## Abstract

We study the well-posedness of a convection problem in a pseudomeasure-type space PM^{a}, without assuming that the gravitational field is bounded. Considering the PM^{a} space with right homogeneity, the existence of self-similar solutions is proved. Finally, an analysis about asymptotic stability is made.

## 1. Introduction

In this paper, we study the following system related to a generalized convection problem on ^{n}:(1.1)(1.2)(1.3)(1.4)and(1.5)with *γ*>0 and *n*≥2. In (1.1)–(1.5), the unknowns *u*(*x*, *t*)∈^{n}, *π*(*x*, *t*)∈ and *θ*(*x*, *t*)∈ represent, respectively, the velocity field, the pressure and the temperature of fluid. The initial velocity and the initial temperature are denoted by *u*_{0}(*x*) and *θ*_{0}(*x*), respectively. Moreover, we assume that the initial data of the velocity *u*_{0} satisfy the condition div *u*_{0}=0 in the distributional sense. In (1.1), *f* is the gravitational field at a point *x*, *f*_{1} represents an external force and in (1.3), *h*_{1} is the reference temperature. The parameters *ρ*, *ν*, *κ* and *Χ* are positive physical constants that represent density, kinematic viscosity, the coefficient of volume expansion and thermal conductance. The Riesz potential operator (−*Δ*)^{r} is defined as usual through the Fourier transform as , where . We use the standard notations *∇*, *Δ* and div for gradient, Laplacian and divergence operators, respectively. The *i*th components of (*u*.*∇*)*u* and (*u*.*∇*)*θ*, in Cartesian coordinates, are given, respectively, byIn the case *γ*=1, system (1.1)–(1.5) corresponds to the model that governs the motion of fluid and the diffusion of heat. Indeed, it was first pointed out by Boussinesq that there are many situations of practical occurrence in which system (1.1)–(1.5) (*γ*=1) is obtained after some simplifications; therefore, to consider the convection phenomena mathematically, the model equations derived from the Boussinesq approximation (Landau & Lifshitz 1968; Chandrasekhar 1981) are usually applied. This approximation says that the density variations are neglected, with the exception of the gravitational term, where they are assumed to be proportional to the temperature variation. Equation (1.2) comes from the well-known continuity equation *ρ*_{t}+div (*ρu*)=0, which, due to the homogeneity of the fluid (*ρ*(*x*, *t*)=const.), is equivalent to the incompressibility condition div *u*=0.

Several papers are devoted to the existence and uniqueness of solutions to the non-stationary problem (1.1)–(1.5); see, for instance, Cannon & DiBenedetto (1980), Hishida (1997), Ferreira & Villamizar-Roa (2006*a*) and papers cited therein. In Cannon & DiBenedetto (1980), using singular integral operators, a construction of solutions of class with suitable exponents *p* and *q* was made. In Hishida (1997), the convection problem in an exterior domain of ^{3}, in the framework of *L*^{(p,∞)} spaces, was analysed. Later, in Ferreira & Villamizar-Roa (2006*a*,,*b*), the problem (1.1)–(1.5) was considered and the well-posedness and asymptotic behaviour of solutions, in the framework of *L*^{(p,∞)} spaces, including the existence of self-similar solutions, were analysed.

In this paper, we study the problem (1.1)–(1.5) considering initial data in a pseudomeasure-type space PM^{a}, introduced in Cannone & Karch (2004),where *a*∈ is a given parameter. In Cannone & Karch (2004), the existence of singular solutions for the three-dimensional Navier–Stokes equations with initial data in the space PM^{2}, and the asymptotic stability of small solutions, was studied. An important characteristic of the PM^{a} space is that it contains homogeneous functions of degree *a*−*n*. In particular, the field −*x*/|*x*|^{1+2γ} belongs to the PM^{n−2γ} space. From a physical standpoint, if we take *f* in (1.1) as being the gravitational field , where *G* is the gravitational constant, we can interpret the convection problem (1.1)–(1.5) as a mathematical version of the Bénard problem in ^{n}. Fractional powers of the Laplacian operator, which correspond to a lesser dissipation, can model physical real phenomena, and in this case, to maintain the right homogeneity of the equation and hence obtain the existence of self-similar solutions, we consider the generalized gravitational field . Like examples of fluid mechanics models with fractional dissipation were analysed in the papers of Constantin & Wu (1999), Córdoba & Córdoba (2004), Córdoba *et al*. (2005) and Wu (2006).

Observing equation (1.1), the coupling term *κθf* carries out an important role on the restrictions of the functional spaces where the solution is searched. In general, when the convection problem is studied, it is assumed that the gravitational field *f* belongs to the space *f*∈*L*^{∞}(*Ω*) (see Cannon & DiBenedetto 1980; Hishida 1997). For instance, in Hishida (1997), the author takes *Ω*∌0, an exterior domain, and . This hypothesis is not verified by the homogeneous field *f* in *Ω*=^{n}. For the Navier–Stokes equations, the non-existence of a coupling term allows the existence of a solution in the Banach space , where *a*=2 in such a way that the norm of the subspace is invariant by the scaling property of the three-dimensional Navier–Stokes equations. On the other hand, considering the convection problem (1.1)–(1.5) with *f* being a homogeneous function of degree −2*γ*, to obtain the existence in , the restriction *n*>4*γ*−1 is necessary, which in the case *γ*=1 is equivalent to the condition *n*>3. In order to overcome this difficulty and include the case *n*=3, besides using a fractional dissipation (*γ*), we study the existence of (1.1)–(1.5) in a space of vector functions where the velocity field *u* and the temperature *θ* are taken in different functional spaces, without losing the existence of self-similar solutions. These subjects are analysed in this paper; indeed, new aspects around the convection problem (1.1)–(1.5) are considered in this work. Firstly, we show the existence and uniqueness of small and large mild solutions in pseudomeasure-type PM^{a} space, discuss some aspects about the smoothness and posteriorly analyse our results in the context of the homogeneous field . Finally, we study the asymptotic stability and the existence of self-similar solutions of (1.1)–(1.5) in PM^{a} spaces; consequently, if we take small perturbations of initial data, we obtain a vanishing criterion.

From another point of view, we show that PM^{a} spaces allow the existence of initially singular solutions. These solutions are instantaneously smoothed out if they are small enough initially. We show the existence of global solutions in PM^{a} space for which we do not know whether or not the singularity persists (see remark 2.8).

Recently, we have known of the existence of the paper by Karch & Prioux (2008), related to problems (1.1)–(1.5), *x*∈^{3}, in PM^{a} spaces. Karch & Prioux assumed the field *f* as being a constant function that was small enough, and they obtained the existence of a self-similar global solution (*u*, *θ*) in class , 1<*r*<2. Note that when *f* is a constant, that is, *f* is a homogeneous function of degree 0, the scaling of system (1.1)–(1.5) is different to the case when the field *f* is a homogeneous function of degree 2 (or −2*γ*); thus, from the point of view of scaling invariant techniques, our problem is different to the one considered by Karch & Prioux. Indeed, it is necessary to analyse the invariance of the scaling by choosing a new existence class that is different to the functional setting of Karch & Prioux, and therefore our results are entirely complementary.

The outline of this paper is given as follows. The basic properties of PM^{a} spaces will be reviewed in §2. In the same section, we show the results of well-posedness. Finally, in §3, we analyse the existence of self-similar solutions and give a result of the asymptotic stability of solutions.

## 2. Preliminaries, definitions and some results

In this section, we introduce the functional spaces that will be used to construct the solutions to system (1.1)–(1.5). We list some facts about convolution and give the notion of solution at these spaces. We start recalling that the Leray projector of a smooth vector field *u* is given by . Moreover, we remark that is a matrix *n*×*n* with elements , where *R*_{j} (*j*=1, 2, …, *n*) are the Riesz transforms that are pseudodifferential operators defined as . In this way, in order to prove the continuity of Leray operator on the PM^{a} spaces, it is sufficient to prove the continuity of the Riesz transform *R*_{j} (*j*=1, …, *n*).

*The Riesz transform R*_{j} (*j*=1, 2, …, *n*) *is continuous at* PM^{a} *spaces,* *a*∈ *with* ‖*R*_{j}‖=1.

Estimating directly the norm ‖.‖_{a} of *R*_{j}, we havewhich proves the lemma. ▪

Now we recall a fact about convolution, which will be useful to carry out some estimates in PM^{a} spaces.

**(convolution of singular kernels,** **Lieb & Loss 2001****).** *Let* 0<*α*<*n,* 0<*β*<*n and* 0<*α*+*β*<*n*. *Then we have*(2.1)

Now, we describe our results of well-posedness of system (1.1)–(1.5) in PM^{a} spaces. We start with the definition of time-dependent functional spaces needed to study the initial-value problem (1.1)–(1.5). Spaces of scalar-value and vector-value distributions will be denoted in the same way.

Let , 0<*T*≤∞, *a*≤*q*, *r*<∞, and . We define the following Banach spaces of time-dependent distributions:andwith the norms defined, respectively, asandwhich are weakly continuous in the distributional sense at *t*=0. When *T*=∞, we denote the spaces and simply by *E*_{r,q} and *E*_{r}, respectively.

With no loss of generality, we will take the constants *ρ*, *ν* and *Χ* in (1.1)–(1.5) to be equal to one. Moreover, we will take *h*_{1} and *f*_{1} to be zero. For a general case, our results remain valid with slight modifications. Therefore, applying the Leray projector in equation (1.1), and using and div (*u*)=0, system (1.1)–(1.5) is formally reduced to the following one:(2.2)(2.3)(2.4)and(2.5)where the velocity filed satisfies div (*u*)=0. Let us recall that applying the divergence operator in equation (1.1), we obtain the elliptical equation , and therefore the pressure *π* may be recovered by

Now, with the help of Duhamel's principle, we introduce the notion of a solution for system (2.2)–(2.5) in Fourier variables.

Let 0<*T*≤∞ and 1/2<*γ*≤1. A mild solution to system (2.2)–(2.5), with initial data *y*_{0}=(*u*_{0}, *θ*_{0}) and div *u*_{0}=0, in PM^{a} spaces, is a time-dependent distribution such that(2.6)and(2.7)for all 0<*t*<*T*, which satisfies div *u*=0 and *y*(*t*)⇀*y*_{0} when *t*→0^{+}, in the distributional sense.

Above, we denote by the matrix *n*×*n* with components , which is the symbol of the operator . Let us write *y*_{0}=(*u*_{0}, *θ*_{0}), *y*_{1}=(*u*_{1}, *θ*_{1}) and *y*_{2}=(*u*_{2}, *θ*_{2}). From now on, we will use the notation given below.(2.8)(2.9)and(2.10)where(2.11)and(2.12)

Now we state our main results about the well-posedness of system (2.2)–(2.5).

*Let* 1/2<*γ*≤1*, a*=*n*−(2*γ*−1)*, r*>*n*/2*, q*+*r*>*n,* *a*≤*q,* *r*<*n* *and* 0<*b*<*n*. *Let y*_{0}=(*u*_{0}, *θ*_{0}), *with* div *u*_{0}=0 *as any vector function in the* PM^{a} *space*.

(

*Small solutions*)*Let n*<*b*+*q*<*n*+*a and b*+*q*−*n*≤*r*≤*b*+*q*−*a*+1.*Assume that**with the norm**sufficiently small*.*Then there exists a constant ϵ*>0*, such that if**,**the initial-value problem*(*2.2*)–(*2.5*)*has a global solution**in the sense of**definition 2.4*.*Moreover,**if**is sufficiently small,**then the solution is unique*.(

*Regularization*)*Moreover,**if a*≤*q*and*r*<*d*<*n,**there exists*0<*ϵ*_{d}≤*ϵ**, such that if**, then the previous solution**verifies*.(

*Large solutions*)*Let a*<*r*<*n,**b*+*r*>*n and h*≥0*,**such that*.*Assume that**with*(*sufficiently small when*).*Then there exists*0<*T*_{1}≤*T, such that the initial-value problem*(*2.2*)–(*2.5*)*has a mild solution*.*Furthermore,**in the previous cases,**if we assume**, with*2*γ*−1<*p*<*n,**then there exists*0<*ϵ*_{p}≤*ϵ,**such that if**, the previous solutions**also verify*,*with T*=∞*in the first case*.

*Let* *and f*=−*G*(*x*/|*x*|^{1+2γ}) *be the generalized gravitation field* (*note that if γ*=1*, f is the Newtonian gravitation field*).

(

*Small solutions*)*Let n*>4*γ*−2*and assume that κG and*‖*y*_{0}‖_{a}*are sufficiently small, where κ is the constant of*(*1.1*)*, which represents the coefficient of volume expansion*.*Then the initial-value problem*(*2.2*)–(*2.5*)*has a global solution*.*On the other hand,**if n*>4*γ*−1*, then the initial-value problem*(*2.2*)–(*2.5*)*has a global solution in*.(

*Large solutions*)*Let*.*Then there exists*0<*T*≤∞*, such that the initial-value problem*(*2.2*)–(*2.5*)*has a mild solution*.

Note that, in part iv of theorem 2.5, we did not assume any smallness hypothesis on the initial data in the norm ‖.‖_{p}. Ever since 1/2<*γ*≤1, in the absence of the gravitational field, that is *f*=0, we obtain a generalization of the results of Cannone & Karch (2004) for the case of fractional dissipation. Furthermore, all the theorems given previously remain true if we consider system (2.2)–(2.5) with an external field *f*_{1} and an external force *h*_{1} to be non-null with suitable smallness conditions on the respective norms.

An interesting point related to the solutions of the convection problem (2.2)–(2.5) is to know if they are solutions in the classical sense. Indeed, we can adapt the arguments of Kato (1992) in order to prove that the regularized solutions given by theorem 2.5 are

*C*^{∞}-smooth instantly and they are the solutions of system (2.2)–(2.5) for*t*>0, in the classical sense. We do not know whether the respective small solutions (first part) have the same property. We observe that for*t*>0, the regularized solutions lie in with*a*≤*q*,*r*<*d*and , where denotes the well-known Lebesgue space; this last fact does not hold for small solutions.We can obtain an analogous version of the regularized solutions for the large solutions and therefore, if we take 0<

*T*<∞ as small enough, the solutions are also*C*^{∞}-smooth instantly and they are the solutions of system (2.2)–(2.5) in the classical sense.

### (a) Proofs of theorem 2.5 and corollary 2.6

In this section, we will develop the proofs of theorem 2.5 and corollary 2.6. For this, we recall the following lemma in a generic Banach space, which can be found in Cannone & Planchon (1999) and Ferreira & Villamizar-Roa (2006*a*). For a generalization of that lemma, in the case of a *p*-nonlinearity, the reader is referred to Ferreira & Villamizar-Roa (2006*b*). The proof is also based on the standard Picard iteration technique completed by the Banach fixed point theorem.

*Let X be a Banach space with norm* ‖.‖_{X},*F* : *X*→*X* *a linear continuous map with norm τ*<1 *and B* : *X*×*X*→*X* *a continuous bilinear map, that is, there exists a constant K*>0*, such that for all x*_{1} *and x*_{2} *in X, we have* . *Then*, *if* *,* *for any vector y*∈*X, y*≠0*, such that* *,* *there exists a solution x*∈*X for the equation* *, such that* . *The solution x is unique in the ball* . *Moreover,* *the solution depends continuously on y in the following sense:* *if* *,* *and* *, then*

We will also need the following preliminary lemmas.

*Let* 0<*T*≤∞*,* 0<*a*≤*q* and *r*<*n. If* *, then**where C is a positive constant that is equal to* 1 *when q*=*r*=*a,* *and* *when t*→0^{+}*,* *in a distributional sense*.

Estimating directly the norm ‖.‖_{q} of *G*_{γ}(*t*), we have

Analogously, we have that .

To complete the proof, we need to prove the weak continuity in *t*=0. Note that for , we haveand the proof is finished. ▪

Now, we show the continuity of the bilinear form *B*(. , .) defined by (2.10).

*Let B*(. , .) *be the bilinear form defined by* (*2.10*), 1/2<*γ*≤1*,* *, r*<*n,* *r*>*n*/2*, r*+*q*>*n,* *and* . *Then*,

*There exists a positive constant**, such that*(2.13)*If*2*γ*−1<*p*<*n,**then there exists a positive constant**,**such that*(2.14)*If*0<*T*<∞*and a*<*r*<*n*,*then there exists a positive constant**, such that*(2.15)

We will omit the proof of estimate (2.14) because we can get it in a way analogous to the inequality (2.13); therefore, we just need to prove the inequalities (2.13) and (2.15). For this, let 0<*b*<*n*, 0<*l*<*n* and *b*+*l*−*n*>0. Using elementary properties of the Fourier transform and proposition 2.2, we obtain(2.16)

Consequently, we haveTherefore for , we obtain(2.17)whereThen we have three cases.

*First case*. If *l*=*q*, , *j*=*a* and noting that when *b*=*a*, we have

*Second case*. If *l*=*q*, and *j*=*a*, we obtain

*Third case*. If we take in the expression *I*(*ξ*,*t*) of (2.17), we getFollowing analogous arguments, we can prove thatandand hence we conclude the proof of the lemma. ▪

*Let* *,* 1/2<*γ*≤1*,* 0<*b,* *q*<*n,* *,* *and F*(.) *be as defined in* (*2.9*). *Then*

*there exists a positive constant K*_{r}*, such that*(2.18)*if*2*γ*−1<*p*<*n,**then there exists a positive constant K*_{p}*, such that*(2.19)*if*0<*T*<∞*and a*<*r*<*n,**then there exist h*≥0*, such that**,**and a positive constant K*_{r,T}*satisfying*(2.20)

We will omit the proof of estimate (2.19), since we can obtain it in a way analogous to the inequality (2.18). Working as in lemma 2.11, we find thatwhere

We analyse the convergence of *I*(*ξ*,*t*) in two cases.

*First case*. If ,

*Second case*. If , then denoting , we have

On the other hand, if *h*≥0 and , we can estimate(2.21)wherewhich proves (2.20). ▪

*Small solutions*. Using lemma 2.10, we can take the initial data (*u*_{0},*θ*_{0}) as sufficiently small such that the hypothesis of lemma 2.9 is verified. In lemma 2.11, the inequality (2.13) implies the continuity of the bilinear form in the*E*_{r,q}space. Now, using inequality (2.18) of lemma 2.12, we obtain the continuity of the linear term*F*(.) whose norm depends on the size of the norm . Consequently, a direct application of lemma 2.9 in the Banach space*E*_{r,q}, implies the well-posedness of the integral equations (2.6) and (2.7) in*E*_{r,q}. On the other hand, we need to show that , as*t*→0^{+}in the distributional sense, but we omit the proof because this follows as in the second part of the proof of lemma 2.10.*Large solutions*. The proof of the third part of theorem 2.5 is also a direct application of lemmas 2.9 and 2.10, inequality (2.15) of lemma 2.11 and inequality (2.20) of lemma 2.12. Indeed, we can take*T*>0 as sufficiently small in such a way that the conditions of lemma 2.9 are verified without a smallness assumption on the data.*Regularization*. Using the hypotheses*a*≤*q*and*r*<*d*<*n*, we have the continuity of the bilinear form*B*(. , .) and the linear term*F*(.) in the space*E*_{d,d}. Consequently, a direct application of lemma 2.9 completes the well-posedness of the integral equations (2.6) and (2.7). Initial data are taken in the same space as in the small solutions and therefore the proof of the regularization of solutions is finished.

The final part of theorem 2.5, that is, , 0<*T*≤∞, for initial data with 2*γ*−1<*p*<*n*, can be proved as follows. We consider the case *T*=∞ corresponding to the case of small solutions (the case 0<*T*<∞ follows using analogous arguments). Since the solution given by lemma 2.9 is obtained by the following sequence:where *y*_{k}=(*u*_{k}, *θ*_{k}) and *k*∈; we can use lemmas 2.10–2.12 in order to obtain the existence of a positive constant *K*^{*}(*f*) such thatandNow, let us choose 0<*ϵ*_{p}≤*ϵ* and take *f* small enough in its respective norm so that and assume that . The proof of the first part of theorem 2.5 (see lemma 2.9) shows that . Therefore,Let us denote and , then the sequence satisfies Taking , we can writeand thus,Finally, lemmas 2.11 and 2.12 imply thatNow, we denote . Since and , then and consequently *Γ*=0. Therefore, the sequence {*y*_{k}} is a Cauchy sequence in the space and thus it converges to some . The uniqueness of the limit in the distributional sense gets the desired conclusion. ▪

*Small solutions*. Note that under the assumptions on *b*, *q*, *n* and *γ*, and taking *r*=*a*, the hypothesis of the first part of theorem 2.5 is satisfied. As , then and moreover . If *κG* is small enough then is small, and hence we can apply the first part of theorem 2.5 in order to obtain the existence of small solutions for the Bénard problem.

*Large solutions*. As , then *h*=0. Moreover, as *a*<*r*<*n* then . Now, applying the third part of theorem 2.5, we obtain the intended result. Note that we did not need smallness assumptions on *G*. ▪

## 3. Self-similar solutions and stability in PM^{a} spaces

### (a) Self-similar solutions

Let us consider that is a smooth function and that the pair is a smooth solution of the convection problem (2.2)–(2.5). We can see that the pair of functions is also a solution of system (2.2)–(2.5). The particular solutions of system (2.2)–(2.5), satisfying(3.1)for any *t*>0, *x*∈^{n} and *λ*>0, are called self-similar solutions of the system. We can check that taking *t*→0^{+} formally in (3.1), should be a homogeneous function of degree −(2*γ*−1).

This fact gives the hint that a suitable space to find self-similar solutions should be the one containing homogeneous functions with that exponent, and this gives another justification to study the convection problem in PM^{a} spaces.

The aim of this subsection is to describe the principal results of the existence of self-similarity solutions in PM^{a} spaces.

*Let* *and* . *Assume that y*_{0} *is a homogeneous vector function of degree* −(2*γ*−1)*, that is,* *for all* *x*∈^{n}*, x*≠0 *and all λ*>0. *If f satisfies the assumptions of* *theorem 2.5* (*respectively* *corollary 2.6*) *and the scale relation**then the solution given by* *theorem 2.5* (*respectively* *corollary 2.6*) *is self-similar,* *that is,* *, for all* *x*∈^{n}*, x*≠0 *and all* *λ*>0.

The proof of theorem 2.5 is based on lemma 2.9, in which the solution is obtained by successive approximations. Hence, we consider the following Picard iteration:where *k*=1, 2, …. We can verify that *y*_{1}(*t*, *x*) satisfies . By an induction process, we can prove that *y*_{k} satisfies , for all *k*. Consequently, as the mild solution *y*(*t*, *x*) is obtained as the limit of sequence , we have that *y*(*t*, *x*) must verify , for all *λ*>0, all *t*>0 and *x*∈^{n}. ▪

### (b) Asymptotic stability in PM^{a} spaces

In the present section, we show some stability properties of mild solutions when the initial velocity and the temperature are perturbed. Our results now read as below.

*Let y*=(*u*, *θ*) *and w*=(*v*, *ϕ*) *be two small global solutions of* (*2.2*)–(*2.5*) *as in* *theorem 2.5,* *corresponding to the initial conditions y*_{0}=(*u*_{0}, *θ*_{0}) *and w*_{0}=(*v*_{0}, *ϕ*_{0})∈(PM^{a})^{n+1}*, respectively*. *If* *and* *, then*(3.2)

*Moreover,* *if* *with l*<*a**, then* (*3.2*) *holds*.

Note that as a consequence of theorem 3.2, if , then the respective solution *y*=(*u*, *v*) decays to zero, that is,

Let us define . Subtracting the integral equations in definition 2.4 for *w* from the analogous expression for *y*, taking the norm in the first coordinate and in the second coordinate of the resulting system, we obtain(3.3)and(3.4)where the small constant *δ* will be chosen later.

Assume that and *α*_{r}=0 (the case *r*>*a* follows in an analogous way). In the first integral on the r.h.s. of (3.3), we change the variables *s*=*tz* and use the identityin order to estimate it byWe deal with the second integral of the inequality (3.3), estimating it directly bywhere *τ* is the constant given by the application of lemma 2.9. On the other hand, we estimate the term as

Therefore, we havefor all *t*>0. We also haveNow, we defineWe will show that *A*=0. Using the dominated convergence theorem, we obtainSincewe have

From the proof of the first part of theorem 2.5, we know thatand therefore, summing (3.3) and (3.4), computing in the resulting inequality and using the last inequalities, we obtainIf we take *δ*>0 as sufficiently small, since and *A* is non-negative, then *A*=0. This completes the proof of the first part of theorem 3.2. In order to prove the second part, we use lemma 2.10 and a density argument to obtainand therefore (3.2) holds. ▪

## Acknowledgments

The second author E.J.V.R. was partially supported by COLCIENCIAS, Colombia, Proyecto COLCIENCIAS-BID III etapa.

## Footnotes

- Received December 12, 2007.
- Accepted March 13, 2008.

- © 2008 The Royal Society