RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION A GEOMETRIC APPROACH

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1 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION A GEOMETRIC APPROACH NIKOLA POPOVIĆ AND PETER SZMOLYAN Abstract. The present work is a continuation of the geometric singular perturbation analysis of the Lagerstrom model problem which was commenced in [PS04]. We establish the same framework here, reinterpreting Lagerstrom s equation as a dynamical system which is subsequently analyzed by means of methods from dynamical systems theory as well as of the blow-up technique. We show how rigorous asymptotic expansions for the Lagerstrom problem can be obtained using geometric methods, thereby establishing a connection to the method of matched asymptotic expansions. We explain the structure of these expansions and demonstrate that the occurrence of the well-known logarithmic switchback) terms therein is caused by a resonance phenomenon.. Introduction Singular perturbation problems in general and singularly perturbed differential equations in particular are characterized by the presence of at least two fundamentally different scales. The existence of these independent scales gives one a small parameter and thus permits one to use perturbation methods. The aim of these methods is to obtain uniformly valid approximations. It is, however, the essence of a singular perturbation problem that a straightforward perturbation fails to be uniformly valid. Indeed, it is typical of singular perturbation techniques that one works with approximations which are valid in restricted domains only. Traditionally, this type of problems has been treated using the method of matched asymptotic expansions: one proceeds by constructing two or more) asymptotic expansions which together cover the entire domain, although neither is uniformly valid there. To obtain a uniformly valid approximation on the entire domain, these individual expansions have to be matched; the essence of matching lies in comparing two expansions on a suitable domain of overlap. An excellent account of the fundamental notions and concepts in perturbation theory is given in [LC72]. More recently, an alternative approach to such problems, known as geometric singular perturbation theory, has emerged see [Fen79] or [Jon95] for details and references). This approach, which in general requires certain hyperbolicity assumptions, is based on methods from the theory of dynamical systems, in particular on invariant manifold theory. A classical singular perturbation problem from fluid dynamics occurs in the asymptotic treatment of viscous flow past a solid at low Reynolds number, see e.g. [vd75]. Though first attempts at clarification date back to [Sto5], it was not until a century later that the conceptual structure of the problem was at last resolved in 99 Mathematics Subject Classification. 34E05, 37Cxx. Key words and phrases. Singular perturbations; Asymptotic expansions; Switchback.

2 2 NIKOLA POPOVIĆ AND PETER SZMOLYAN [Kap57, KL57, PP57]. To illustrate the mathematical ideas and techniques used by Kaplun in his asymptotic treatment of low Reynolds number flow, [Lag66] proposed an analytically rather simple model problem which was subsequently analyzed by Lagerstrom himself as well as by numerous other workers, see [Lag88] and the references therein. Much of the interest in Lagerstrom s model problem has been directed at the development of matched asymptotic expansions to describe its solutions, see e.g. [KC8, Lag88], or [HTB90]. Our goal is to show how such expansions can be obtained using geometric methods, thereby establishing a connection between the two approaches. As observed already by [Fen79], for layer-type problems finding outer solutions is equivalent to computing expansions of slow manifolds. However, the well-developed geometric theory is not applicable at points where hyperbolicity is lost. In several instances it has been possible to extend the geometric approach past such points using the blow-up technique. Blow-up is essentially a sophisticated rescaling which allows one to analyze the dynamics near a singularity; details can be found in [DR96]. In particular, blow-up has been employed by [KS0] and [vgks] to give a detailed geometric analysis of the singularly perturbed planar fold. Apart from deriving asymptotic expansions of slow manifolds continued beyond the fold point, they also explained the structure of these expansions and gave an algorithm for the computation of their coefficients. Our line of attack is very similar to theirs. A distinctive feature of asymptotic expansions for the singularly perturbed planar fold as well as for Lagerstrom s model is the occurrence of logarithmic terms. The nature of the expansions in these and similar problems is both complicated and unexpected, as the governing equations typically give no immediate hint of the presence of such terms; indeed, this is why they are often so tricky to obtain. Traditionally, logarithmic terms have been accounted for under the notions of switchback and integrated effects. We show that the occurrence of logarithms in the expansions for Lagerstrom s model equation is caused by a resonance phenomenon. At this point we conjecture that similar resonance phenomena are responsible for the occurrence of logarithmic terms in many other singular perturbation problems, at least after reinterpretation in a dynamical systems framework. This article is organized as follows: Section 2 contains some background information on the Lagerstrom model problem as well as on the blow-up transformation used in our analysis; in Section 3, we derive asymptotic expansions for Lagerstrom s model, whereas Section 4 briefly indicates how these expansions are related to the classical ones known from the literature. This work should be regarded as a continuation of [PS04]. 2. Background information 2.. Lagerstrom s model equation. In its simplest form, Lagerstrom s model equation is given by the non-autonomous second-order boundary value problem a) b) ü + n u + u u = 0, x uε) = 0, u ) =,

3 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 3 with n N, 0 < ε x, and the overdot denoting differentiation with respect to x. Equivalently, by introducing the rescaling 2) ξ = x ε, one can write ) as 3a) 3b) u + n u + εuu = 0, ξ u) = 0, u ) =, with ξ and the prime denoting differentiation with respect to ξ. Originally, ) and 3) are the versions of the model which was first introduced in [Lag66] and [KL57] to elucidate certain basic ideas used in the asymptotic treatment of viscous flow past a solid at low Reynolds number. In the following, we will only consider the cases n = 2 and n = 3, which represent the physically relevant settings of flow in two and three dimensions, respectively. For more background information and further references on Lagerstrom s model equation, we refer to [PS04]. As in [PS04], replacing ξ [, ) by η := ξ 0, ], appending the trivial) equation ε = 0, and setting u = v yields 4) u = v, v = n )ηv εuv, η = η 2, ε = 0 in extended phase space, with boundary conditions given by 5) u) = 0, η) =, u ) = ; obviously, 5) means that v ) = 0 for the solution to 4), whereas v) still is to be determined. For ε > 0 fixed, let V ε be defined by 6) V ε := { 0, v, ) v [v, v] }, where 0 v < v <, and let the point Q be given by Q :=, 0, 0) see again [PS04]). Note that V ε and Q correspond to the inner and outer boundary conditions in 3b), respectively. The saturation of V ε under the flow induced by 4) we denote by M ε. Correspondingly, the manifolds V and M in extended phase space are defined by V := ε [0,ε V 0] ε {ε} and M := ε [0,ε M 0] ε {ε}; here, the parameter ε is not fixed, but is allowed to vary in some interval [0, ε 0 ] with ε 0 > 0 small. The equilibria of 4) are located on the line l := { u, 0, 0, ε) u R +, ε [0, ε 0 ] } ; obviously, Q l. For ε > 0, the one-dimensional strongly stable manifold of Q, which we call Wε ss, can be computed explicitly due to the simple structure of 4) for η = 0, whence e.g. 7) vu) = ε u 2 ) ; 2 here we have used v) = 0. The following result can be found in [PS04]: Proposition 2. [PS04]). Let k N be arbitrary, and let ε > 0. ) There exists an attracting two-dimensional center manifold W c ε of 4) which is given by {v = 0}.

4 4 NIKOLA POPOVIĆ AND PETER SZMOLYAN v v ε W s ε V ε W ss ε Q W c ε u M ε η Figure. Geometry of system 4) for ε > 0 fixed. 2) For u, v, and η sufficiently small, there is a stable invariant C k -smooth foliation F s ε with base W c ε and one-dimensional C k -smooth fibers. Given Proposition 2., one can define the stable manifold Wε s of Q as 8) Wε s := Fε s P ), P Υ where Υ := {, 0, η) 0 η }, i.e., as a union of fibers F s ε F s ε with base points in the weakly stable orbit Υ. The situation is illustrated in Figure. The main result in [PS04] is the following theorem on the existence and local) uniqueness of solutions to 4) and, consequently, to )): Theorem 2.2 [PS04]). For ε 0, ε 0 ] with ε 0 > 0 sufficiently small and n = 2, 3, there exists a locally unique solution to 4),5). The proof is constructive, and is performed by tracking M ε through phase space and showing that its intersection with W s ε is transverse under the resulting flow. As we are only interested in small values of ε, we took a perturbational approach: given transversality for ε = 0, we concluded that the intersection remained transverse for ε > 0 small. On a technical level, the tracking was done along singular orbits of 4) connecting V 0 to Q. These orbits, which we denoted by Γ, were used as templates for orbits of the full problem obtained for ε > 0. However, due to the non-hyperbolic character of the problem for ε = 0, we could not deduce the existence of a stable manifold W s 0 from standard invariant manifold theory. It was shown in [PS04], however, that W ss and W s can still be smoothly defined down to ε = 0 by using blow-up techniques.

5 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 5 Φ Φ ε ε K 2 ū, v u, v η K η Figure 2. The blow-up transformation Φ The blow-up transformation. The polar) blow-up transformation Φ introduced in [PS04] to analyze the dynamics of 4) near l is given by { R B R 4, 9) Φ : ū, v, η, ε, r) ū, r v, r η, r ε), where B := S 2 R and S 2 denotes the two-sphere in R 3, i.e., S 2 = { v, η, ε) v 2 + η 2 + ε 2 = }. Note that obviously Φ l) = R S 2 {0}, which is the blown-up locus obtained by setting r = 0. Moreover, for r 0, i.e., away from Φ l), Φ is a C -diffeomorphism. We will only be interested in r [0, r 0 ] with r 0 > 0 small. The reason for introducing 9) is that degenerate equilibria, such as those in l, are often amenable to analysis by means of blow-up techniques. The blow-up is a singular) coordinate transformation whereby the degenerate equilibrium is blown up to some n-sphere. Transverse to the sphere and even on the sphere itself one often gains enough hyperbolicity to allow a complete analysis by standard techniques. For a general discussion of blow-up we refer to [DR9] and to [Dum93], whereas applications to singular perturbation problems can be found in [DS95] and [DR96] as well as in [KS0] and [vgks]. The vector field on R B, which is induced by the vector field corresponding to 4), is most conveniently studied by introducing different charts for the manifold R B. In the following, we will be concerned only with two charts K and K 2 corresponding to η > 0 and ε > 0 in 9), respectively, see Figure 2. The reason is that these two charts correspond precisely to the inner and outer regions in the classical approach, see [PS04]. The directional blow-up in the direction of positive η i.e., in K ) is given by { R 4 R 4, 0) Φ : u, v, r, ε ) u, r v, r, r ε ),

6 6 NIKOLA POPOVIĆ AND PETER SZMOLYAN whence ) u = u, v = r v, η = r, ε = r ε. After transformation to K and desingularization, the equations in 4) have the following form: 2) u = v, v = 2 n)v ε u v, r = r, ε = ε. The desingularization, which is necessary to obtain a non-trivial flow for r = 0, is performed by dividing out the common factor r on both sides of the equations; it corresponds to a rescaling of time, leaving the phase portrait unchanged. The equilibria of 2) are easily seen to lie in l := { u, 0, 0, 0) u R +}. A simple computation shows that the corresponding eigenvalues are given by, 0, and both for n = 3 and for n = 2; these eigenvalues obviously are in resonance. In fact, it is these resonances in K which are responsible for the occurrence of logarithmic switchback terms in the Lagerstrom model, as will become clear later on. Similarly, in chart K 2, 9) is given by { R 4 R 4, 3) Φ 2 : u 2, v 2, η 2, r 2 ) u 2, r 2 v 2, r 2 η 2, r 2 ) respectively 4) u = u 2, v = r 2 v 2, η = r 2 η 2, ε = r 2, which is simply an ε-dependent rescaling of the original variables, since r 2 = ε. Desingularizing once again, we obtain for the blown-up vector field in K 2 5) u 2 = v 2, v 2 = n)η 2 v 2 u 2 v 2, η 2 = η 2 2, r 2 = 0; these equations are simple insofar as r 2 occurs only as a parameter. The equilibria of 5) are given by l 2 := { u 2, 0, 0, r 2 ) u2 R +, r 2 [0, r 0 ] }. The following observation, which is valid in both cases alike, is the starting point of our analysis: Proposition 2.3 [PS04]). Let k N be arbitrary. ) There exists an attracting three-dimensional center manifold W c 2 of 5) which is given by {v 2 = 0}. 2) For u 2, v 2, η 2, and r 2 sufficiently small, there is a stable invariant C k -smooth foliation F s 2 with base W c 2 and one-dimensional C k -smooth fibers. Let W2 ss denote the fiber F2 s Q 2 ) F2 s with base point Q 2 :=, 0, 0, 0); note that for any r 2 = ε [0, ε 0 ] fixed, Q 2 and W2 ss correspond to the original Q and its stable fiber Wε ss, respectively.

7 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 7 Remark. Just as was the case with Wε ss, W2 ss also is known explicitly: given v 2 ) = 0, one obtains from 5) with η 2 = 0 that 6) v 2 u 2 ) = ) u ; hence, W2 ss is independent of both ε and n. As in [PS04], let the orbit γ 2 be defined by 7) γ 2 ξ 2 ) := {, 0, ξ 2, 0) ξ2 0, ) }, and let Γ 2 := γ 2 {Q 2 }; note that γ 2 ξ 2 ) Q 2 as ξ 2. With Proposition 2.3 it then follows: Proposition 2.4 [PS04]). The manifold W s 2 defined by 8) W s 2 := P 2 Γ 2 F s 2 P 2 ) is an invariant, C k -smooth manifold, namely the stable manifold of Q 2. Tracking the manifold W s 2 along the singular orbit Γ to the inner boundary in K defines a global manifold W s which determines the solution to 4),5) as given by Theorem 2.2, see Figure 3. The relation between charts K and K 2 on their overlap domain can be described as follows: Lemma 2.5 [PS04]). Let κ 2 denote the change of coordinates from K to K 2, and let κ 2 = κ 2 be its inverse. Then, κ 2 is given by 9) u 2 = u, v 2 = v ε, η 2 = ε, r 2 = r ε, and κ 2 is given by 20) u = u 2, v = v 2 η 2, r = r 2 η 2, ε = η 2. Remark 2 Notation). Let us introduce the following notation: for any object in the original setting, let denote the corresponding object in the blow-up; in charts K i, i =, 2, the same object will appear as i when necessary. Moreover, as in [PS04] we define the sections Σ in, Σ out, and Σ in 2, where 2a) 2b) 2c) Σ in := { u, v, r, ε ) u 0, v 0, ε 0, r = ρ }, Σ out := { u, v, r, ε ) u 0, v 0, r 0, ε = δ }, Σ in 2 := { u 2, v 2, η 2, r 2 ) u2 0, v 2 0, r 2 0, η 2 = δ } with 0 < ρ, δ arbitrary, but fixed; obviously κ 2 Σ out 3. Rigorous asymptotic expansions ) = Σ in 2. We now set out to derive asymptotic expansions for the function v ε := v ξ= as defined by the unique solution to 4),5) given in Theorem 2.2, see Figure 4. It is well-known that to leading order, the method of matched asymptotic expansions gives 22) v ε = ε ln ε γ + )ε + Oε 2 )

8 8 NIKOLA POPOVIĆ AND PETER SZMOLYAN W s r 2 u 2, v 2 a) M η 2 ε V ε > 0 { V ε Γ r u, v W s r 2 u 2, v 2 b) M η 2 ε V ε > 0 { V ε Γ r u, v Figure 3. Geometry of the blown-up system for a) n = 3 and b) n = 2.

9 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 9 ε ε v ε v ε v v r P P V V r a) b) Figure 4. v ε for a) n = 3 and b) n = 2. for n = 3 and 23) v ε = ln ε + γ ) ln ε) 2 + O ln ε) 3 for n = 2, respectively, see e.g. [Lag88]. Classically, the somewhat surprising occurrence of logarithmic terms in these expansions has been accounted for under the notion of switchback; we will show that these terms are in fact due to a resonance phenomenon. Incidentally, note that v ε equals du dξ ξ=, the analogue of the drag on the solid, which is a quantity of considerable interest in the original fluid dynamical problem. Our approach is rigorous in the sense that the expansions we compute are approximations to a well-defined geometric object, namely, to the invariant manifold W s introduced in the previous section. Roughly speaking, our strategy is the following: we begin by computing expansions for W2 s in K 2 ; these expansions, when translated to K and evaluated at the inner boundary there, will provide us with expansions for v ε. We again distinguish between n = 3 and n = 2 here, the case n = 3 being considerably simpler. Remark 3. This strategy, which is slightly different from the strategy applied in [PS04], is somewhat more efficient as far as computing expansions for v ε is concerned. Later on, we will indicate how asymptotic solution expansions for 4),5) can be obtained. The complicated structure of the expansions in K arises as W s passes near the line of equilibria l in K. As indicated above, the logarithmic terms in 22) respectively 23) are due to the resonant eigenvalues, 0, and which occur in K, see [PS04]. We now present a simple argument to substantiate our claim: for n = 3, consider the equations in K given by 2). After introducing the new variable ṽ = e ξ v,

10 0 NIKOLA POPOVIĆ AND PETER SZMOLYAN one obtains for the first two equations in 2) 24) u = e ξ ṽ, ṽ = ε u ṽ. Integration of 24) yields 25) ξ u ξ ) = u 0 + e ξ ṽ ξ ) dξ, 0 ξ ṽ ξ ) = v 0 ε 0 e ξ u ξ )ṽ ξ ) dξ, 0 where we have used ε = ε 0 e ξ and u 0, v 0, and ε 0 are constants. Note that near l, v and ε are small, which implies ṽ = O) there. Hence, a Picard iteration scheme can be applied to 25), with the starting point given by u 0), ) ṽ0) = u 0, v 0 ). In fact, one easily sees that 25) defines a contraction operator for u and ṽ in L [ ] 0, ln ε ε 0, which ensures convergence of the scheme. 2 A straightforward computation gives u ) = u 0 + v 0 e ξ ) 26a), ṽ ) = v 0 + ε 0 u 0 v 0 e ξ ) 26b), u 2) = u ) 26c) + ε 0 u 0 v 0 ξ e ξ), 26d) ṽ 2) = ṽ ) + ε 0 v ξ e ξ) + 2 ε 0 u 2 0 v 0 2e ξ + e 2ξ) + 2 ε 0 u 0 v ξ 4e ξ + e 2ξ) for the first two iterates in 25). As ξ = ln ε ε 0, this then generates a logarithmic term in ε after rewriting 26c) as a function of ε. Similarly, the products of powers of ξ and e ξ which occur for higher iterates in 25) will give rise to products of powers of ln ε and ε after those iterates have been rewritten in terms of ε. In fact, it is thus possible to obtain successive approximations to the transition map Π from Σ in to Σ out for 2), see [KS0]; the above computation gives the leading order behaviour of Π. Note that it is precisely the resonant terms in 2) which cannot be eliminated by a normal form transformation and which preclude the existence of a linearizing transformation for 2) proper, see e.g. [CLW94]. Without loss of generality, we set ξ0 = 0 here. 2 Incidentally, the introduction of ṽ in 2) is required for the proof of contractivity of 25).

11 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 3.. The case n = Expansions in chart K 2. As the equations in K 2 are completely independent of r 2, we can simply omit the last equation in 5), which leaves us with the essentially three-dimensional system 27) u 2 = v 2, v 2 = 2η 2 v 2 u 2 v 2, η 2 = η 2 2. Remark 4. In contrast to what is usually done in the literature, we do not intend to derive asymptotic expansions for the solutions to 4) here, but rather for the manifold W s as defined in Section 2. As the solutions to 4),5) clearly do depend on ε, however, any ansatz aimed at obtaining solution expansions would of course have to take into account this dependence on ε. Note that in our approach, ε enters only in chart K, see below. Given Proposition 2.3, we can make an ansatz for the expansion of W2 s of the form 28) v 2 u 2, η 2 ) = C j η 2 )u 2 ) j, where 29) C j η 2 ) := j! j=0 j u j 2 v 2 u 2, η 2 ) u2=, see [vgks]. Hence, N 30) v 2u 2, η 2 ) C j η 2 )u 2 ) j = O u 2 ) N+) j=0 for any N N, and the above estimate is uniform for η 2 bounded. Remark 5. An equally valid ansatz would be to set 3) u 2 v 2, η 2 ) = D j η 2 )v j 2, with 32) D j η 2 ) := j! j v j 2 j=0 u 2 v 2, η 2 ) v2=0. However, the reason for considering 28) and not 3) is that ultimately we are interested in deriving an expansion for v ε. Given Lemma 2.5, it is therefore the ansatz in 28) we have to use. Rewriting 27) with η 2 as the independent variable and omitting the subscript 2, we obtain 33a) du dη = v η 2, 33b) dv dη = 2 η v + u η 2 v + v η 2 ;

12 2 NIKOLA POPOVIĆ AND PETER SZMOLYAN inserting 28) into 33b) yields [ )] dc j 34) dη u )j C j ju ) j η 2 C k u ) k j=0 = 2 η C j u ) j + η 2 j=0 k=0 j=0 C j u ) j+ + η 2 j=0 C j u ) j, where we have used 33a). Collecting powers of u in 34) gives a recursive sequence of differential equations for the coefficient functions in 28), C C C η η + ) 35a) η + 2 = 0, C j C j 2 + ) j + η η η 2 C C j = η 2 C j + 35b) η 2 kc k C l, j 2, with initial conditions k+l=j+ k,l 2 36) C 0) =, C 2 0) = 2, C j0) = 0, j 3; note that C 0 0 due to v) = 0. 36) is obtained from Remark, as W ss is given by 37) vu, 0) = u ) 2 u )2. We will first explicitly solve these equations for j = and afterwards derive the general form of the solution for j arbitrary. Remark 6. Most of the following computations have been performed with the help of the computer algebra package Maple, see e.g. [Cor02]. From 35a) it follows that 38) C η) = η e η Ẽ η ) γ e, η where Ẽk is in general defined by 3 39) Ẽ k z) := η 2 e zτ τ k dτ, z C, Rz) > 0, k N, see e.g. [AS64], and γ R is some constant to be determined. With 36) and de l Hôspital s rule we obtain lim η 0 C η) = for γ = 0; hence indeed γ = 0. Due to 40) Ẽ 2 η ) = e η η Ẽ η ), we can write 4) C η) = ηe η Ẽ 2 η ). 3 Here Rz) denotes the real part of z.

13 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 3 As we are interested in 28) for η which corresponds to the overlap domain between the two charts K and K 2 ), we expand Ẽ2 ) η about η = to obtain an indication as to what the C j s might look like in general: 42) Ẽ 2 η ) = + γ)η + η ln η + γ 2 2γ + 3 ) η 2 2 which implies 43) C η) = ηe η γklη k ln η) l. k,l=0 + 2γη 2 ln η + η 2 ln η) 2 + O η 3), We will show that C j can in fact be expanded as in 43) for any j N. To that end, note that e.g. for j = 2, equation 35b) becomes 44) C 2 C ) + 3e η η η ηẽ2η ) C 2 = e η ηẽ2η ), which has the solution [ 45) C 2 η) = exp 3 e η η Ẽ 2 η ) dη ]η 3 Ẽ 2 η ) ) dη + γ2 η 2 e η exp [ 3 e η η Ẽ 2 η ) dη ]; here we have used 4). 45) obviously cannot be integrated in closed form. Still, one can derive the following result concerning the structure not only of C 2, but of any C j with j 2: Proposition 3.. For j, the solution C j η) to 35),36) can be written as 46) C j η) = ηe η γ j kl η k ln η) l. k,l=0 Here, γ j kl R are constants to be determined from 36). Proof. The proof is by an induction argument: for i =, the assertion is obviously valid, see 43); let us assume that it holds for i =,..., j. For the homogeneous solution to 35b) one finds 47) Cj hom η) = γ j η 2 e η exp the integrand in 47) can be expanded as [ j + ) e η η Ẽ 2 η ) dη ]; 48) e η ηẽ2η ) = η + γη 2 η 2 ln η + O η 3), whence 49) [ [ η exp j + ) + γη 2 η 2 ln η + O η 3)] ] dη = O η j ), j 2. For 47), the claim now follows from 42), 49), and the following lemma:

14 4 NIKOLA POPOVIĆ AND PETER SZMOLYAN Lemma 3.2. For any α, β Z, z α+ ln z) β 50) z α ln z) β dz = α + β α + z α ln z) β dz, α ln z) β+ β +, α =. For the particular solution, note first that by the induction hypothesis, the righthand side of 35b) can be written as 5) η 2 C j + η 2 A particular solution to 52) C j C j η is given by 53) C part j η) = k+l=j+ k,l ) η kc k C l = e η e η m,n=0 γ j mnη m ln η) n. + j + ) ηẽ2η ) C j = e η η m ln η) n [ η m 2 ln η) n exp j + ) η 2 e η exp [ j + ) e η η Ẽ 2 η ) ] dη dη e η η Ẽ 2 η ) dη ]; With 42), 49), and Lemma 3.2, this concludes the proof, as m, n 0 and j 2. We can even obtain a somewhat more precise result on the structure of C j, j. Let k, l) denote the index of a term η k ln η) l in 46); given this notation, we have the following Proposition 3.3. A term with index k, l) can occur in 46) only if l k. Proof. The proof is again by induction: for i =, the assertion is obvious from 4) and 42). Given the assertion for i =,..., j, it follows immediately from 49) and Lemma 3.2 that it holds for the homogeneous part 47) of C j, as well. To prove the assertion for 53), we proceed as follows: as in [vgks] we define a map, say, ιm, n), which assigns to the index of a term in 5) the set of indices of the terms it generates in 53). By 49) and the proof of Proposition 3. one then easily sees that with Lemma 3.2, 54) { {m + m, n + n ), m + m, n + n ),..., m + m, 0)}, m j ιm, n) = {j + m +, n + n + )}, m = j; here m, n N with n m. This completes the proof, as n m by assumption Expansions in chart K. Given Proposition 3., we are able to derive asymptotic expansions for W s 2 in K 2 for η 2. To get the desired expansion for

15 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 5 v ε, however, we need to know what these expansions correspond to in K. First, note that with Lemma 2.5, 28) becomes 55) ε v u, ε ) = where 56) A j ε ) = e ε ε A j ε )u ) j, j=0 α j kl εk ln ε ) l ; k,l=0 l k here α j kl = ) l γ j kl. It remains to show that 56) does indeed make sense for ε 0 which is equivalent to η 2 in K 2 ). To that end, we assume that a curve of initial conditions in Σ out of the form 57) u, v, ε ) = u out, v out u out ), δ ), v out ) = 0 is given, and we investigate the corresponding invariant manifold consisting of segments of solutions of 2). By variation of constants integrating backwards from Σ out, this manifold can be represented as follows: 58a) 58b) 58c) u ξ, u out ) = u out Ξ ξ v ξ, u out ) δ ) = ε vout u out e ξ ε ξ ) = εe ξ, + e ξ Ξ ξ v ξ, u out ) dξ, e ξ ε ξ )u ξ, u out ) v ξ, u out ) dξ, where Ξ = ln δ ε. We have the following result: ) u out be C k -smooth for some k N. Then, for j = v u, ε ) exists and is continuous for ε [0, δ] and u β with Proposition 3.4. Let v out 0,..., k, j u j β > 0 sufficiently small. Proof. Changing the integration variable to ε in 58), we obtain 59a) 59b) δ u ε ) = u out v u ε ), ε ) dε ε ε, v u, ε ) = δ ε v out + ε δ ε ε u ε )v u ε ), ε ) dε ; here ε = εe ξ. Then, 59) together with 60) u ε ) u out + v out δε ), v u ε ), ε ) δ ε vout

16 6 NIKOLA POPOVIĆ AND PETER SZMOLYAN implies that v remains continuous in u, ε ) for ε 0. Differentiating 59) formally with respect to u yields 6a) 6b) v u, ε ) u = duout du = δ dv out ε du + δ ε δ ε ε as we have dε dε = ε ε, it follows that 62) whence 63) ε v u ε ), ε ) u dε, [ du ε ) v u ε ), ε ) + u ε ) v u ε ), ε ] ) dε ; du u du ε ) du = v u ε ), ε ) v u ε ), ε ), v u ε ), ε ) = v u ε ), ε ) v u ε ), ε ) u u ε ) v u ε ), ε ). This, together with 6a), gives us a formula for dvout du 64) dv out du in sum, we obtain 65) + v u, ε ) u ε v u, ε ) = dvout du out + = δ dv out ε δ Suppose now that v out ε ε du out = dvout du out du out du, δ v u ε ), ε ) v u ε ), ε ) v u, ε ) ε u ε ) ε dε δ v u, ε ) v u ε ), ε ) + ε u ε ) [ v u ε ), ε ) + u ε ) v u ε ), ε ) u ε ) u out ) ) v u ε ), ε ) ε dε ] v u ε ), ε ) dε. ) is C -smooth; using a standard fixed point argument, one can show that 65) has a unique solution vu,ε) u which is continuous in u, ε ). This concludes the proof for k = ; the argument for k 2 is similar. Inspired by 55) and 56), we feel induced to attempt an expansion of v u, ε ) as 66) v u, ε ) = a ij u )ε i ln ε ) j, j i see Figure 5; the requirement that j i in the summation in 66) is a consequence of Proposition 3.3. The coefficient functions a ij, 0 j i, will be determined uniquely by the demand for smoothness in u and by the requirement that, seen as a double expansion, 66) should agree with 55). To that end, let us introduce u as the independent variable in 2), whence 4 67) 4 Again the subscript will be omitted. dv = εu, du dε du = ε v. ;

17 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 7 ε ε v u, ε ) = X A j ε )u ) j j=0 Γ P out Σ out Q u v P v ε = v 0, ε) = P in v u, ε ) = j i X a ij 0)ε i ln ε) j j i X a ij u )ε i ln ε ) j Figure 5. Strategy for deriving an expansion for v ε in K n = 3). Remark 7. With 3) instead of 28) in K 2, we would now have 68) uv, ε) = b ij v)ε i ln ε) j j i instead of 66). One can easily check that the following considerations would then go through just the same, with only a few minor adjustments required. By proceeding just as in K 2 and multiplying the resulting equations with v, we obtain 69) a ijε i ln ε) j a kl ε k ln ε) l + a ijε i ln ε) j i ln ε + j) j i k,l=0 l k = εu) a ij ε i ln ε) j, j i where = d du now. We do not bother to look for the general solution to 69) right now, but will for the moment only consider the first few terms in 66); hence e.g.

18 8 NIKOLA POPOVIĆ AND PETER SZMOLYAN for 0 j i 2, 70a) 70b) 70c) 70d) 70e) 70f) a 00a 00 = a 00, a a 00 + a 00a + a = a, a 0a 00 + a 00a 0 + a 0 + a = a 0 ua 00, a 22a 00 + a a + a 00a a 22 = a 22, a 2a 00 + a a 0 + a 0a + a 00a 2 + 2a 2 + 2a 22 = a 2 ua, a 20a 00 + a 0a 0 + a 00a a 20 + a 2 = a 20 ua 0. Note that these equations can be solved recursively: 70a) yields either a 00 0 or a 00 = ; however, for 55) and 66) to agree when seen as double expansions, we have to take the latter, see 4), whence 7) a 00 = u + α 00. Here, α 00 is a constant which is to be determined from 55); indeed, it follows from 4) and 42) that an expansion for v is given by 72) vu, ε) = [ + γε + ε ln ε + Oε 2 ) ] u ) + O u ) 2). To obtain agreement to lowest order between 55) and 66), we hence have to take α 00 =. By plugging 7) into 70b) and solving the resulting equation 73) a u ) + a = 0, one then has 74) a = α u ). Similarly, 7) and 74) together with 70c) give 75) a 0u ) + a 0 = u α )u ), which has the solution 76) a 0 = u ) 2 α 0 )u ) α )u ) ln u ); for 76) to be smooth, α has to be chosen such that the ln u )-terms in 77) vanish, which implies α =. The requirement that 55) and 66) should agree then gives α 0 = + γ, see 72): 77) a 0 = u ) 2 + γu ). For 70d), 70e), and 70f), one obtains by the same procedure 78a) 78b) 78c) a 22 = u ) 2 u ), a 2 = 2γ + )u ) 2 2γ )u ), a 20 = u ) α 20 )u ) 2 γ 2 γ + )u ), where α 22 = and α 2 = 2γ + have again been chosen such that the ln u )- terms in 78b) and 78c) cancel, and α 20 has to be determined by comparing the leading-order terms in 55) and 66), see Figure 6.

19 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 9 ε Γ P out Σ out Q u P in Σ in v Figure 6. Overlap domain shaded) of expansions 55) and 66). Remark 8. The above procedure is in fact closely related to the approach one would classically take when matching 55) and 66). As these two expansions have to agree on the overlap domain between the two charts K and K 2, it is there one would have to define an intermediate variable. Note that in [HTB90], say, logarithmic switchback is handled using a modified version of the block matching principle introduced by [vd75]: terms are matched in blocks according to the powers of ε they contain, with no distinction being made for any additional logarithmic factors. Our approach seems to justify this principle, as the logarithmic factors in 66) are determined simply by the requirement that a ij be smooth. One expects, of course, that the above procedure can be carried out to any order in i and j in 69), where for i fixed one starts with j = i and then proceeds recursively down to j = 0. That this is indeed possible is contained in the following result: Proposition 3.5. There exist unique smooth functions a ij u ) such that 55) and 66), seen as double expansions, are the same. Proof. We proceed just as in computing the first few coefficients of 66) above: for fixed i we successively solve 69), starting with j = i. By induction we will establish 79a) 79b) a ij = i k= k= α ij k u )k, j i, i+ a i0 = αk i0 u ) k, j = 0 for any i and 0 j i, with some constants α ij k R to be chosen appropriately. Indeed, for i = the claim follows from 74) and 77) by inspection. Let us assume that 79) is valid for a kl, where k =,..., i and l k. By plugging 7) into

20 20 NIKOLA POPOVIĆ AND PETER SZMOLYAN 69) and collecting powers of ε ln ε, one obtains 80) a iju ) + ia ij = j + )a i,j+ ua i,j k+m=i l+n=j l k i, n m i here we have used a 00 =. The homogeneous solution to 80) is given by 8) α ij u ) i, 0 j i a kla mn ; with some constant α ij ; to complete the proof, we have to consider the following cases: for j = i, equation 80) becomes 82) a iiu ) + ia ii = which has the solution k= k+m=i k,m a kka mm, i 83) a ii = α ii u ) i + αk ii u ) k, as by 79) 84) k+m=i k,m i a kka mm = k= α ii k u ) k and a term α ii k u )k gives a particular solution of the form 85) αii k i k u )k, k i ; the constant α ii remains to be determined in one of the next steps. for j = i which is indeed representative of all further cases), one obtains 86) a i,i u ) + ia i,i = ia ii ua i,i 87a) k+m=i k,m [ a kk a m,m + a kk a ] m,m, where the homogeneous solution is again given by 8). As for the inhomogeneity, note that terms of the form iαk ii u )k in ia ii generate particular solutions of the form iα ii u ) i ln u ), k = i, 87b) 88a) 88b) iαii k i k u )k, k i. Similarly, for the terms α i,i k uu ) k in ua i,i one obtains α i,i i u ) i u ) ln u ) ), k = i, α i,i i 2 i k)i k ) u )k + i k)u ), k i 2.

21 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 2 By the induction hypothesis, for the remaining terms one has 89) 90a) 90b) k+m=i k,m [ a kk a m,m + a kk a m,m ] = i k=0 α i,i k u ) k for some constants α i,i k R. Just as above, the terms α i,i k u ) k give rise to terms of the form α i,i i u ) i ln u ), k = i, αi,i k i k u )k, k i. In sum, one thus has i 9) a i,i = α i,i u ) i + α i,i k u ) k k= + iα ii + α i,i i + α i,i ) i ln u )u ) i, where α ii is now chosen such that 9) is smooth, i.e., such that the ln u )-terms cancel, and α i,i still is at our disposal. for j i 2 in general, one repeats the same procedure, i.e., one solves 80) and subsequently fixes α i,j+ appropriately so as to eliminate any ln u )-terms in a ij, guaranteeing the smoothness of a ij. in the final step, for j = 0, additional terms of O u ) i+) are generated as claimed due to the terms α i,0 i uu ) i and α i i0u )i+ in the right-hand side of 80) giving 92) α i,0 i and u + ln u ) ) u ) i 93) α i0 i uu ) i in a i0, respectively. One is then left with α i and α i0, which one chooses such as to make sure that a i0 is smooth and that 55) and 66) agree if both are seen as double expansions, which concludes the proof. We are now ready to formulate the main result of this section, namely, to give an expansion of v ε for n = 3: Proposition 3.6. For ε 0, ε 0 ] with ε 0 > 0 sufficiently small, v ε = v 0, ε) can be expanded as 94) v ε = ε ln ε γ + )ε + 2ε 2 ln ε) 2 + 4γε 2 ln ε + Oε 3 ). Proof. By plugging 7), 74), 77), 78a), and 78b) into 66), one obtains the following expansion for v : 95) v u, ε ) = u ) + u )ε ln ε + [ u ) 2 + γu ) ] ε + [ u ) 2 u ) ] ε 2 ln ε ) 2 + [ 2γ + )u ) 2 2γ )u ) ] ε 2 ln ε + O ε 3 ). The assertion now follows with u = 0 and ε = ε in 95).

22 22 NIKOLA POPOVIĆ AND PETER SZMOLYAN Remark 9. Note that the expansion in 94) is equally valid in the original setting of 4), i.e., v ε = v ε, as is easily seen by performing the appropriate blow-down transformation, which is trivial here. Analogous results have been obtained in the literature, see e.g. [Lag88] The case n = 2. Although the case n = 2 is potentially more difficult, we can use the same strategy as before to compute expansions for v ε. Not surprisingly, however, computationally the analysis is more involved now, due to the extensive switchback which arises in the matching process for n = Expansions in chart K 2. As the situation in K 2 is very similar to that for n = 3, we will not go into too many details: given the ansatz 96) v 2 u 2, η 2 ) = C j η 2 )u 2 ) j, which we plug into 27) rewritten with η 2 as the independent variable, 97a) 97b) j=0 du dη = v η 2, dv dη = v η + u η 2 v + v η 2, we obtain a recursive sequence of equations for C j, j, by comparing powers of u : C C C η η + ) 98a) η + = 0, C j C j + ) j + η η η 2 C C j = η 2 C j + 98b) η 2 kc k C l, j 2. The initial conditions are again given by k+l=j+ k,l 2 99) C 0) =, C 2 0) = 2, C j0) = 0, j 3; moreover, C 0 0 just as for n = 3. From 98a), we have 00) C η) = ηe η Ẽ η ) + γ ; as lim η 0 C η) = for γ = 0, we conclude that γ = 0 again. The expansion of Ẽ η ) about η = is given by 0) Ẽ η ) = ln η γ) η ln η γ) η 2 ln η γ) 2 whence 02) C η) = ηe η γklη k ln η γ) l ; k,l=0 + η 2 ln η γ) 3 + O η 3),

23 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 23 here γ is Euler s constant, as before. For j = 2, 98b) gives 03) C 2 C 2 + ) + 3e η η η ηẽη ) C 2 = e η ηẽη ), which has the solution 04) C 2 η) = η 2 Ẽ η ) ) 2 dη + γ2 ηe η Ẽ η ) 3. Although 04) cannot be integrated in closed form, we still have the following result, which is similar to the one derived for n = 3: Proposition 3.7. For j, the solution C j η) to 98),99) can be written as 05) C j η) = ηe η γ j kl η k ln η γ) l. k,l=0 Here, γ j kl R are constants to be determined from 36). Proof. The proof is very similar to that of Proposition 3.: for i =, the assertion holds by 00); let it be valid for i =,..., j. The homogeneous solution to 98b) is given by 06) Cj hom ηe η η) = γ j Ẽ η ), j+ where Ẽ η ) j can be expanded as 07) Ẽ η ) j = ln η γ) j j + )η ln η γ) j j + )η 2 ln η γ) j j + )j + 2)η 2 ln η γ) j 3 + O η 3). Given the induction hypothesis, the right-hand side of 98b) has the following form: 08) η 2 C j + η 2 k+l=j+ k,l kc k C l = e η m,n=0 γ j mnη m ln η γ) n ; a particular solution of 98b) corresponding to a term e η η m ln η γ) n is given by 09) C part j η) = η m ln η γ) n Ẽ η ) j+ dη ηe η Ẽ η ) j. With the substitution η = η γ and 07), the integrand in 09) can be written as 0) γ m mn n ln η ) η m n ; m =m+ n =n j the claim now follows from Lemma 3.2 with η again replaced by η. Note, however, that we can state no analogue to Proposition 3.3 here; in fact, it seems that any index pair k, l) can occur in 66) now.

24 24 NIKOLA POPOVIĆ AND PETER SZMOLYAN Expansions in chart K. As in the case n = 3, in order to obtain an expansion for v ε, it now remains to translate 28) to K, see Figure 7. Lemma 2.5 again gives ) ε v u, ε ) = see 55), where 2) A j ε ) = e ε ε with α j kl = )l γ j kl. A j ε )u ) j, j=0 k,l=0 α j kl ε k ln ε + γ) l Remark 0 Transcendentally small terms). Terms in powers of ε, as well as terms in powers of ε multiplied by powers of ln ε + γ), are smaller than all positive powers of ln ε +γ) : they are said to be beyond all orders of ln ε +γ), or to be transcendentally small terms. In the original setting of flow around a circular cylinder, [Ski75] showed how to calculate a few of these terms. He pointed out, however, that they are in fact negligible: it is the only very slight asymmetry in the flow field which indicates the relative insignificance of these inertial terms for low Reynolds numbers, see [Ski75] for a detailed analysis. As for n = 3, using variation of constants we can again write 3a) 3b) 3c) u ξ, u out ) = u out v ξ, u out ) = v out ε ξ ) = εe ξ Ξ ξ u out v ξ, u out ) dξ, ) + Ξ ξ ε ξ )u ξ, u out ) v ξ, u out ) dξ, for the manifold consisting of segments of solutions to 2) given the initial curve 57). In analogy to Proposition 3.4 we now have ) Proposition 3.8. Let v out u out be C k -smooth for some k N. Then, for j = 0,..., k, j v u j u, ε ) exists and is continuous for ε [0, δ] and u β with β > 0 sufficiently small. Proof. The proof is the same as for n = 3, with the relevant relations given by 4a) 4b) and 5) v u, ε ) u + u ε ) = u out δ v u, ε ) = δ ε v out + ε δ v u, ε ) now, respectively. = dvout du out δ ε + ε v u ε ), ε ) dε ε, ε u ε )v u ε ), ε ) dε δ v u ε ), ε ) v u ε ), ε ) v u, ε ) ε u ε ) [ v u ε ), ε ) + u ε ) v u ε ), ε ) u ε ) ) ε dε ] v u ε ), ε ) dε

25 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 25 ε ε v u, ε ) = X A j ε )u ) j j=0 Γ P out Σ out P Q u P in v u, ε ) = X a ij u ) ε i ln ε + γ) j X ε i v ε = v 0, ε) = a ij 0) ln ε + γ) j v Figure 7. Strategy for deriving an expansion for v ε in K n = 2). To derive an expansion for v u, ε ) of the form ε i 6) v u, ε ) = a ij u ) ln ε + γ) j, we have to consider 7) dv du = εu, dε du = ε v, which yields 8) a ij ε i ln ε + γ) j k,l=0 a kl ε k ε i ln ε + γ) l + a ij ln ε + γ) j = εu i ) j ln ε + γ a ij ε i ln ε + γ) j.

26 26 NIKOLA POPOVIĆ AND PETER SZMOLYAN Collecting powers of εln ε + γ) in 8), we obtain the following sequence of equations for a ij with 0 i + j 3: 9a) 9b) 9c) 9d) 9e) 9f) 9g) 9h) 9i) 9j) a 00a 00 = 0, a 0a 00 + a 00a 0 = 0, a 0a 00 + a 00a 0 + a 0 = ua 00, a 02a 00 + a 00a 02 + a 0a 0 a 0 = 0, a a 00 + a 00a + a 0a 0 + a 0a 0 + a = ua 0, a 20a 00 + a 00a 20 + a 0a 0 + 2a 20 = ua 0, a 03a 00 + a 00a 03 + a 02a 0 + a 0a 02 2a 02 = 0, a 2a 00 + a 00a 2 + a a 0 + a 0a + a 0a 02 + a 02a 0 + a 2 a = ua 02, a 2a 00 + a 00a 2 + a 20a 0 + a 0a 20 + a a 0 + a 0a + 2a 2 = ua, a 30a 00 + a 00a 30 + a 20a 0 + a 0a a 30 = ua 20. Here we have proceeded by diagonalization: as we do not have such precise information on the structure of 05) as we had for n = 3, a simple recursion will not work. We thus have to take a different approach, comparing coefficients in 8) for i + j = p constant. Let us illustrate the procedure by explicitly solving the first few equations in 9): from 9a) we have a 00 0 or a 00 = 0; however, for ) and 6) to agree, we have to take the former, which substantially simplifies all further arguments. Equation 9b) is vacuous, as indeed 0 = 0, whereas 9c) then immediately yields a 0 0. Given a 00 0, 9d) implies either a 0 0 or a 0 =. Here, the requirement that ) and 6) agree to lowest order fixes a 0 =, whence 20) a 0 = u + α 0 for some α 0 R. In fact, with 00) and 0) one finds [ 2) vu, ε) = ln ε + γ ε ln ε + γ + ε ln ε + γ) 2 + ε 2 2 ln ε + γ 5 ε 2 4 ln ε + γ) 2 ε 2 ] + ln ε + γ) 3 + Oε3 ) u ) + O u ) 2), which implies α 0 =. Plugging 20) into 9e) then gives 22) a = u ) 2 u ); moreover, it follows from 9f) that a 20 0, as well. Again with 20), equation 9g) becomes 23) a 02u ) a 02 = 0, which has the solution 24) a 02 = α 02 u ) for some constant α 02. With reference to 2) one can fix α 02, whence α 02 = 0. As for a 2 and a 2, one easily obtains from 9h) and 9i) that 25) a 2 = 2u ) 2 + u

27 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 27 and 26) a 2 = 2 u )3 + u ) 2 + u ), 2 respectively, whereas a As for n = 3, we can now prove the following general result: Proposition 3.9. There exist unique smooth functions a ij u ) such that ) and 6), seen as double expansions, are the same. Proof. The proof differs significantly from the one we gave for n = 3, although it is again by induction, now on the sum i + j instead of on i alone, however. We will show 27a) 27b) 27c) i+j a ij = α ij k u )k, i, j, k= j a 0j = α 0j k u )k, j 2, k= a i0 0, i 2; indeed, for i + j = 2, the assertion is obvious from 22) and 24). Let 27) be valid for i + j = 2,..., p ; we have to show that it is valid for i + j = p, as well. Collecting powers of εln ε + γ) in 8), we obtain 28) [a kla mn + a mna kl ] + ia ij j )a i,j = ua i,j. k+m=i l+n=j Plugging a 00 0 and 20) into 28) gives the following equation for a 0p : 29) a 0pu ) p )a 0p = [a 0ka 0l + a 0la 0k ]; k+l=p+ k,l 2 by the induction hypothesis, the above sum can be written as p 2 30) k= α 0p k u )k. As terms of the form α 0p k u )k generate particular solutions of the form 3) α0p k k p + u )k, k p 2 in 29) and the homogeneous solution is given by 32) α 0p u ) p, for a 0p the claim follows. Note that the constant α 0p has to be chosen such that ) and 6) agree when seen as double expansions; the smoothness of a 0p is granted irrespective of the choice of α 0p. For a ij with i + j = p and i, 28) yields 33) ia ij = ja i,j ua i,j [a kla mn + a mna kl ]; k+m=i l+n=j

28 28 NIKOLA POPOVIĆ AND PETER SZMOLYAN due to the fact that a 00 0, this completely determines a ij. Moreover, it follows from the induction hypothesis that a ij is of the desired form and that a p0 0, respectively, which concludes the proof. Remark. The above proof shows that once the leading-order behaviour in 6) is determined, the transcendentally small terms in 6) are given as solutions not of differential, but of algebraic equations. Our approach thus immediately provides us with these terms, whereas in the classical approach, quite cumbersome computations are required for their determination, as matching is typically done only up to transcendentally small quantities there. We can now give an expansion of v ε for n = 2: Proposition 3.0. For ε 0, ε 0 ] with ε 0 > 0 sufficiently small, v ε = v 0, ε) can be expanded as 34) v ε = ) ln ε + γ + O ln ε + γ) 2. Proof. As for n = 3, 6) gives 35) v u, ε ) = u ) ln ε + γ + [ u ) 2 u ) ] ε ln ε + γ [ + 2 u ) 3 + u ) 2 + ] 2 u ε 2 ) ln ε + γ + O whence we obtain the assertion with u = 0. Remark 2. As ln ε + γ) can be expanded as 36) ln ε + γ = γ ) j ln ε ln ε j=0 for 0 < ε < ε 0 sufficiently small, v ε can be written as 37) v ε = ln ε + γ ) ln ε) 2 + O ln ε) 3, ln ε + γ) 2 which agrees with the expansion found e.g. in [HTB90]. In fact, as was pointed out by [LC72], the expansion is more compact if it is telescoped, i.e., arranged in powers of ln ε + γ). However, this arrangement is not helpful numerically, as 34) becomes undefined for ε = , whereas 37) allows for values of ε up to. ), 4. Asymptotic solution expansions 4.. Expansions in chart K. Given the expansion for v ε from Proposition 3.6, which we can write as 38) v ε = β ij ε i ln ε) j j i

29 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 29 with constants β ij R, it makes sense to set up expansions 39) u r, ε ) = a ij r )ε i ln ε ) j, v r, ε ) = j i b ij r )ε i ln ε ) j, j i where we regard both u and v as functions of r, ε ) now. The ansatz in 39) is closer to the classical approach than what was done in the previous sections; however, as the techniques we apply are very similar to the ones used before, we only sketch the procedure here, leaving out most of the details. Rewriting 2) with r as the independent variable now and omitting the subscript again, we obtain 40) du dr = v r, dv dr = v r + εuv r, dε dr = ε r. With 40), we get 4a) 4b) j i j i [ a ijε i ln ε) j a ] ij r εi ln ε) j i ln ε + j) = r [ b ijε i ln ε) j b ] ij r εi ln ε) j i ln ε + j) = r + r j i b ij ε i ln ε) j, j i b ij ε i ln ε) j j i a ij ε i ln ε) j b kl ε k ln ε) l ; k,l=0 l k comparing powers of ε ln ε yields the following recursive sequence of differential equations, 42a) 42b) ra ij ia ij j + )a i,j+ = b ij, rb ij ib ij j + )b i,j+ = b ij + k+m=i l+n=j l k, n m a kl b mn, with the initial conditions given by 43) a ij ) = 0, b ij ) = β ij, 0 j i. Remark 3. For v, it follows directly from 38) that the ansatz in 39) is plausible, whereas for u, it can be justified a posteriori using 42a).

30 30 NIKOLA POPOVIĆ AND PETER SZMOLYAN For the first few coefficients in 39, we thus have 44a) 44b) 44c) 44d) 44e) 44f) ra 00 = b 00, rb 00 = b 00, ra a = b, rb b = b, ra 0 a 0 a = b 0, rb 0 b 0 b = b 0 + a 00 b 00 ; with Proposition 3.6, the solutions to 44) are easily found to be given by 45a) 45b) 45c) a 00 = r, b 00 = r, a = r r), b = r 2, a 0 = γ)r r) + 2r 2 ln r, b 0 = r + γr) 2r 2 ln r, which allows us to state the following result: Proposition 4.. For n = 3, the solution to ) can be expanded as 46) uξ, ε) = ξ + ε γ ln ε) ) ε + ) ln ξ + Oε 2 ) ξ ξ inner expansion); here ξ is as defined in 2). Proof. The result is immediate from 39) and 45) after one has applied the appropriate blow-down transformations r = ξ and ε = εξ, as 47) u r, ε ) = r r r )ε ln ε + γ)r r )ε + Similarly, for n = 2 the equations du dr = v r, dv 48) dr = εuv r, dε dr = ε r in combination with an ansatz of the form ε i 49) u r, ε ) = a ij r ) ln ε + γ) j, v r, ε ) = lead to the recursive sequence of equations 50a) 50b) ra ij ia ij + j )a i,j = b ij, rb ij ib ij + j )b i,j = the initial conditions are again given by k+m=i l+n=j + 2r 2 ln r ε + O ε 2 ). ε i b ij r ) ln ε + γ) j a kl b mn ; 5) a ij ) = 0, b ij ) = β ij, i, j 0

31 RIGOROUS ASYMPTOTIC EXPANSIONS FOR LAGERSTROM S MODEL EQUATION 3 for some β ij R. To leading order one finds 52) a 0 = ln r, b 0 =, whence one obtains the following Proposition 4.2. For n = 2, the inner expansion of the solution to ) is given by 53) uξ, ε) = ln ξ ) ln εξ + γ + O ln εξ + γ) 2. Proof. The proof is analogous to that for n = 3. Remark 4. By expanding ln εξ + γ) for 0 < ε < ε 0 small, one can write 54) uξ, ε) = ln ξ ) ln ε + γ + O ln ε + γ) 2 or 55) uξ, ε) = ln ξ ln ε + γ ln ξ ) ln ξ)2 + ln ε) 2 ln ε) 2 + O ln ε) 3, which are the expansions usually found in the literature, see e.g. [LC72] or [HTB90] Expansions in chart K 2. To derive solution expansions in K 2, we would have to proceed as above, i.e., we would set out by determining the structure of the coefficients in 39) respectively 49) in general, which we would then use to rearrange 39) and 49) with respect to a new basis. It is these expansions which would provide us with the proper ansatz for the corresponding expansions in K 2. In sum, we expect to obtain the following result which we cite for reference only, see [LC72]: Proposition 4.3. The outer solution expansion for ) is given by 56) ux, ε) = εe 2 x) + Oε 2 ) for n = 3 and by 57) ux, ε) = + for n = 2, respectively; here E k is defined by 58) E k z) := z ) ln ε + γ E x) + O ln ε + γ) 2 e t t k dt, z C, Rz) > 0, k N. Acknowledgements. The authors thank Astrid Huber, Martin Krupa, and Martin Wechselberger for numerous helpful discussions. This work has been supported by the Austrian Science Foundation under grant Y 42-MAT. References [AS64] M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Number 55 in National Bureau of Standards Applied Mathematical Series. United States Department of Commerce, Washington, 964. [CLW94] S.N. Chow, C. Li, and D. Wang. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge, 994. [Cor02] R.M. Corless. Essential Maple 7: an introduction for scientific programmers. Springer- Verlag, New York, 2002.