sequence of functions, bounded in Lp ðR2 Þ and vanishing 0 outside U, then the restrictions to U of the Kfn have a subsequence converging in the Lp -norm. Moreover, if f 2 Lp ðR2 Þ vanishes 2;p 2 outside U, then Kf 2 W loc ðR Þ and verifies the following Poisson equation
DKf ¼ f; almost everywhere in R2 . Lemma 2.4. Let U R2 be a bounded domain and let f0 2 Lp ðR2 Þ vanish outside U. If f0 is a strict maximizer of energy function E with respect to F U ðf0 Þ, then f0 is a strict maximizer for E on F U ðf0 Þw . And, for any d > 0 such that f0 maximizes E strictly on F U ðf0 Þw \ Bd ðf0 Þ, there exists k > 0 such that
supfEðnÞjn 2 F U ðf0 Þw \ @Bd ðf0 Þg < Eðf0 Þ k; p
ð2:3Þ p
where Bd ðf0 Þ ¼ ff 2 L ðUÞ : kf f0 kp 6 dg and @Bd stands for the boundary of Bd in L ðUÞ. Proof. From Lemma 2.3, operator K is a compact linear operator, then it is easy to prove that E : Lp ðUÞ ! R is weakly sequentially continuous. Therefore by Corollary 3.4 in [16], we can suppose f0 is the strict maximizer of E relative to F U ðf0 Þw . Let d > 0 be as stated in the lemma and suppose there exists no such k satisfying (2.3). Then, we can choose a maximizing sequence fhn g for E relative to F U ðf0 Þw \ @Bd ðf0 Þ, converging weakly namely h 2 F U ðf0 Þw \ Bd ðf0 Þ with EðhÞ ¼ Eðf0 Þ. Since f0 is the strict maximizer, h ¼ f0 and khn kp ¼ kf0 kp . So, by uniform convexity of Lp ðUÞ, the convergence is strong, which is impossible since khn hkp ¼ d. h
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Lemma 2.5. Let f 2 Lp ðR2 Þ vanish outside a bounded domain. Then, there exist constants C 1 and C 2 independent of x such that
jr? Kf ðxÞj 6 C 2 kf kp ;
jKf ðxÞj 6 C 1 kf kp ;
8x 2 R 2 :
ð2:4Þ
Proof. Applying Hölder’s inequality, we obtain the first inequality. For the second, we have from Appendix A of [21]
rKf ðxÞ ¼
Z
1 2p
xy
R2
jx yj2
f ðyÞdy;
8x 2 R2 :
Let gðx Þ denote the Schwarz-symmetrization of jf j with respect to x. Using Hardy’s inequality [23,24], we deduce
jr? Kf ðxÞj 6 where
1 2p
Z Br ðxÞ
1 gðx yÞdy; jx yj
pr2 ¼ jsuppðf Þj for some r > 0. Since kgðx Þkp ¼ kf kp and p0 < 2, Hölder’s inequality yields (2.4). h
3. Transport equation The purpose of this section is to collect some known results corresponding to linear transport equation with a given velocity field, and to prove some new ones related to the rearrangement topics. Let v 2 L1loc ð0; 1; ðL1loc ðR2 ÞÞ2 Þ be a vector field in R2 ; for the sake of convenience, hereafter we will write L1loc ð0; 1; L1loc ðR2 ÞÞ. The transport equation with velocity v is the following equation
@ t f þ v rf ¼ 0;
on ð0; 1Þ R2 :
p 2 L1 loc ð0; 1; L ðR ÞÞ
We say f 2 distribution sense
Z 0
1
Z R2
wt f þ
Z R2
ð3:1Þ p
2
is a solution of (3.1) with initial value f0 2 L ðR Þ if it satisfies the following equation in the
wð0; xÞf0 ðxÞ þ
Z
1
Z
0
R2
fr ðv wÞ ¼ 0;
8w 2 Dð½0; 1Þ R2 Þ:
ð3:2Þ
We have the following result from [18]. Lemma 3.1. Assume
0
v 2 L1loc ð0; 1; L1loc ðR2 ÞÞ \ L1loc ð0; 1; Lploc ðR2 ÞÞ such that r v 2 L1loc ð0; 1; L1 ðR2 ÞÞ. Let f0 2 Lp ðR2 Þ, then
p 2 (i) there exists a solution f of (3.1) in L1 loc ð0; 1; L ðR ÞÞ corresponding to the initial value f0 . R R p p d (ii) dt R2 jfj þ R2 r v jfj ¼ 0 almost everywhere on ð0; 1Þ and in the case that r v ¼ 0, kfðt; Þkp is independent of t. 0 1 1 1 1 2 2 2 v (iii) f 2 Cð½0; T; Lp ðR2 ÞÞ if v 2 L1 ð0; T; W 1;p loc ðR ÞÞ and 1þjxj 2 L ð0; T; L ðR ÞÞ þ L ð0; T; L ðR ÞÞ for T > 0.
The following theorem is essential in the proof of the nonlinear stability for vortices that will be stated in the next theorem. ðR2 ÞÞ be a given vector field with r v ¼ 0. Let f0 2 Lp ðR2 Þ Theorem 3.2. Let U R2 be a bounded domain and v 2 L1loc ð0; 1; W 1;p loc with bounded support and suppose f is a solution of the following transport equation
@ t f þ v rf ¼ 0;
on ð0; 1Þ R2 ;
with initial value fð0; Þ ¼ f0 such that suppðfðt; :ÞÞ U for t P 0. Then fðt; Þ is a rearrangement of fð0; Þ for t > 0. Proof. Let f be a solution of the transport equation with the velocity field v, by (Section 5, Theorem 1, [20]) and r v ¼ 0, f solves
ft þ r ðv fÞ ¼ 0: To prove the assertion, it is enough to show that the following integral is independent of t
Z bðfÞ;
ð3:3Þ
U
for every bounded Lipschitz function b : R ! R. Consequently, fðt; Þ is a rearrangement of fð0; Þ. Let b be a bounded Lipschitz function on R. Then from (Theorem 3.2 [13]]) we have
@ t bðfÞ þ r ðbðfÞv Þ bðfÞr v ¼ b0 ðfÞð@ t f þ r ðfv Þ fr v Þ which, as r v ¼ 0 and f satisfies the transport equation, yields
@ t bðfÞ þ r ðbðfÞv Þ ¼ 0:
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F. Bahrami et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 541–546
Applying a test function of the form vðtÞ/ðxÞ, where
Z
1
Z R2
0
ðbðfÞ bð0ÞÞvt / þ
Z 0
1
Z R2
v 2 Dð0; 1Þ and / 2 DðR2 Þ, we obtain
ðbðfÞ bð0ÞÞvv r/ ¼ 0:
ð3:4Þ
Let / 2 DðR2 Þ such that / ¼ 1 on U. Since bðfÞ bð0Þ ¼ 0 out of U and / ¼ 1 on U, we deduce
Z 0
1
Z U
ðbðfÞ bð0ÞÞvt ¼ 0;
from which we conclude that
R U
8v 2 Dð0; 1Þ; bðfÞ is independent of t. h
We end this section with a definition that will be provided as a sufficient condition for velocity fields to have nonlinear stability in the next section. Definition 3.3. Let v 2 L1loc ð0; 1; L1loc ðR2 ÞÞ be a given velocity field and f0 2 Lp ðR2 Þ. We say the pair ðv ; f0 Þ is admissible if the corresponding transport equation with v has a solution with initial value f0 such that for every t P 0; suppðfðt; ÞÞ U for some bounded domain U R2 .
4. Nonlinear stability In this section we intend to present the nonlinear stability theorem for steady solutions of (2.2). To do this, first we prove the existence of steady solutions. Proposition 4.1. Let f0 2 Lp ðR2 Þ with bounded support be a maximizer of energy functional E relative to F ðf0 Þ. Then, f0 is a steady solution of the vorticity Eq. (2.2).
Proof. Let f0 be a maximizer of energy functional E relative to F ðf0 Þ and W ¼ Kf0 þ g. We claim each level set of W has zero T ^ such that La^ ¼ fxjWðxÞ ¼ a ^ g suppðf0 Þ has positive measure. Then, measure on suppðf0 Þ. To this end, suppose there exist a since f0 þ h ¼ DW, by Theorem 7.10 in [22] we deduce that f0 þ h vanishes on La^ , which is a contradiction. On the other hand, by Lemma 2.2 we have
f0 ¼ u W; almost everywhere in R2 for some increasing function u unknown a priori. Now, in view of the fact that level sets of W have zero measure and suppðf0 Þ is bounded, with a slight modification we can follow the proof presented for Lemma 6 in [17] to deduce the result. h Theorem 4.2. Let f0 2 Lp ðR2 Þ with bounded support be the strict maximizer of energy functional E relative to F ðf0 Þ. Let p ? 2 q 2 L1 loc ð0; 1; Lloc ðR ÞÞ be an energy-conserving solution of (2.2) such that the pair ðu; f0 Þ with u ¼ r ðKq þ gÞ is admissible, then for every > 0 there exists d > 0 such that if kqð0; Þ f0 kp < d, kqðt; Þ f0 kp < for every t > 0. Proof. Let f0 be the strict maximizer of energy functional E, from the proposition above it is the steady solution of (2.2). Next, we shall prove its stability. Let q be an energy-conserving solution of (2.2) such that suppðqðt; ÞÞ X for every t P 0, where X is a bounded domain. Let 0 < < 1, choose 0 < d1 < =2 such that f0 maximizes E strictly on F X ðf0 Þw \ Bd1 ðf0 Þ. From Lemma 2.4 there exists k > 0 such that
supfEðnÞ j n 2 F X ðf0 Þw \ @Bd1 ðf0 Þg < Eðf0 Þ k: By Lemma 2.5, the energy functional E is uniformly continuous on bounded sets of Lp ðR2 Þ. Therefore, there exists 0 < d < d1 such that if f1 ; f2 2 Br ð0Þ with r ¼ kf0 kp þ 1 and kf1 f2 kp < d then
jEðf1 Þ Eðf2 Þj < k=2: ?
ð4:1Þ p
2
2 W 1;p loc ðR Þ,
Let u ¼ r ðKq þ gÞ where g ¼ Kh for h 2 L ðR Þ having bounded domain. By Lemma 2.3, uðt; Þ 2 then from 0 2 Sobolev embedding uðt; Þ is continuous in R2 for any t P 0. Also, uðt; Þ 2 W 1;p loc ðR Þ since p > 2. Due to the fact that r u ¼ 0 and (2.4), u satisfies all conditions of Lemma 3.1. Thus, under our assumption that ðu; f0 Þ is an admissible pair, there exists f 2 Cð0; 1; Lp ðR2 ÞÞ so that suppðf ðt; ÞÞ U; t P 0 and f satisfies the linear transport equation ft þ r ðfuÞ ¼ 0 on ð0; 1Þ R2 with initial value f ð0; Þ ¼ f0 where U is a bounded domain including X. Since q is the solution of (2.2), by linearity we deduce
ðf qÞt þ r ððf qÞuÞ ¼ 0: As r u ¼ 0, through Lemma 3.1, we have f q 2 Cð½0; 1Þ; Lp ðR2 ÞÞ and from Theorem 1 and Lemma 2.1 we deduce
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F. Bahrami et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 541–546
kf ðt; Þ qðt; Þkp ¼ kf ð0; Þ qð0; Þkp ¼ kf0 qð0; Þkp < d:
ð4:2Þ
Since q is an energy-conserving solution of (2.2),
jEðf ðt; ÞÞ Eðf0 Þj 6 jEðf ðt; ÞÞ Eðqðt; ÞÞj þ jEðqð0; ÞÞ Eðf0 Þj: By using (4.1) and (4.2), both terms on the right-hand side can be estimated by k=2, which gives
jEðf ðt; ÞÞ Eðf0 Þj 6 k: It follows that f ðt; Þ R @Bd1 ðf0 Þ for all t, therefore the continuity of kf ðt; Þkp yields f ðt; Þ 2 Bd1 ðf0 Þ for every t > 0. Hence
kqðt; Þ f0 kp 6 kqðt; Þ f ðt; Þkp þ kf ðt; Þ f0 kp < d þ d1 < : The proof is completed.
h
Employing Lemma 2.5 and the result of the above theorem we can deduce the following lemma that state the stability for stream function and velocity of the unsteady flow. Corollary 4.3. Let f0 ; q and u be given as in Theorem 2, and let W ¼ Kq þ g be the stream function and u ¼ r? ðKq þ gÞ the velocity field of the flow. Then for every > 0 there exists d > 0 such that if kqð0; Þ f0 kp < d,
jWðt; xÞ W0 ðxÞj 6 C 1 ;
juðt; xÞ u0 ðxÞj 6 C 2 ;
for x 2 R and t P 0 where constants C 1 ; C 2 are as Lemma 2.5. Here W0 ¼ Kf0 þ g is the stream function and u0 ¼ r? ðKf0 þ gÞ the velocity of the steady flow with vorticity f0 . 2
5. Conclusion We have proved the nonlinear stability of stationary barotropic flows that maximimize the energy in a set of rearrangements of the potential vorticity field. The problem is defined with a localized topographic feature, such as a seamount, and the stationary flow has the form of a vortex attached to the seamount. The existence of such stationary flows was previously proved by Nycander and Emamizadeh [6]. To our knowledge, this is the first stability proof for maximum energy flows in an infinite domain. Previous proofs were either for minimum energy flows [14], or for maximum energy flows in a bounded domain [14,17]. The infinite domain caused technical difficulties. To overcome these, the stability theorem was restricted to time-dependent solutions such that the corresponding time-dependent velocity field does not transport the potential vorticity of the stationary solution to infinity (i.e. out of some bounded domain). References [1] Izrailsky YuG, Kozlov VF, Koshel KV. Some specific features of chaotization of the pulsating barotropic flow over elliptic and axisymmetric sea-mounts. Phys Fluids 2004;16:3173–90. [2] Kozlov VF. Background currents in geophysical hydrodynamics. Atmos Oceanic Phys 1995;31:229–34. [3] Carnevale GF, Frederiksen JS. Nonlinear stability and statistical mechanics of flow over topography. J Fluid Mech 1987;175:157–81. [4] Arnol’d VI. Mathematical Methods of Classical Mechanics. Springer; 1978. [5] Nycander J, LaCasce JH. Stable and unstable vortices attached to seamounts. J Fluid Mech 2004;507:71–9. [6] Nycander J, Emamizadeh B. Variational problem for vortices attached to seamounts. Nonlinear Anal TMA 2003;55:15–24. [7] Emamizadeh B, Bahrami F. Existence of seamount steady vortex flows. ANZIAM J 2005;47:75–88. [8] Emamizadeh B, Bahrami F. Existence of solutions for the Barotropic-vorticity equation in an unbounded domain. Rocky MT J Math 2006;36:1135–47. [9] Bahrami F, Nycander J. Existence of energy minimizing vortices attached to a flat-top seamount. Nonlinear Anal RWA 2007;8:288–94. [10] Burton GR, Nycander J. Stationary vortices in three-dimensional quasi-geostrophic shear flow. J Fluid Mech 1996;389:255–74. [11] Bahrami F, Nycander J, Alikhani R. Existence of energy maximizing vortices in a three-dimensioanl quasigeostrophic shear flow with bounded height. Nonlinear Anal RWA 2010;11:1589–99. [12] Emamizadeh B, Mehrabi MH. Existence of solutions to the three dimensional Baratropic-vorticity equation. Methods Appl Anal 2007;14(1):77–86. [13] Bouchut F. Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch Rational Mech Anal 2001;157:75–90. [14] Arnol’d VI. On an a priori estimate in the theory of hydrodynamic stabillity. Am Math Soc Transl Series 1969;2(79):267–9. [15] Burton GR. Rearrangements of functions, maximization of convex functionals and vortex rings. Math Ann 1987;276:225–53. [16] Burton GR. Variational problems on classes of rearrangements and multiple configurations for steady vorticies. Ann Inst Henri Poincarè 1989;6:295–319. [17] Burton GR. Global nonlinear stability for steady ideal fluid flow in bounded planar domains. Arch Rational Mech Anal 2005;176:149–63. [18] DiPerna RJ, Lions PL. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math 1989;98:511–47. [19] Emamizadeh B. Steady vortex in a uniform shear flow of an ideal fluid. Proc Roy Soc Edinburgh 2000;130:801–12. [20] Evans LC. Partial Differential Equations. American Mathematical Society; 1998. [21] Frankel LE. An introduction to maximum principles and symmetry in elleptic problems. Cambridge University Press; 2000. [22] Gilbarg D, Trudinger NS. Elliptic Partial Differential Equations of Second Order. Springer-Verlag; 1998. Second edition. [23] Lieb EH, Loss M. Analysis. American Mathematical Society; 1997. [24] Hardy GH, Littlewood JE, Pólya G. Inequalities. Cambridge University Press; 1934.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Stability investigation for steady solutions of the barotropic vorticity equation in R2 Fariba Bahrami a,⇑, Hadi Taghvafard a, Jonas Nycander b, Abbasali Mohammadi a a b
Faculty of Mathematical Sciences, University of Tabriz, 29 Bahman St., Tabriz 51665-163, Iran Department of Meteorology, Stockholm University, 106 91 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 25 April 2012 Received in revised form 20 July 2012 Accepted 30 July 2012 Available online 6 August 2012
a b s t r a c t In this paper, we investigate the p-norm stability for vortices of geophysical flows over a surface of variable height that are strict maximizers of the kinetic energy relative to all isovortical flows. In this note, stability means nonlinear stability in the p-norm on the space of vorticity. Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Nonlinear stability Vortices Rearrangements of a function Transport equation Barotropic vorticity equation Energy conserving solutions
1. Introduction The present study considers the p-norm stability for vortices of geophysical flows over a surface of variable height (e.g. a seamount in an ocean or a mountain in an atmosphere). The basic equation governing such flows is the two-dimensional barotropic vorticity equation
@q þ ½q; W ¼ 0; @t
ð1:1Þ
where W is the stream function, h is the height of the bottom topography relative to some reference value and q is given by
q ¼ DW þ h:
ð1:2Þ 2
Here, the symbol ½; denotes the Jacobian. The domain of the flow is R . The expression DW þ h in Eq. (1.2) is the quasi-geostrophic approximation of the potential vorticity. This approximation requires that h is small compared to the depth of the ocean. Much work has been devoted to finding stationary flow solutions of Eqs. (1.1) and (1.2). Such solutions must satisfy
q ¼ u W;
ð1:3Þ
where u is an arbitrary function. ⇑ Corresponding author. E-mail addresses: [email protected] (F. Bahrami), [email protected] (H. Taghvafard), [email protected] (J. Nycander), [email protected] (A. Mohammadi). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.07.024
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The simplest case is to assume that q ¼ const. Because of the nonuniform topography hðxÞ, where x 2 R2 , solving Eq. (1.2) then gives a nontrivial flow W in the form of a ‘Taylor column’: a stationary anticyclone above the seamount. This flow can be combined with a uniform external flow with arbitrary strength and direction, since such an external flow has zero potential vorticity [1]. Solutions of this kind have also been found for more complex model equations than (1.1), (1.2), allowing for large amplitude topography or baroclinicity [2]. More solutions can be obtained by assuming that u in Eq. (1.3) is a general linear function [3]. In this case it is possible to prove existence of stationary solutions by using well-known properties of linear eigenvalue problems. However, in most stationary solutions u is nonlinear. Existence of families of such solutions can be proved by using a variational principle. According to this principle, any potential voriticity field that maximizes or minimizes the energy in the set of rearrangements of a given function is a stationary solution of Eq. (1.1) [4]. In an application of this principle, a heuristic proof for the existence of both energy maximizing and energy minimizing families of localized vortex solutions of Eq. (1.1) was given by Nycander and LaCasce, who also studied the stability of these vortices numerically [5]. Existence of the energy maximizing vortices was proved rigorously by Nycander and Emamizadeh [6], and Emamizadeh and Bahrami [7,8] later considered the same problem in the half- and quarter-planes, respectively. A corresponding existence proof for energy minimizing vortices was given by Bahrami and Nycander [9]. For the existence of three-dimensional quasi-geostrophic vortices see [10–12]. Once the existence of a stationary flow has been proved, it is essential to study its stability. A fundamental stability theorem was proved by Arnol’d [14]. Essentially, he proved that if u in Eq. (1.3) is a monotonic decreasing function, the flow is nonlinearly stable. (He did not include a nonuniform topography h, but his theorem can easily be extended to include this.) He also proved a second therorem, which guarantees stability when u is a monotonic increasing function, but this is only valid in a bounded domain. A monotonic decreasing u corresponds to a potential vorticity field that minimizes the energy in a set of rearrangements, while an increasing u corresponds to a maximum energy flow. Physically, it is very natural that a minimum energy state is stable, and such cases are much more common than maximum energy states. A maximum energy flow cannot exist if the system studied allows waves that can transport away energy to infinity, for example gravity waves (for the shallow water equations) or Rossby waves (if the background potential vorticity is nonuniform). However, in the system (1.1), (1.2) studied here, there are no such waves, and maximum energy flows therefore exist, as proved by Nycander and Emamizadeh [6]. They have the form of an anticyclonic localized vortex attached to a seamount. These flows should be expected to be stable for the same reason that minimum energy flows are stable. Our purpose is to prove this. For a localized vortex attached to a seamount, an infinite domain is the natural setting, and we note that our proof is valid in an infinite domain, in contrast to Arnold’s second theorem [14]. Nevertheless, the unbounded domain causes major technical difficulties. To overcome these difficulties, we consider solutions q of equation ð1:1Þ staying in a bounded domain for all time. However, the corresponding stream functions W are defined in R2 . We provide sufficient conditions that under these assumptions the steady solutions of ð1:1Þ are stable. Here, stability means nonlinear stability in the p-norm on the space of vorticity. 2. Preliminary results Henceforth, we assume p 2 ð2; 1Þ and p0 is the conjugate exponent of p; that is, 1=p þ 1=p0 ¼ 1, and points in R2 are denoted by x ¼ ðx1 ; x2 Þ. We use the notation Br ðxÞ for the open ball at point x in R2 with radius r > 0. For A R2 ; jAj stands for the Lebesgue measure of A. For a measurable and non-negative function f on R2 , the strong support (or simply support) of f is denoted by suppðf Þ ¼ fxjf ðxÞ > 0g. If f and g are measurable and non-negative functions that vanish outside sets of finite measure in R2 , we say f is a rearrangement of g if
jfx 2 R2 : f ðxÞ P agj ¼ jfx 2 R2 : gðxÞ P agj; for every real a > 0. Let f 2 Lp ðR2 Þ be a non-negative function vanishing outside a bounded set. The set of all rearrangements with bounded support of f is denoted by F ðf Þ. For a bounded domain X R2 , the subset of F ðf Þ comprising functions vanishing outside of X is denoted by F X ðf Þ and the closed convex hull in Lp ðXÞ of F X ðf Þ is denoted by F X ðf Þw . For a non-negative f 2 Lp ðR2 Þ having bounded support, we set
Kf ðxÞ ¼
1 2
Z
log
R2
1 f ðyÞdy; jx yj
and g ¼ Kh where h 2 Lp ðR2 Þ is a non-negative function with bounded support. Now we can rewrite the vorticity Eq. (1.1) in the following form
qt þ r ðquÞ ¼ 0 in R2 ;
ð2:1Þ
?
u ¼ r ðKq þ gÞ; where r denotes the gradient in the space variable, and \ denotes a clockwise rotation through a right angle. Here q represents vorticity, and u represents velocity. The kinetic energy at time t is given by
F. Bahrami et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 541–546
Eðqðt; ÞÞ ¼
1 2
Z
qðt; ÞKqðt; Þ þ
R2
Z
543
gðÞqðt; Þ;
R2
whenever it is well defined. In case q is time independent, we write EðqÞ for q 2 Lp ðR2 Þ with bounded support. p 2 Suppose q0 2 Lp ðR2 Þ having bounded support, we say q 2 L1 loc ð0; 1; Lloc ðR ÞÞ is a solution of Eq. (2.1) with initial value q0 , if there exists a bounded set X R2 containing suppðhÞ and suppðqðt; ÞÞ for every t P 0; and also it satisfies the following equation in the distribution sense
Z
1
0
Z R2 ?
Z
wt q þ
R2
wð0; xÞq0 ðxÞdx þ
Z
1 0
Z R2
qu rw ¼ 0;
ð2:2Þ
u ¼ r ðKq þ gÞ; for all test functions w 2 C 1 ð½0; 1Þ R2 Þ with compact support in ½0; 1Þ R2 , we can denote this set of functions by Dð½0; 1Þ R2 Þ. Let f 2 Lploc ðR2 Þ, we say f is a steady solution of Eq. (2.1) if
Z
R2
fu rw ¼ 0;
for w 2 DðR2 Þ. In the sequel, we have some lemmas that are crucial to prove the main results. First, we begin by a result from Burton [15]. Lemma 2.1. Let U be a bounded domain in R2 and g 2 Ls ðUÞ for some s P 1, then we have
(i) kf ks ¼ kgks ; for f 2 F U ðgÞ, (ii) F U ðgÞw is weakly sequentially compact in Ls ðUÞ. The following lemma was proved by Nycander and Emamizadeh [6]. Lemma 2.2. There exists f0 2 Lp ðR2 Þ with bounded support such that it maximizes the energy functional E with respect to F ðf0 Þ, and if we set W ¼ Kf0 þ g, then W satisfies the following semilinear elliptic partial differential equation
DW ¼ u W þ h; almost everywhere in R2 for some increasing function u unknown a priori. The following lemma was proved by Emamizadeh [19]. Lemma 2.3. Let p > 2 and U be a bounded open subset of R2 . Then 0
K : Lp ðUÞ ! Lp ðUÞ is a compact linear operator; the compactness is in the sense that if ffn g is a sequence of functions, bounded in Lp ðR2 Þ and vanishing 0 outside U, then the restrictions to U of the Kfn have a subsequence converging in the Lp -norm. Moreover, if f 2 Lp ðR2 Þ vanishes 2;p 2 outside U, then Kf 2 W loc ðR Þ and verifies the following Poisson equation
DKf ¼ f; almost everywhere in R2 . Lemma 2.4. Let U R2 be a bounded domain and let f0 2 Lp ðR2 Þ vanish outside U. If f0 is a strict maximizer of energy function E with respect to F U ðf0 Þ, then f0 is a strict maximizer for E on F U ðf0 Þw . And, for any d > 0 such that f0 maximizes E strictly on F U ðf0 Þw \ Bd ðf0 Þ, there exists k > 0 such that
supfEðnÞjn 2 F U ðf0 Þw \ @Bd ðf0 Þg < Eðf0 Þ k; p
ð2:3Þ p
where Bd ðf0 Þ ¼ ff 2 L ðUÞ : kf f0 kp 6 dg and @Bd stands for the boundary of Bd in L ðUÞ. Proof. From Lemma 2.3, operator K is a compact linear operator, then it is easy to prove that E : Lp ðUÞ ! R is weakly sequentially continuous. Therefore by Corollary 3.4 in [16], we can suppose f0 is the strict maximizer of E relative to F U ðf0 Þw . Let d > 0 be as stated in the lemma and suppose there exists no such k satisfying (2.3). Then, we can choose a maximizing sequence fhn g for E relative to F U ðf0 Þw \ @Bd ðf0 Þ, converging weakly namely h 2 F U ðf0 Þw \ Bd ðf0 Þ with EðhÞ ¼ Eðf0 Þ. Since f0 is the strict maximizer, h ¼ f0 and khn kp ¼ kf0 kp . So, by uniform convexity of Lp ðUÞ, the convergence is strong, which is impossible since khn hkp ¼ d. h
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Lemma 2.5. Let f 2 Lp ðR2 Þ vanish outside a bounded domain. Then, there exist constants C 1 and C 2 independent of x such that
jr? Kf ðxÞj 6 C 2 kf kp ;
jKf ðxÞj 6 C 1 kf kp ;
8x 2 R 2 :
ð2:4Þ
Proof. Applying Hölder’s inequality, we obtain the first inequality. For the second, we have from Appendix A of [21]
rKf ðxÞ ¼
Z
1 2p
xy
R2
jx yj2
f ðyÞdy;
8x 2 R2 :
Let gðx Þ denote the Schwarz-symmetrization of jf j with respect to x. Using Hardy’s inequality [23,24], we deduce
jr? Kf ðxÞj 6 where
1 2p
Z Br ðxÞ
1 gðx yÞdy; jx yj
pr2 ¼ jsuppðf Þj for some r > 0. Since kgðx Þkp ¼ kf kp and p0 < 2, Hölder’s inequality yields (2.4). h
3. Transport equation The purpose of this section is to collect some known results corresponding to linear transport equation with a given velocity field, and to prove some new ones related to the rearrangement topics. Let v 2 L1loc ð0; 1; ðL1loc ðR2 ÞÞ2 Þ be a vector field in R2 ; for the sake of convenience, hereafter we will write L1loc ð0; 1; L1loc ðR2 ÞÞ. The transport equation with velocity v is the following equation
@ t f þ v rf ¼ 0;
on ð0; 1Þ R2 :
p 2 L1 loc ð0; 1; L ðR ÞÞ
We say f 2 distribution sense
Z 0
1
Z R2
wt f þ
Z R2
ð3:1Þ p
2
is a solution of (3.1) with initial value f0 2 L ðR Þ if it satisfies the following equation in the
wð0; xÞf0 ðxÞ þ
Z
1
Z
0
R2
fr ðv wÞ ¼ 0;
8w 2 Dð½0; 1Þ R2 Þ:
ð3:2Þ
We have the following result from [18]. Lemma 3.1. Assume
0
v 2 L1loc ð0; 1; L1loc ðR2 ÞÞ \ L1loc ð0; 1; Lploc ðR2 ÞÞ such that r v 2 L1loc ð0; 1; L1 ðR2 ÞÞ. Let f0 2 Lp ðR2 Þ, then
p 2 (i) there exists a solution f of (3.1) in L1 loc ð0; 1; L ðR ÞÞ corresponding to the initial value f0 . R R p p d (ii) dt R2 jfj þ R2 r v jfj ¼ 0 almost everywhere on ð0; 1Þ and in the case that r v ¼ 0, kfðt; Þkp is independent of t. 0 1 1 1 1 2 2 2 v (iii) f 2 Cð½0; T; Lp ðR2 ÞÞ if v 2 L1 ð0; T; W 1;p loc ðR ÞÞ and 1þjxj 2 L ð0; T; L ðR ÞÞ þ L ð0; T; L ðR ÞÞ for T > 0.
The following theorem is essential in the proof of the nonlinear stability for vortices that will be stated in the next theorem. ðR2 ÞÞ be a given vector field with r v ¼ 0. Let f0 2 Lp ðR2 Þ Theorem 3.2. Let U R2 be a bounded domain and v 2 L1loc ð0; 1; W 1;p loc with bounded support and suppose f is a solution of the following transport equation
@ t f þ v rf ¼ 0;
on ð0; 1Þ R2 ;
with initial value fð0; Þ ¼ f0 such that suppðfðt; :ÞÞ U for t P 0. Then fðt; Þ is a rearrangement of fð0; Þ for t > 0. Proof. Let f be a solution of the transport equation with the velocity field v, by (Section 5, Theorem 1, [20]) and r v ¼ 0, f solves
ft þ r ðv fÞ ¼ 0: To prove the assertion, it is enough to show that the following integral is independent of t
Z bðfÞ;
ð3:3Þ
U
for every bounded Lipschitz function b : R ! R. Consequently, fðt; Þ is a rearrangement of fð0; Þ. Let b be a bounded Lipschitz function on R. Then from (Theorem 3.2 [13]]) we have
@ t bðfÞ þ r ðbðfÞv Þ bðfÞr v ¼ b0 ðfÞð@ t f þ r ðfv Þ fr v Þ which, as r v ¼ 0 and f satisfies the transport equation, yields
@ t bðfÞ þ r ðbðfÞv Þ ¼ 0:
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Applying a test function of the form vðtÞ/ðxÞ, where
Z
1
Z R2
0
ðbðfÞ bð0ÞÞvt / þ
Z 0
1
Z R2
v 2 Dð0; 1Þ and / 2 DðR2 Þ, we obtain
ðbðfÞ bð0ÞÞvv r/ ¼ 0:
ð3:4Þ
Let / 2 DðR2 Þ such that / ¼ 1 on U. Since bðfÞ bð0Þ ¼ 0 out of U and / ¼ 1 on U, we deduce
Z 0
1
Z U
ðbðfÞ bð0ÞÞvt ¼ 0;
from which we conclude that
R U
8v 2 Dð0; 1Þ; bðfÞ is independent of t. h
We end this section with a definition that will be provided as a sufficient condition for velocity fields to have nonlinear stability in the next section. Definition 3.3. Let v 2 L1loc ð0; 1; L1loc ðR2 ÞÞ be a given velocity field and f0 2 Lp ðR2 Þ. We say the pair ðv ; f0 Þ is admissible if the corresponding transport equation with v has a solution with initial value f0 such that for every t P 0; suppðfðt; ÞÞ U for some bounded domain U R2 .
4. Nonlinear stability In this section we intend to present the nonlinear stability theorem for steady solutions of (2.2). To do this, first we prove the existence of steady solutions. Proposition 4.1. Let f0 2 Lp ðR2 Þ with bounded support be a maximizer of energy functional E relative to F ðf0 Þ. Then, f0 is a steady solution of the vorticity Eq. (2.2).
Proof. Let f0 be a maximizer of energy functional E relative to F ðf0 Þ and W ¼ Kf0 þ g. We claim each level set of W has zero T ^ such that La^ ¼ fxjWðxÞ ¼ a ^ g suppðf0 Þ has positive measure. Then, measure on suppðf0 Þ. To this end, suppose there exist a since f0 þ h ¼ DW, by Theorem 7.10 in [22] we deduce that f0 þ h vanishes on La^ , which is a contradiction. On the other hand, by Lemma 2.2 we have
f0 ¼ u W; almost everywhere in R2 for some increasing function u unknown a priori. Now, in view of the fact that level sets of W have zero measure and suppðf0 Þ is bounded, with a slight modification we can follow the proof presented for Lemma 6 in [17] to deduce the result. h Theorem 4.2. Let f0 2 Lp ðR2 Þ with bounded support be the strict maximizer of energy functional E relative to F ðf0 Þ. Let p ? 2 q 2 L1 loc ð0; 1; Lloc ðR ÞÞ be an energy-conserving solution of (2.2) such that the pair ðu; f0 Þ with u ¼ r ðKq þ gÞ is admissible, then for every > 0 there exists d > 0 such that if kqð0; Þ f0 kp < d, kqðt; Þ f0 kp < for every t > 0. Proof. Let f0 be the strict maximizer of energy functional E, from the proposition above it is the steady solution of (2.2). Next, we shall prove its stability. Let q be an energy-conserving solution of (2.2) such that suppðqðt; ÞÞ X for every t P 0, where X is a bounded domain. Let 0 < < 1, choose 0 < d1 < =2 such that f0 maximizes E strictly on F X ðf0 Þw \ Bd1 ðf0 Þ. From Lemma 2.4 there exists k > 0 such that
supfEðnÞ j n 2 F X ðf0 Þw \ @Bd1 ðf0 Þg < Eðf0 Þ k: By Lemma 2.5, the energy functional E is uniformly continuous on bounded sets of Lp ðR2 Þ. Therefore, there exists 0 < d < d1 such that if f1 ; f2 2 Br ð0Þ with r ¼ kf0 kp þ 1 and kf1 f2 kp < d then
jEðf1 Þ Eðf2 Þj < k=2: ?
ð4:1Þ p
2
2 W 1;p loc ðR Þ,
Let u ¼ r ðKq þ gÞ where g ¼ Kh for h 2 L ðR Þ having bounded domain. By Lemma 2.3, uðt; Þ 2 then from 0 2 Sobolev embedding uðt; Þ is continuous in R2 for any t P 0. Also, uðt; Þ 2 W 1;p loc ðR Þ since p > 2. Due to the fact that r u ¼ 0 and (2.4), u satisfies all conditions of Lemma 3.1. Thus, under our assumption that ðu; f0 Þ is an admissible pair, there exists f 2 Cð0; 1; Lp ðR2 ÞÞ so that suppðf ðt; ÞÞ U; t P 0 and f satisfies the linear transport equation ft þ r ðfuÞ ¼ 0 on ð0; 1Þ R2 with initial value f ð0; Þ ¼ f0 where U is a bounded domain including X. Since q is the solution of (2.2), by linearity we deduce
ðf qÞt þ r ððf qÞuÞ ¼ 0: As r u ¼ 0, through Lemma 3.1, we have f q 2 Cð½0; 1Þ; Lp ðR2 ÞÞ and from Theorem 1 and Lemma 2.1 we deduce
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kf ðt; Þ qðt; Þkp ¼ kf ð0; Þ qð0; Þkp ¼ kf0 qð0; Þkp < d:
ð4:2Þ
Since q is an energy-conserving solution of (2.2),
jEðf ðt; ÞÞ Eðf0 Þj 6 jEðf ðt; ÞÞ Eðqðt; ÞÞj þ jEðqð0; ÞÞ Eðf0 Þj: By using (4.1) and (4.2), both terms on the right-hand side can be estimated by k=2, which gives
jEðf ðt; ÞÞ Eðf0 Þj 6 k: It follows that f ðt; Þ R @Bd1 ðf0 Þ for all t, therefore the continuity of kf ðt; Þkp yields f ðt; Þ 2 Bd1 ðf0 Þ for every t > 0. Hence
kqðt; Þ f0 kp 6 kqðt; Þ f ðt; Þkp þ kf ðt; Þ f0 kp < d þ d1 < : The proof is completed.
h
Employing Lemma 2.5 and the result of the above theorem we can deduce the following lemma that state the stability for stream function and velocity of the unsteady flow. Corollary 4.3. Let f0 ; q and u be given as in Theorem 2, and let W ¼ Kq þ g be the stream function and u ¼ r? ðKq þ gÞ the velocity field of the flow. Then for every > 0 there exists d > 0 such that if kqð0; Þ f0 kp < d,
jWðt; xÞ W0 ðxÞj 6 C 1 ;
juðt; xÞ u0 ðxÞj 6 C 2 ;
for x 2 R and t P 0 where constants C 1 ; C 2 are as Lemma 2.5. Here W0 ¼ Kf0 þ g is the stream function and u0 ¼ r? ðKf0 þ gÞ the velocity of the steady flow with vorticity f0 . 2
5. Conclusion We have proved the nonlinear stability of stationary barotropic flows that maximimize the energy in a set of rearrangements of the potential vorticity field. The problem is defined with a localized topographic feature, such as a seamount, and the stationary flow has the form of a vortex attached to the seamount. The existence of such stationary flows was previously proved by Nycander and Emamizadeh [6]. To our knowledge, this is the first stability proof for maximum energy flows in an infinite domain. Previous proofs were either for minimum energy flows [14], or for maximum energy flows in a bounded domain [14,17]. The infinite domain caused technical difficulties. To overcome these, the stability theorem was restricted to time-dependent solutions such that the corresponding time-dependent velocity field does not transport the potential vorticity of the stationary solution to infinity (i.e. out of some bounded domain). References [1] Izrailsky YuG, Kozlov VF, Koshel KV. Some specific features of chaotization of the pulsating barotropic flow over elliptic and axisymmetric sea-mounts. Phys Fluids 2004;16:3173–90. [2] Kozlov VF. Background currents in geophysical hydrodynamics. Atmos Oceanic Phys 1995;31:229–34. [3] Carnevale GF, Frederiksen JS. Nonlinear stability and statistical mechanics of flow over topography. J Fluid Mech 1987;175:157–81. [4] Arnol’d VI. Mathematical Methods of Classical Mechanics. Springer; 1978. [5] Nycander J, LaCasce JH. Stable and unstable vortices attached to seamounts. J Fluid Mech 2004;507:71–9. [6] Nycander J, Emamizadeh B. Variational problem for vortices attached to seamounts. Nonlinear Anal TMA 2003;55:15–24. [7] Emamizadeh B, Bahrami F. Existence of seamount steady vortex flows. ANZIAM J 2005;47:75–88. [8] Emamizadeh B, Bahrami F. Existence of solutions for the Barotropic-vorticity equation in an unbounded domain. Rocky MT J Math 2006;36:1135–47. [9] Bahrami F, Nycander J. Existence of energy minimizing vortices attached to a flat-top seamount. Nonlinear Anal RWA 2007;8:288–94. [10] Burton GR, Nycander J. Stationary vortices in three-dimensional quasi-geostrophic shear flow. J Fluid Mech 1996;389:255–74. [11] Bahrami F, Nycander J, Alikhani R. Existence of energy maximizing vortices in a three-dimensioanl quasigeostrophic shear flow with bounded height. Nonlinear Anal RWA 2010;11:1589–99. [12] Emamizadeh B, Mehrabi MH. Existence of solutions to the three dimensional Baratropic-vorticity equation. Methods Appl Anal 2007;14(1):77–86. [13] Bouchut F. Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch Rational Mech Anal 2001;157:75–90. [14] Arnol’d VI. On an a priori estimate in the theory of hydrodynamic stabillity. Am Math Soc Transl Series 1969;2(79):267–9. [15] Burton GR. Rearrangements of functions, maximization of convex functionals and vortex rings. Math Ann 1987;276:225–53. [16] Burton GR. Variational problems on classes of rearrangements and multiple configurations for steady vorticies. Ann Inst Henri Poincarè 1989;6:295–319. [17] Burton GR. Global nonlinear stability for steady ideal fluid flow in bounded planar domains. Arch Rational Mech Anal 2005;176:149–63. [18] DiPerna RJ, Lions PL. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math 1989;98:511–47. [19] Emamizadeh B. Steady vortex in a uniform shear flow of an ideal fluid. Proc Roy Soc Edinburgh 2000;130:801–12. [20] Evans LC. Partial Differential Equations. American Mathematical Society; 1998. [21] Frankel LE. An introduction to maximum principles and symmetry in elleptic problems. Cambridge University Press; 2000. [22] Gilbarg D, Trudinger NS. Elliptic Partial Differential Equations of Second Order. Springer-Verlag; 1998. Second edition. [23] Lieb EH, Loss M. Analysis. American Mathematical Society; 1997. [24] Hardy GH, Littlewood JE, Pólya G. Inequalities. Cambridge University Press; 1934.