## Abstract

This study proves the existence of a steady vortex ring of an ideal fluid in Poiseuille flow. The method that was used is a variational method proposed by Benjamin (Benjamin 1976 *The alliance of practical and analytical insight into the nonlinear problems of fluid mechanics*, vol. 503, pp. 8–29), in which a steady vortex ring can be obtained as a maximizer of a functional that is related to kinetic energy and the impulse over the set of rearrangements of a prescribed function.

## 1. Introduction

A steady vortex ring of an ideal fluid flow occupying the entire space , is a bounded axisymmetric region of vorticity in an otherwise irrotational flow. Benjamin (1976) observed that when a flow of an ideal fluid is in a steady state, the kinetic energy *E* and impulse *I*_{2} in the *z*-direction of cylindrical coordinates are preserved, and that in the presence of axisymmetry, the vorticity can be seen as a rearrangement of its initial state. Benjamin claimed also that the maximizer of a certain functional related to the kinetic energy *E* and impulse *I*_{2} over the set, where the vorticity being a rearrangement of a prescribed function *ζ*_{0}, should give rise to a steady vortex ring of the flow. Recently, Burton (1987) developed the required functional analysis supporting Benjamin's variational method. In this paper, Burton's theory has been used to prove the existence of a steady vortex ring in Poiseuille flow. The flow is written in terms of a Stokes stream function with respect to cylindrical coordinates, which is symmetric about *z*-axis direction and approaches at infinity −(*λ*/4)*r*^{4}, representing a flow of velocity *λr*^{2} in *z*-direction, where *λ* is a parameter number corresponding to the speed of the flow. Hence, the vorticity is described by , where is the Laplacian operator in three-dimensions with respect to the cylindrical coordinates. The vorticity in the region *r*>0 is non-negative and *Ψ* satisfies the equation(1.1)where here is an increasing function. This equation for arbitrary *ϕ* represents the relationship that should exist between the vorticity and the Stokes stream function, when the flow is in a steady state (see Lamb 1932, p. 245). The main result shows that for all sufficiently small positive *λ*, a functional that is related to the kinetic energy and the other functional (which is called the generalized impulse *I*_{4}) has a maximizer belonging to , the set of all rearrangements of a given function (*p*>2) having support of finite volume. Therefore, it will be shown that there exists a solution (*Ψ*, *ϕ*) for (1.1) for which the vorticity is a rearrangement of *ζ*_{0}, where *Ψ* satisfies the boundary conditions and as .

Recently, the problem of the existence of steady vortex rings of an ideal fluid in uniform flow has been studied by Badiani & Burton (2001). By using the Benjamin variational method, they showed that the vorticity belongs to the weak closer of the set . For different points of view, the existence theorem for steady vortex rings in uniform flow proved by diverse variational or topological methods may be found in Fraenkel & Berger (1975), Ni (1980), Friedman & Turkington (1987) and Ambrosetti & Struwe (1989).

By the symmetry in the *r*-direction, this problem can be reduced to one in the half-plane and −∞<*z*<∞. Hence, the variational problem suffers from the two mathematical difficulties. The first is in the nature of the set of rearrangements (as a subset of *L*^{p}). The second is in the loss of compactness which arises from the unbounded domain *Π*. To overcome all these difficulties, the problem will first be solved on a bounded domain in *Π* by using Burton's results (1987). The functional, therefore, has a maximizer in this bounded domain approximation. In the second step, the maximizer will be proved to be the same for all sufficiently large bounded domains. Thus, the validity of the solution is established throughout the half-plane *Π*.

## 2. Mathematical formulation

For *ξ*>0 and *X*>0, we define the sets , where *Π* is the half-plane in defined by *r*>0 and −∞<*z*<∞. For *p*≥1, the definition of can be given as follows:where *ν* is a measure on *Π* having density 2*πr* with respect to the two-dimensional Lebesgue measure *μ*_{2} on *Π*. Hence, we use to denote the volume of any measurable cylindrically symmetric subset . The support of any function is the set , thus if there exists a positive constant *C*, such that for all we have , then the set supp *ζ* is bounded. Now for all (*r*, *z*) and in *Π* we set(2.1)then *G* is the Green function for the elliptic partial differential operator,with homogeneous Dirichlet boundary conditions on *Π* (see Lamb 1932, p. 237). If for some *p*>1, then for all , we define the function *Kζ* byMoreover, if *p*>5/2, then *Kζ* is the weak solution in the distribution sense for the problem in *Π*. The kinetic energy and the generalized impulse are given in terms of some functionals and as follows:hence by assuming that , for all *λ* positive, the functional can be defined asAccording to Benjamin's approach, to show the existence of a steady vortex ring in Poiseuille flow, we shall maximize the functional *Φ*_{λ} over the set where the vorticity *ζ* is a rearrangement of a given function.

Let *ζ*_{1} and *ζ*_{2} be two non-negative measurable functions vanishing outside a set of finite volume in *Π*. We say that *ζ*_{1} is a rearrangement of *ζ*_{2} or *ζ*_{2} is a rearrangement of *ζ*_{1} with respect to *ν* measure if we have for all *t*>0. In the case where *p*≥1, then it follows that ; hence . Any measurable function *ζ* on *Π* will be called Steiner-symmetric if *ζ* satisfiesLet be a non-negative measurable function, which vanishes outside a set having a finite volume. We define the Steiner-symmetrization of *ζ* to be the essentially unique rearrangement *ζ*^{s} satisfyingfor almost every *t*>0, where *μ*_{1} denotes the one-dimensional Lebesgue measure.

Now with all these notations and formulations, we are able to present our main result as follows.

*Let* 5<*p*<∞ *and let* *be a non-negative function having support of finite volume. Let* *be the set of all rearrangements of ζ*_{0} *on Π with respect to ν. Then there exists a positive number Λ, such that the functional Φ*_{λ} *attains a maximum value relative to* *for all* . *Furthermore, if ζ is a maximizer and* *, then we have*(2.2)*almost everywhere in Π, for some increasing function ϕ*.

In this theorem, the maximizer *ζ* will be shown to give rise to a solution *Ψ* of the boundary-value problem for axisymmetric steady vortex rings in Poiseuille flow. Thus, *Ψ* is just a function of (*r*, *z*) only in cylindrical coordinates and satisfies (2.2), *Ψ*(*r*, *z*)=0, when *r*=0 and *Ψ*(*r*, *z*)→0 as . Therefore, the function is the Stokes stream function of a steady ideal fluid flow, whose velocity in is given byHence, the vorticity in terms of *Ψ* is given by

## 3. Description of the method of the proof

Let 1≤*p*<∞, let *q* be the conjugate exponent of *p* and let be a non-negative function having support of finite volume. Let be a bounded symmetric domain, such that . Let be the set of all rearrangements of *ζ*_{0} on *D* with respect to *ν*. We explain now a strategy to prove that the functional *Φ*_{λ} attains a maximum value relative to . The first step of the proof is to show that for every bounded domain , the functional *Φ*_{λ} attains a maximum value relative to , and that, if *ζ*_{D} is any maximizer (*ζ*_{D} must be Steiner-symmetric by Riesz's inequality (Lieb & Loss 1997, theorem 3.7)) and , then *Ψ*_{D} satisfies (2.2) almost everywhere in *D*. This is an application of Burton's result given as follows.

*Let Φ be a real strictly convex functional on* *, sequentially continuous in the weak topology. Then, Φ attains a maximum value relative to* . *If* *is a maximizer and* *is a subgradient of Φ at* *, then* *almost everywhere for some increasing function ϕ*.

*Second step*. We pass to the unbounded domain *Π*, by deriving some estimates for the function , it will be shown that(3.1)where *δ* is a positive constant independent of *λ* and *D*. To prove this step, it will be sufficient for us to show that for any symmetric bounded domain depending on *λ*, the volume of the set, where tends to ∞ when *λ*→0. Hence, the inequality (3.1) follows.

*Last step*. By using the same estimates, it will be also shown that there exists a bounded domain independent of *D* that is Steiner symmetric for whichHence, by using the fact *ϕ* is an increasing function, it follows then thatTherefore, *ζ*_{D} is a maximizer for *Φ*_{λ} relative to .

## 4. Estimates for the Green function *G*

We begin our study with some results concerning the Green function *G*, which allow us to find some properties and estimates for the function *Kζ*, where for some *p*.

*For* *, we set**Then there exists* *which depends on ϵ, such that*

First, for all it is easy to show that(4.1)Setting , then for all and we have ; hence it follows thatBy using (4.1) and the fact that cos *θ*<1, the result isIf we now set tan *θ*/2=*u*, then and ; hence by using the fact that for all , from above we getTherefore, we deduce that there exists a number which depends on *ϵ* for whichThis completes the proof. ▪

*Let* *be two points in Π, let* , , *and* . *Then there exists a number* *which depends on a and b, such that*

Since the integral in (2.1) is even, then we havewhere . We now writeThen, by setting in the right-hand side of this equality, we obtainwhere *f*(*ϵ*) is defined as in lemma 4.1. Thus, by substituting , , and using lemma 4.1 we findwhere is the same number that has been found in lemma 4.1, but here it depends on *a* and *b*. This completes the proof. ▪

*Let the assumptions about* , *, a, b, ρ and* *be the same as in* *lemma 4.2*. *Then*(4.2)*where* *is an arbitrary number*.

Since , , then and . Also, it can be shown that , hence by lemma 4.2 it follows thatNow, for any , we have ; hence by using the fact and lemma 4.2, we obtain

▪## 5. Estimates for *Kζ*

Henceforth in this section, let *p*>2, let *q* be the conjugate exponent of *p* and let be a function, such that . Let be any arbitrary number and finally let *M* denote any positive number depending on *α*, *q* and *S*. Now, with these assumptions, we have the following estimates.

*There exists M*>0*, such that for all* *we have*(5.1)

Let *a*, *b*, *ρ* and be numbers defined as in lemma 4.3, then by using the right-hand side of the estimate (4.2), for all and we have(5.2)Now, we have(5.3)where we have used Hölder's inequality to obtain the last line. Therefore, by taking , and combining (5.2) with (5.3), we can choose a positive number *M* depending on *q*, *α* and for which (5.1) holds. This completes the proof. ▪

*If ζ is Steiner-symmetric, then for all* *there exists M*>0*, such that*

We adapt the same method that Rebah (2002) has used to prove lemma 2.19. Indeed, let *a*, *b*, *ρ* and be numbers defined as in lemma 4.3. Then the estimate (4.2) of lemma 4.2 yieldswhere is any arbitrary number and . Thus by setting , we have(5.4)If ζ is Steiner-symmetric, then ; hence(5.5)Since and , then it follows that if *z*>0 and if *z*<0. Therefore, from (5.5), we obtain(5.6)It now remains just to find an estimate for the first term in the right-hand side of (5.4). We recall that if *f*, *g* are two functions in and and denote the rearrangements of *f* and *g*, respectively, as decreasing functions, then Hardy *et al*. (1934) proved the inequality(5.7)holds. Now since is decreasing function with respect to , then it follows from (5.7) that(5.8)where is the rearrangement of that is symmetric decreasing about . If , then it follows that , where . If , then for all . Therefore, by using (5.8) and Hölder's inequality we have(5.9)where . Now, since is just an arbitrary number, then we take to ensure that , hence by (5.9), we obtain(5.10)Thus, the desired result follows from (5.4), (5.6) and (5.10), provided . ▪

The last lemma in this section concerns the operator *K*. For a bounded domain *Ω* in *Π*, we recall that the Hilbert space was defined to be the completion of with scalar product,

*Let* *and* . *Then for* *there is a unique* *that is a weak solution of* *in Ω. Furthermore:*

*is a bounded linear operator,**is symmetric, strictly positive and compact operator,**If**, then*.

## 6. Properties of the functional *Φ*_{λ}

In this section, all the estimates of the function *Kζ* and properties that were proved in the last sections, will be used to derive some properties for the functionals *Φ*_{λ} and *E*. In what follows we shall use 1_{Ω} to denote the characteristic function of a subset *Ω* in *Π* defined by if and if .

*Let λ*>0*, let* *and let* *be a non-negative function having support of finite volume. Let* *be the set of rearrangements of ζ*_{0} *having bounded support in Π. Then there exists a positive number X which depends on λ and* *, such that if* *and does not vanish almost everywhere for r*>*X, we can choose a positive number ξ for which we have* *and* *, where* *and* *is some function, such that* .

Let *k*>1 be an arbitrary number and let *q* be the conjugate exponent of *p*. Then from lemma 5.1, we can choose a positive number *M* depending on *k*, *p* and , such that for all *r*>1. Hence, by taking and , we find that(6.1)Now, for , we write , where and . From lemma 5.3, the operator *K* is strictly positive, then this yields that *Φ*_{λ} is strictly convex; hence by assuming that , applying (6.1) and using the convexity of *Φ*_{λ} we findTo prove the second part of this lemma, we assume that *ζ* does not vanish almost everywhere for *r*>*X*. Let *a*>0 so that . Let *ξ* and *ϵ* be two positive numbers chosen, such that and *ϵ*<*X*; we can then find a measurable subset *A* that satisfies , and finally . Now, the Theorem of Isomorphism of Measures (see Halmos 1950) shows that there exists an isomorphism ; hence by setting we find that is a rearrangement of supported by the measurable set *A*. Thus, we have and moreover, the function that is defined by is a rearrangement of *ζ*. It now remains just to prove that if *ϵ* is small enough, then . To do this, we need just to show that as . Indeed, by lemma 5.3, the operator *K* is symmetric, so by using the form of *ζ*_{2} and (6.1) we find thatThus, by lemma 5.1,(6.2)because *ϵ* is small enough. Also, by lemma 5.3 the operator *K* is positive; it follows then that(6.3)Thus, by (6.2) and (6.3) we find that as . Therefore, we deduce that for small *ϵ*>0,which completes the proof. ▪

In order to prove theorem 2.1 in §2, lemma 6.1 can be used to replace a maximizing sequence for *Φ*_{λ} relative to the set , by a sequence of maximizers of *Φ*_{λ} relative to the setsfor various *ξ*, which again give a maximizing sequence relative to .

Lemma 6.3 concerns the relationship between the supremum of *Φ*_{λ} over the set as defined in lemma 6.1, and the supremum of *Φ*_{λ} over . Note that, in general, the set is likely to contain some functions with unbounded support, which are rearrangements of *ζ*_{0}.

*Let the assumption about λ, p, ζ*_{0} *and* *be the same as in* *lemma 6.1*. *Let* *be the set of rearrangements of ζ*_{0} *on Π. Then, we have*

To prove this lemma, we follow the same argument that we have used in proving lemma 6.1, so we should first prove that for a given function *ζ* in , there exists a positive function supported by the region , such that , where *X* is a positive number that is defined in lemma 6.1. The second step is to show that for arbitrary *δ*>0, there exists a rearrangement of supported in the region , where for some *Z*>0, such that(6.4)We set , then the first step follows from lemma 6.1. For the second step, we need first to build . Indeed, let and *Z* be two positive numbers chosen so that , where ; so there exists a measurable set *B* that satisfies , and . Thus, it follows from the above mentioned Theorem of Isomorphism of Measures spaces that there is an isomorphism ; hence, by setting , we find that is a rearrangement of supported by the measure set *B*, where . It remains just to prove that, for *δ*>0 arbitrary, by suitable choice of *ϵ* and *Z* we can ensure that (6.4) holds. Indeed, we may write as , where . By lemma 5.3, the operator *K* is positive, then this yields(6.5)Also, with the same argument, we haveNow setting , and using lemma 5.1 and (6.5), we find(6.6)where *M* is the same constant that has been found in lemma 5.1. Note that and as . Then, for given *ϵ*>0, we can find with , such that if , then and . Thus, for all , we have . Now, for *δ*>0 arbitrary, we set , then it follows from (6.6) that for all and . Therefore, by using the first step we deduce that for all *δ*>0 arbitrary, we can choose a rearrangement having bounded support, such that (6.4) holds. Hence, the lemma is proved. ▪

*Let X*>2*, let p*>2 *and let q be the conjugate exponent of p. Let* *be a non-negative Steiner-symmetric function, such that* . *Let* *and* *be any two positive sequences which satisfy* *and* . *Let* . *Then*

Since , then ; hence, by Hölder's inequality,(6.7)Therefore, the result follows by letting . This completes the proof. ▪

*Let k*>1 *and let the assumptions about k, X, p, q, ζ, S be the same as in* *lemma 6.4*. *Let* . Then

By lemma 5.3, *K* is strictly positive, this implies that . Thus, to prove this lemma, it will be sufficient to find a function *f*, such that as , and such that . Indeed, let *X*>2 be fixed, let *q* be the conjugate exponent of *p*, let , where is an integer number which will be chosen later, let and be some sequences of sets defined by and , where and are some positive sequences chosen so that , , , if *n*=1, and(6.8)(6.9)(6.10)if *n*>1. Also, for , we set(6.11)According to lemma 6.4, we have(6.12)Moreover, by setting , then by (6.9) it follows . Therefore, by (6.12) we obtain(6.13)Next, by lemma 5.2, there exists a positive constant *M*_{1} independent of *X*, such that(6.14)Now, if *n*=1, then ; hence, by (6.11) and (6.14), we find(6.15)Next, by (6.8) we have . Hence, by (6.11) and (6.15),(6.16)Again by (6.8), . Then from above we have(6.17)Next, proceeding to higher orders *n*, it is easy to show that(6.18)By Höler's inequality and (6.10),Hence,(6.19)Therefore, by combining (6.13) and (6.19) with (6.18), we gethence, by choosing *n* so that , we obtainTherefore, by letting the result follows immediately. ▪

*Let α*>0*, let* *, let* *, let* *, let* *, let q be the conjugate exponent of p and let* *be a non-negative function having support of finite volume. Let X*>2*, let* *and let* . *Then we have**where*(6.20) *and M is a positive number depending on k and q*.

For , let *X*_{j} be a sequence defined by if and , where and . Then, we can write , where and . Thus,(6.21)Now, by Badiani & Burton (2001, lemma 2.8) we have(6.22)where *M*_{1} is a positive number depending on *q*. Also, by using lemma 5.1,(6.23)where *M*_{2} is a positive number depending on *k* and *q*. Thus, by combining (6.23) and (6.22) with (6.21), it follows that(6.24)where . Now, clearly(6.25)If , then ; hence it follows that(6.26)Also if , then . Therefore, we find(6.27)Now for all , *a*_{j} satisfies . Thus by (6.24)–(6.27), we getbecause . Therefore, the proof follows immediately from the above inequality. ▪

## 7. Proof of our main result

In this section, with all the estimates and properties of the operator *K*, and the functionals *Φ*_{λ} and *E* that have been derived in §§5 and 6, we will be able to prove our main result. We begin with a lemma giving a lower bound for the volume of the setwhere *ζ*∈*L*^{p}(*Π*, *ν*) for some *p*, is a non-negative function.

*Let λ>*0*, let p≥*1 *and let* *be a non-negative function. Let F(ζ) defined as in* *(6.20)*. *Then for all* *we have*

By using lemma 4.3, for all *R*=(*r*, *z*)∈*Π* we havebecause *r*^{2}<1+|*R*|^{3}. Therefore, the result follows fromThis completes the proof. ▪

Let *λ*>0, let *p*>5, let be the conjugate exponent of *p* and let *a*>0 be, such that . Let (*ζ*_{0}) be the set of all rearrangements of *ζ*_{0} on *Π*. Let *X* be a positive number chosen as in lemma 6.1. Also, let (*ζ*_{0}) be the set defined as in lemma 6.1 and let ^{s}(*ζ*_{0}) be the set of all Steiner-symmetric functions in (*ζ*_{0}). Now by lemma 6.3, a maximizing sequence for the functional *Φ*_{λ} relative to (*ζ*_{0}) can be chosen in (*ζ*_{0}). Hence, if we assume that , then by lemma 6.1, for each , we can choose *f*_{j} having bounded support, for which we have , and . Now, let us consider a sequence chosen for which *ξ*_{j}≥*j* and for all . Also define the setsNow, by lemma 5.3 the operator is symmetric, compact and strictly positive; hence the functional *Φ*_{λ} is weakly sequentially continuous and strictly convex on . Therefore by theorem 3.1 the functional *Φ*_{λ} attains a maximum value relative to (*ξ*_{j}, *X*). If *ζ* is a maximizer of *Φ*_{λ} relative to (*ξ*_{j}, *X*), then it follows from Riesz's & Lieb (see Lieb & Loss 1997, theorem 3.7) that *Φ*_{λ}(*ζ*^{s})≥*Φ*_{λ}(*ζ*). Thus, there exists a maximizer for *Φ*_{λ} relative to and it may be chosen in the setFor each , letting denotes a maximizer for *Φ*_{λ} which belongs to the set , then from above we obtain ; hence is a maximizing sequence for the functional *Φ*_{λ}. Therefore, by applying theorem 3.1 again, there is an increasing function *ϕ*_{j}, such that(7.1)almost everywhere in *Π*(*ξ*_{j}, *X*), for fixed *λ*.

We are going now to show that if *j*→∞, *λ* is small and positive, then the support of is a subset of the set where . Hence, by (7.1), we need just to prove that if *j*→∞ and *λ*∈(0, *Λ*), where *Λ* is a small positive number, then there exists a positive *δ*>0, depending only on *Λ* for whichIndeed, by lemma 7.1,(7.2)where is defined as in (6.20). Thus, this leads to finding a lower bound depending on *λ* for for which the right-hand side in (7.2) tends to ∞ as *λ*→0. To do this, let *λ*_{0} be a positive number chosen so that , where *ζ*_{1} is some rearrangement of *ζ*_{0} having support in ×(0, *X*). We can then find , such that if *j*≥*j*_{0} and *λ*∈(0, *λ*_{0}), then . Hence it follows that(7.3)Now, by lemma 6.1 , where *M*_{1} is a positive constant depending on *q* and *k*≥1 is an arbitrary number which will be chosen later. Also, by lemma 5.2, there exists a positive number *M*_{2} depending on *q*, such that(7.4)provided |*z*|>2. Let *λ*_{1} be a positive number chosen so that and let be such that for all *j*≥*j*_{1}, where . Next, setting , , , and . Then we can write . Henceforth, we assume that *j*≥max{*j*_{0}, *j*_{1}} and 0<*λ*<min{*λ*_{0}, *λ*_{1}}. Now, from (7.3) we may write(7.5)By (7.4) we have(7.6)Also by (7.4), it follows thatbecause . Hence, we can find *λ*_{2}>0, such that if , then(7.7)Thus, by (7.5)–(7.7), the result isBy lemma 6.5, we can choose *λ*_{3}>0, such that if , then(7.8)Next, set *τ*=(2/3)^{8}, *q*_{0}=9−4*τ*, *q*_{1}=12−8*τ* and . Since is Steiner-symmetric and , then from (7.8) it follows thatThus, by lemma 6.6, there exists a positive constant *M*_{3} independent of *X*, such that(7.9)Now since , then by combining (7.2) with (7.9), we obtain(7.10)where *M* is a positive constant which depends only on , *α*, *M*_{1}, *M*_{2} and *M*_{3}. Since , then it follows that *q*_{1}−*qq*_{0}>0. We can thus assume that to ensure that the right-hand side in the inequality (7.10) tends to ∞ as *λ*→0, because *k* is just an arbitrary number. Therefore we can choose *λ*_{4}>0 and *δ*>0 independent of *λ*, such that if and *ξ*_{j}→∞ as *j*→∞, then from (7.10) we have , whereWe setThe fact that is an increasing function of and (7.1) implies that apart from a set of zero measure,Now, by (7.4) all points satisfy . Thus there exists a positive number *Z*(*λ*)>2, such that |*z*|<*Z*(*λ*) for all . Now for all 0<*λ*<*Λ* we set and let be such that *ξ*_{j}≥*r*(*λ*) for all *j*≥*j*_{2}; then is bounded by the rectangle *Π*(*r*(*λ*), *X*). Therefore, maximizes the functional *Φ*_{λ} relative to (*ξ*_{j}, *X*) for all . Thus, maximizes the functional *Φ*_{λ} relative to (*ζ*_{0}) for all *λ*∈(0, *Λ*). Therefore, maximizes *Φ*_{λ} relative to (*ζ*_{0}), provided *λ*∈(0, *Λ*). By now writing and using (7.1) we get(7.11)almost everywhere in *Π*(*r*(*λ*), *X*). It remains only to extend (7.11) to . We can assume that for all and consider the function which is defined by if *t*>*δ* and *ϕ*(*t*)=0 if *t*≤*δ*. Since outside *Π*(*r*(*λ*), *X*) and is an increasing function of almost everywhere in *Π*(*r*(*λ*), *X*), then *ϕ* is an increasing function on *Π*. Therefore by setting *Ψ*≔*Kζ* we havealmost everywhere in *Π* for some increasing function *ϕ* and *λ* positive and small. This completes the proof of theorem 2.1. ▪

## Footnotes

- Received June 22, 2005.
- Accepted November 8, 2005.

- © 2006 The Royal Society