TRANSONIC EVAPORATION WAVES IN A SPHERICALLY SYMMETRIC NOZZLE XIAOBIAO LIN AND MARTIN WECHSELBERGER Abstract. This paper stdies the liqid to vapor phase transition in a cone shaped nozzle. Using the geometric method presented in [3, 34], we frther develop reslts on sbsonic and spersonic evaporation waves in [4] to transonic waves. It is known that transonic waves do not exist if restricted solely to the slow system on the slow manifolds [4]. Ths we consider the existence of transonic waves that inclde layer soltions of the fast system that cross or connect to the sonic srface. In particlar, we are able to show the existence and niqeness of evaporation waves that cross from spersonic to sbsonic regions and evaporation waves that connect from the sbsonic region to the sonic srface and then contine onto the spersonic branch via the slow flow.. Introdction In this paper, we investigate the liqid to vapor phase transition in a spherically symmetric cone-shaped nozzle. Sbsonic evaporation processes, sch as fel injection into a combstion engine, show p in many important engineering problems and are stdied in many research laboratories arond the world. The ring formation observed in a shock tbe experiment [33] is an example of a spersonic process, as arged by Fan in []. Or focs is on transonic evaporation waves. While we are not aware of any experiments where transonic evaporation waves have been observed, we expect that or theoretical reslts might be sefl in some physical and engineering process when the speed of the flid in a nozzle approaches sonic speed. The stdy of nozzle flows was pioneered by Corant & Friedrichs [6] and T.-P. Li [6]. Transonic flow withot liqid-gas phase transition has been stdied by many athors, see recent articles [3, 4, 5, 3, 35, 36]. Among them [35] dealt with sbsonic and sbsonic to sonic flows throgh infinitely long nozzles. For transonic flows modeled by reactiondiffsion eqations, W. Li et al. [7, 8, 7] considered one dimensional standing waves for a simplified model of gas flows in a nozzle with general variable cross sectional area a(x). To the best of or knowledge, transonic flows that inclde a reaction-diffsion eqation describing evaporation or condensation have not been rigorosly analysed mathematically. Liqid-vapor phase transitions have been mch stdied sing a van der Waals pressre fnction [5]. The van der Waals model reqires detailed modeling of the evaporation of individal droplets [8, 30] or of the ncleation process by which vapor condenses. Figre (a) shows the graph of pressre p as a fnction of specific volme v (the inverse of density ρ) with the temperatre held constant. The decreasing branch of the pressre fnction p for v < α corresponds to the liqid state while the other decreasing branch for v > β corresponds to the vapor state of the flid. There is a spinodal region, α < v < β, in which pressre Acknowledgement: XBL was partially spported by NSF grant DMS-070 and MW by ARC grants DP00775 and FT000309.
p p!=0!= p eq p eq m " # (a) M v m (b) M v Figre. Figre (a) shows the van der Waals pressre fnction p(v) where the Maxwell line p = p eq connects v = m to v = M on p = p(v). Figre (b) shows the fnctions p(λ, v), λ = 0,, where the Maxwell line p = p eq connects v = m on λ = 0 to v = M on λ =. is, anomalosly, an increasing fnction of v, i.e., a decreasing fnction of density ρ. This region is considered to be highly nstable and cannot be observed in experiments. In reality, this part of the pressre fnction p is replaced by the Maxwell line p = p eq which is defined by the eqal area rle ; see Figre (a). The flid can be in any liqid/vapor state for the eqilibrim pressre p eq, see [9]. An alternative model proposed by Fan [9] introdces a variable 0 λ to represent the mass fraction of vapor, where λ = 0 represents the pre liqid state, and λ = represents the pre vapor state. Since Fan s model allows intermediate vales of λ, it can be sed to stdy experiments in which mixtres of the liqid and vapor states occr []. In Fan s model, pressre p is a fnction of (λ, v). The graphs of p(0, v) and p(, v) are shown in Figre (b). This model was motivated by, among other works, [33, 0, 3]. An important featre of Fan s model [9, 0, ] is a reaction-diffsion eqation that describes the phase transition between liqid and vapor. This model was proposed nder the assmption that the heat capacity of the flid nder consideration is high (called a retrograde flid), reslting in evaporation that is mainly cased by the pressre change, not the temperatre change. Fan s model consists of a viscose p-system that describes the motion of compressible liqid-vapor mixtre and a reaction-diffsion eqation that describes the liqid to vapor phase transition: (.) ρ t + (ρ) = 0, (ρ) t + (ρ() + p(λ, ρ)i) = η ( + T ) + η (( )I), (λρ) t + (λρ) = w(λ, ρ) γ + µ (ρ λ), where ρ > 0 is the density of the flid, R 3 the velocity vector of the flid, λ [0, ] is the mass fraction of vapor. All the constants in (.) are small parameters. Among them η is the shear viscosity of the flid, η is a linear combination of the shear and the volme viscosity coefficients that is related to dilation of the flid, and µ is the diffsion coefficient.
λ =0 λ = r 0 r i Figre. Spherically symmetric nozzle geometry: the cone s bondary is shown in solid bold. Flid is injected at r = r 0 and discharged at r = r. The dashed crve at r = r i indicates an internal bondary layer of a standing evaporation wave where a transition from a liqid state (λ = 0) to a vapor state (λ = ) happens. r Assmption. The pressre p(λ, ρ) is a fnction of the density ρ and the mass fraction of vapor λ which satisfies (.) p ρ > 0, p ρρ > 0, p λ > 0, p(λ, 0) = 0, p(λ, ) =. A typical pressre fnction which satisfies all the conditions in (.) is given by (.3) p(λ, ρ) = C( + λ)ρ k, C > 0, k >. The last eqation of system (.) comes from a simplified model proposed by Fan [9] where the vapor initiation term has been omitted and only the vapor growth term w(λ, ρ) is inclded. The fnction w is defined as (.4) w(λ, ρ) := (p(λ, ρ) p eq )λ(λ )ρ, and w/γ represents the vapor prodction rate where γ is the typical reaction time. If p = p eq then w = 0 and the mixtre cold be in any liqid/vapor configration state 0 λ. Necessary and sfficient conditions for the existence of phase-changing traveling waves (liqid to vapor or vapor to liqid) in one-dimensional space were proved in [9, 0, ]. Using dynamical systems methods, the proof of the existence of those one-dimensional waves was simplified in [3]. Let r be the radial coordinate of the nozzle. We assme that the flid is injected at the smaller end r = r 0 and discharged at the larger end r = r (see, Figre ). Assming that the cone s bondary is slippery and offering no resistance to tangential flows at the bondary, the bondary effect is therefore negligible. We consider the spherically symmetric soltions 3
which are fnctions of (r, t) only and satisfy the following system: ρ t + (ρ) r + ρ = 0, r (.5) (ρ) t + (ρ + p) r + ρ = ϵ( r + r r ) r, (λρ) t + (λρ) r + λρ = r γ (p p eq)λ(λ )ρ + µ((ρλ r ) r + ρλ r r ), where ϵ is the combined viscosity. A straightforward bt rather lengthy calclation shows that ϵ = η + η. Fan and Lin in [4] stdied the existence of non-transonic evaporation waves that are in either sbsonic or spersonic regions for the entire evaporation process. In this paper we shall extend their reslts to evaporation waves that cross the sonic srface. That is, at r = r 0 the liqid state flid is sbsonic (or spersonic) while at r = r the vapor state flid becomes spersonic (or sbsonic). In the seqel, we call sch waves transonic evaporation waves. We look for standing waves that are stationary soltions of (.5). The evaporation front of a stationary wave remains in the finite nozzle, making the process more sefl. Assme that the flid particles move from the smaller end to the larger end of the nozzle with the speed > 0. If, in the Elerian coordinates, the speed of the wave is zero, then in the Lagrangian coordinates, the wave travels with the speed, i.e., pointing towards the smaller end of the nozzle. Since the speed of sond is p ρ (see, e.g., []), we call the standing wave with < p ρ (or = p ρ or > p ρ ) the sbsonic (or sonic, or spersonic) wave, which really means that the wave speed in the Lagrangian coordinates is sbsonic (or sonic, or spersonic). The region of the phase space where < p ρ (or = p ρ or > p ρ ) will be called the sbsonic (or sonic, or spersonic) region. The sonic region is a codimension one srface in the phase space. The spherically symmetric standing waves are stationary soltions of (.5). Based on the laws of flid dynamics, the typical reaction time γ, the diffsion coefficient of vapor µ and the viscosity ϵ are small parameters, and are proportional to the mean free path. Using this physical fact and to simplify the matter mathematically, we shall assme that γ = ϵ/a, µ = ϵb, where a, b = O(). Then the stationary soltion in spherical coordinates satisfies (ρ) r + ρ = 0, r (.6) (ρ + p) r + ρ r (λρ) r + λρ = ϵ( r + r ) r, = a ϵ (p p eq)λ(λ )ρ + ϵb((ρλ r ) r + ρλ r r ). r In this paper we only consider an evaporation process where p < p eq is a necessary condition for the vapor growth fnction w to be positive. We state this as an assmption for ftre reference: Assmption. The flid described by system (.6) satisfies the condition p < p eq. In the rest of the paper, we analyze this system with the emphasis on transonic evaporation waves. In particlar, we look for standing waves with an internal layer at r = r i, r 0 < r i < r, 4
P 0 γ 0 γ i γ P P 0( ε) γ 0 (ε) Γ(ε) γ (ε) P ( ε) Figre 3. Sketch of a singlar concatenated orbit Γ = γ 0 γ i γ and its corresponding soltion Γ(ε). where the transition from the liqid state to the vapor state of the flid happens (see, Figre ). By formlating appropriate bondary conditions at the injection and discharge sites of the nozzle at r = r 0 and r = r, we prove the existence of niqe sper to sbsonic as well as sb to spersonic evaporation waves. The precise statements are given in Theorem 5. and Theorem 5. for the sper to sbsonic case and in Theorem 5.3 for the sb to spersonic case. In the following, we otline the strctre of the paper that leads to the precise formlations and the proofs of these theorems. A key in or analysis is to identify system (.6) as a singlarly pertrbed system. In, we introdce appropriate slow and fast variables so that system (.6) can be converted into a slow-fast dynamical system and stdied by means of geometric singlar pertrbation theory [4, 6, 9, 34]. In 3, we focs on the (internal) layer problem for the existence of evaporation layer soltions. In general, there may exist two kinds of evaporation transonic layer soltions connecting the pre liqid state λ = 0 to the pre vaporized state λ = : a spersonic to sbsonic layer or a sbsonic to spersonic layer. Each of these internal layers is a heteroclinic soltion that crosses the sonic srface in fast radial time. We prove that spersonic to sbsonic internal layer soltions at r = r i, r 0 < r i < r do exist bt sbsonic to spersonic layer soltions do not. However, we are still able to show the existence of internal sbsonic (or spersonic) to sonic layer soltions. In 4, we stdy the redced oter problem on the critical manifold. We show that the flow on the critical manifold points away from the sonic srface. Ths transonic evaporation waves cannot be constrcted solely by soltions on the slow manifold bt mst consist of internal layers. As a common practice in singlar pertrbation problems, we define in 5 singlar evaporation waves Γ = γ 0 γ i γ which are concatenations of slow (oter) soltions γ 0, γ and an internal layer soltion γ i (see Figre 3). As highlighted above, we are able to prove the existence and niqeness of transonic evaporation waves Γ(ε). In the spersonic to sbsonic case, Γ(ε) is O(ε) close to Γ while in the spersonic to sbsonic case, Γ(ε) is O(ε /3 ) close to Γ. The proof of the spersonic to sbsonic case relies on a generalization of the Exchange Lemma that has been recently proven in [5]. The fractional power of the order of ε in the sbsonic to spersonic case points to a trning point problem that stems from the sonic connection of the internal layer γ i. To prove this reslt we se a geometric desinglarization known as the blow-p techniqe (see, [7,, 3]). This techniqe reveals an intermediate region near the sonic srface which serves as the link between the fast layer soltion to the redced flow on the slow manifold for ε 0 (see also [, 34]). In 6, we 5
otline the proof that is based on the blow-p techniqe (see [3] for details) and describe the flow near the sonic srface. We provide some final remarks in 7.. A geometric singlar pertrbation approach Since ε is a small parameter, we will se singlar pertrbation techniqes [4, 6, 9, 34] to find standing waves for system (.6). To convert the system into a fast-slow form, we introdce the new variables (.) m := ρ, θ := ϵbρλ y, n := m p + ϵ( y + /r), r = y. The dmmy independent variable y 0 allows s to rewrite (.6) as an atonomos system. The change of variables (ρ,,, λ, λ) (m, n,, λ, θ) is non-singlar when > 0, which is the case considered in this paper. This leads to the following slow system on the slow scale y: r y =, (.) m y = m r, n y = m r, ϵ y = n + m + p(λ, m ) ϵ r, ϵλ y = θ bm, ϵθ y = θ b aw(λ, m ) ϵθ r. An eqivalent system on the fast scale z = y/ϵ is given by the fast system r z = ϵ, (.3) m z = ϵ m r, n z = ϵ m r, z = n + m + p(λ, m ) ϵ r, λ z = θ bm, θ z = θ b aw(λ, m ) ϵθ r. 6
The singlar natre of these systems is revealed by taking the limit ϵ 0. For the fast system (.3) this yields the layer problem r z = m z = n z = 0, (.4) z = n + m + p(λ, m ), λ z = θ bm, θ z = θ b aw(λ, m ), for the evoltion of the fast variables (, λ, θ). Note that the slow variables (r, m, n) are parameters in the layer problem. On the other hand, for the slow system (.) the reslt of taking the limit ϵ 0 is the redced problem r y =, m y = m r, (.5) n y = m r, 0 = n + m + p(λ, m ), 0 = θ bm, 0 = θ b aw(λ, m ), which is a differential-algebraic system for the evoltion of the slow variables (r, m, n). The phase space of the redced problem is defined by the algebraic constraints 0 = n + m + p(λ, m ), (.6) 0 = θ bm, 0 = θ b aw(λ, m ), or eqivalently n = m p(λ, m ), (.7) θ = 0, w(λ, m ) = 0. Assming ρ > 0, it follows that w = 0 has three possible soltions: λ = 0, λ = and p = p eq which correspond to the three branches of the three-dimensional critical manifold (or slow manifold in some literatre) S = S 0 S S e. These branches can be expressed as fnctions of the (r, m, ) variables: (.8) S 0,,e := {(r, m, n,, λ, θ) R 6 : λ = Λ(m, ), θ = 0, n = N(m, )}. 7
Under Assmption, the srface S e will not be of interest to s. The fnctions Λ(m, ) and N(m, ) for S 0 and S are given in the following table: Λ(m, ) N(m, ) S 0 0 m p(0, m/) S m p(, m/) An example of the graphs of S 0 and S is shown in Figre 4 where we chose p = ( + λ)ρ as a representative pressre fnction in (.3). For fixed (λ, m), since the fnction p(λ, m/) is concave pward with p(λ, 0) = 0, p(λ, ) =, the graph of N(m, ) is concave downward, has a vertical asymptote as 0, and approaches n = m as. The maximm vale of N(m, ) is reached on the codimension-one sonic srface where = p ρ. Definition. For a fixed m, let n be the maximm vale of N(m, ) on S. For a given pair of m and r, the nmber of eqilibrim points of the layer problem on S 0 or S is a fnction of n only. Since eqilibria do not depend on r, we will often not mention r in the ftre. There is no eqilibrim point on S for n > n. Therefore we shall only consider n n. Depending on n = n or n < n, the nmber of eqilibrim points on S 0 S can be three or for as illstrated in Figre 4. From the layer problem (.4) we find that (.9) z = n + m + p(λ, m ) = n N(m, ). By flipping the graph of N(m, ) to N(m, ) then shifting by the constant n, we can easily determine the sign of z for any (λ, ); see Figre 5. This will be very sefl when constrcting the layer soltions... Stability analysis of the layer problem. We stdy the eigenvales and eigenfnctions for the eqilibrim points of the the layer problem (.4) on S 0 S. The Jacobian of (.4) is given by (.0) J = m + p p λ 0 θ bm 0 bm θ aw b aw λ b The derivatives of p and w with respect to λ and are given in the following table: p p λ w w λ S 0 m p ρ < 0 > 0 0 m (p p eq) S m p ρ < 0 > 0 0 + m (p p eq) On the branches S 0 and S, the Jacobian redces to (.) J =. m ( p ρ ) p λ 0 0 0 bm 0 ±a m (p eq p) b where the mins sign is associated to S 0 and the pls sign to S. The eigenvales {l, l, l 3 } of the matrix J satisfy the characteristic eqation ( m ( (.) ( p ρ ) l) l(l b ) ± a ) b (p p eq) = 0, 8,
n n =! m _ n n = p! S S 0 Figre 4. (A) Shown are fnctions n = N(m, ) for the branches S 0, S (.8) of the three-dimensional critical manifold S. Here we se p(λ, ρ) = ( + λ)ρ for nmerical comptations. Note that S 0 lies above S. (B) Cross section for a fixed m: If n < n, there are for eqilibrim points two on S 0 and two on S. If n = n, there are three eqilibrim points two on S 0 and one on S. where the mins sign belongs to S and the pls sign to S 0. On both branches S 0, S, one eigenvale is given by l = m ( p ρ ). On the branch S and nder Assmption, p < p eq, the other two eigenvales are real and satisfy l < 0 < l 3. On the other branch S 0, if p < p eq and + 4ab(p p eq ) > 0 then the other two eigenvales are real and satisfy 0 < l < l 3. On the other hand, if p < p eq and + 4ab(p p eq ) < 0, the eigenvales l, l 3 are complex conjgates. A direct calclation of the corresponding eigenspace of this complex conjgate pair of eigenvales {l, l 3 } implies that layer soltions will enter the physically nrealistic region λ < 0 as z, de to the oscillations of these layer soltions near S 0 cased by the complex eigenvales. 9
z z= 0 = p ρ λ = λ = 0 _ n < n _ n = n y = m + n 3 4 Figre 5. The two concave p crves correspond to n = N(m, ) with λ = 0, respectively. By choosing λ [0, ] and, we can find z from the figre. For example, given n < n, we have z < 0 if < < 3 and z > 0 if < or > 4. The evaporation waves discssed in this paper occr in a bonded sbset Ω of the phase space, defined as Ω := {(r, m, n,, λ, θ) : ū ū, m m m, r 0 r r }. The set Ω and the positive constants ū, ū, m and m depend on the types of evaporation waves we try to constrct and are not difficlt to determine in each case. Throghot this paper we assme that the variables (, λ, m) satisfy the following condition: Assmption 3. + 4ab(p(λ, m ) p eq) > 0 for ū ū, m m m and λ = 0. Althogh this assmption is only given for λ = 0, it is easy to show the following reslt: Lemma.. If (, λ = 0, m) satisfy Assmption 3, then + 4ab(p(λ, m ) p eq) > 0 for 0 λ. Proof. Since p λ > 0, then we have p(λ, m/) > p(0, m/) for λ > 0. Notice that if ū > 0 is given, Assmption 3 holds if a and b are small enogh sch that ab < ū /(4p eq ). Under Assmption 3, the eigenvales of the Jacobian (.) are real. The corresponding eigenvectors to the eigenvales {l, l, l 3 } are given by ( pλ (.3) v = (, 0, 0) v j =,, bm ) l j l l j, j =, 3, where we assme that l j l, j =, 3 for convenience. We remark that if l = l j for j = or 3, then the expression for v j can be complicated. However, most of the reslts of this paper do not depend on this expression and are still valid. An important observation is that the eigenvale l vanishes if = p ρ, i.e. along the sonic srface of S 0 respectively S. Geometrically, the branches S 0 and S are three-dimensional folded manifolds within the six-dimensional phase space where the sonic srface corresponds to the fold where the layer flow is tangent to the branches S 0 and S along the eigendirection 0
spanned by the nllvector v. Conseqently, the layer problem projected onto the onedimensional nllspace ndergoes a saddle-node bifrcation along the sonic srface as can be seen in Figre 4. Under the Assmptions and 3, the signs of the eigenvales for the layer problem on S 0 and S can be smmarized in the following table: S 0 S sbsonic, < p ρ, +, +,, + sonic, = p ρ 0, +, + 0,, + spersonic, > p ρ +, +, + +,, + Since the critical manifolds consist of eqilibrim points of the layer problem, the fiber dimensions of the associated nstable and stable manifolds of S 0 and S on the sbsonic and spersonic regions, and the center(-stable) manifolds of S 0 and S on the sonic srface can be determined by the signs of eigenvales of the layer problem and are listed in the following table: dim W (S 0 ) dim W s (S 0 ) dim W (S ) dim W s (S ) sbsonic spersonic 3 0 sonic dim W c (S 0 ) = dim W cs (S ) = 3. Existence of evaporation layer soltions Evaporation layer soltions are heteroclinic connections from S 0 to S. Assming that Assmptions and 3 are satisfied, then the types of connections are listed in the following table, where we refer to the fiber dimensions of S 0 and S : dim W (S 0 ) dim W (c)s (S ) sbsonic spersonic 3 transonic (sb to sper) transonic (sper to sb) 3 sb (or spersonic) to sonic (or 3 ) The existence of non-transonic evaporation waves, i.e. sbsonic to sbsonic and spersonic to spersonic waves, has been proved in [4], where the transverse intersection of W (S 0 ) and W s (S ) for sbsonic waves was checked nmerically. Using the method presented in the proof of Lemma 3.3, it can also be proved rigorosly. Definition. For any 0 < α < β, a corresponding rectangle R(α, β) can be defined in the (, λ) plane: R(α, β) := {(, λ) 0 λ, α β}.
F s F r θ F f λ α F b β F k Figre 6. The pentahedron that is based on the rectangle R(α, β) and all its srfaces F x, x = b, f, k, r, s. A pentahedron shaped solid W (Figre 6) in (, λ, θ) space is said to be based on the rectangle R(α, β) if the five srfaces of W are as follows: (3.) F b = {(, λ, θ) : α < < β, 0 < λ <, θ = 0} F r = {(, λ, θ) : α < < β, λ =, 0 < θ < m/} F s = {(, λ, θ) : α < < β, 0 < λ <, θ = mλ/} F k = {(, λ, θ) : = β, 0 < λ <, 0 θ mλ/} F f = {(, λ, θ) : = α, 0 < λ <, 0 < θ < mλ/} Lemma 3.. For a given m, assme that (, λ, m) satisfy Assmption for α β. If W is a pentahedron shaped solid based on the rectangle R(α, β), then at points on the planes F b, F r and F s, the flow of the layer problem (.4) leaves W. Moreover, if (α, 0, 0) and (β, 0, 0) are two eqilibrim points for the layer problem on S 0, then at points on the plane F f, the flow of the layer problem (.4) enters W while at points on the plane F k, the flow of the layer problem (.4) leaves W. Proof. We show that the layer flow leaves W throgh the interior of F x, x = b, r, s, i.e. g n x > 0 evalated in the interior of F x where g denotes the vector field of the layer problem (.4) and n x denotes the corresponding oter normal vector of F x given by (3.) n b = (0, 0, ), n r = (0,, 0), n s = (0, m/, ). We have the following reslts for p < p eq : (n b g) Fb = aw(λ, m ) > 0, α < < β, 0 < λ < (n r g) Fr = θ bm > 0, α < < β, θ > 0 (n s g) Fs = λm a(p p eq )λ(λ ) m 4b = λm ( 4ab(p p eq )(λ ) ) 4b λm ( + 4ab(p p eq ) ) > 0, α < < β, 0 < λ < 4b which follows from Assmption 3 and Lemma..
The oter normal vector of F k and F f are n k = (, 0, 0), n f = (, 0, 0) respectively. We have n + m + p = 0 at the eqilibria (α, 0, 0) and (β, 0, 0). De to the fact that p λ > 0 and λ > 0, we have (n f g) Ff = ( n m p) Ff < 0, θ > 0, 0 < λ < (n k g) Fk = (n + m + p) Fk > 0, θ > 0, 0 < λ <. This proves the last assertion of the lemma. We shold always assme that n n (otherwise, evaporation layers are not possible). If n < n, then there are two eqilibrim points E, E S 0 and two eqilibrim points E+, E+ S. Also E, E+ are on the sbsonic and E, E+ are on the spersonic branches respectively. If n = n, there are two eqilibrim points E, E S 0 and a niqe eqilibrim point E + S. Also E, E +, E are on the sbsonic, sonic and spersonic branches respectively; see Figres 7 and 8. The signs of z can easily be determined from these figres as well, as described in Section. Let s define the -nllsrface in the (, λ, θ)-space along which z = 0 in (.9). Since p λ > 0 it follows that z = 0 can be solved for λ = Λ c (). Frthermore, we know that dλ c d = m (p ρ ) p λ > 0 if < p ρ, = 0 if = p ρ, < 0 if > p ρ. Note that the right hand side of (.9) is independent of θ. Hence, we can define the projection of a -nllsrface along the θ-direction, onto (, λ)-space and call it a -nllcline. For simplicity we sometimes call sch -nllsrface a -nllcline too. In Figres 7(b) and 8(b), crves in (, λ)-space connecting eqilibrim points diagonally inside the rectangle boxes are segments of the -nllcline. If n < n, then there are two trivial transonic heteroclinic orbits from spersonic to sbsonic regions: One satisfies λ = 0, θ = 0 and connects E to E on S 0 ; the other satisfies λ =, θ = 0 and connects E+ to E+ on S. Following the terminology of Haitao Fan [], we will call them wet shock and dry shock. With (λ, θ) given, a wet or dry shock is determined by a scalar ODE for (z) and is easy to find. From Figres 7 and 8, it is clear that z < 0 for all z R for both wet shocks or dry shocks. If n = n, there is only one trivial transonic heteroclinic orbit connecting E to E on S 0, which is a wet shock. All those trivial transonic layers are monotone decreasing in and go from spersonic to sbsonic regions. 3.. Existence of spersonic to sbsonic layer soltions. For n < n, there are for eqilibrim points {E, E +, E +, E } on S 0 S marked by their coordinates < + < + < in Figre 7. Theorem 3.. Assme n < n and the Assmptions and 3 are satisfied with ū < < < ū. Then there exists a one-parameter family of spersonic to sbsonic heteroclinic soltions to the layer problem (.4) connecting E to E + (see Figre 7). This one-parameter family of E E + heteroclinic soltions are bonded by two pairs of heteroclinic soltions. At one end the bondary is a pair of heteroclinic orbits connecting E to E along the line λ = 0 (wet shock) and then connecting E to E + (sbsonic layer). At the other end the bondary is a pair of heteroclinic orbits connecting E to E + (spersonic layer) and then connecting E + to E + along the line λ = (dry shock). 3
(a) z λ = = 0 z E _ E + = p ρ E + E _ λ = 0 _ n < n _ + + _ (b) λ λ = _ E + E + _ (c) λ λ = _ E + E + _ λ = 0 E _ + + E _ λ = 0 E _ + + E _ Figre 7. Figre (a) shows z as a fnction of (, λ) for a fixed n. The arrows in Figre (b) illstrate the signs of z in the (, λ) coordinates. A oneparameter family of layer soltions connecting E to E + are plotted in Figre (c). Those soltions are bonded by layer soltions E E +, E E +, the wet shock E E, and the dry shock E + E +. Proof. Among the -coordinates of the for eqilibrim points, is the minimm and is the maximm. Based on the two-dimensional rectangle R(, ), one can constrct a pentahedron shaped solid W in (, λ, θ)-space as in Definition. Then by Lemma 3., for any point P W, the forward flow throgh P leaves W throgh the srfaces F b, F r, F s, F k. On the other hand the backward flow throgh P can leave W only throgh F f. It is also straightforward to show that the backward layer flow cannot leave W throgh the six edges of W that do not form the bondary of F f. The projection of the -nllsrface onto the (, λ)-plane, the -nllcline, divides the base rectangle R(, ) into three components: the most left one R above the segment of the -nllcline that connects E and E +, the most right one R 3 above the segment of the - nllcline that connects E + and E and the middle one R between the two segments of the -nllcline. Define V (R j ) := {(, λ, θ) : (, λ) R j }, j =,, 3. From Figres 7 (a) and 7 (b), V (R ) is backward invariant with respect to the layer problem. We now prove the existence of a heteroclinic soltion. Note that if a heteroclinic connection exists, it has to be solely in W. The point E = (, 0, 0) is on the spersonic branch of S 0 and is flly repelling. Ths in backward time E is attracting all the points in a neighborhood of E. If the order of the corresponding eigenvales is 0 < l < l < l 3 then the eigenvector v points along one edge of W while the others point not into W (see the definitions of the 4
eigenvectors in (.3)). On the other hand, if 0 < l < l then the eigenvector v points into W as well. Hence the weakest eigendirection of E always points into (or at least along) W. The point E + = ( +,, 0) is on the sbsonic branch of S. Ths W s (E +) is two dimensional and is spanned by the eigenvectors (v, v ). The local stable manifold can be expressed as W s (E +) = {(, λ, θ) θ = θ (, λ), (, λ) < δ 0 }. Consider a semicircle of radis δ in the half plane λ where ϕ is the angle of v measred from the negative -axis: (3.3) v(ϕ) = ((ϕ), λ(ϕ)) = δ(cos ϕ)(, 0) + δ(sin ϕ)(0, ), 0 ϕ π. Then {P (ϕ) = ((ϕ), λ(ϕ), θ(ϕ)) θ(ϕ) = θ ((ϕ), λ(ϕ)), 0 < ϕ < π} describes a smooth arc on the local stable manifold W s (E +). Its projection on the (, λ) plane intersects the -nllcline at 0 < ϕ = ϕ < π/ and the line = + at ϕ = π/. Now let ϕ be the infimm of the angles sch that the backward orbit throgh P (ϕ) will enter V (R ). It is clear that 0 < ϕ < ϕ. For ϕ < ϕ < π, the backward orbit throgh P (ϕ) will enter the region V (R ) which is backward invariant. Since dλ/dz > 0, it follows that λ(z) 0 as z. The invariant set on the srface λ = 0 consists only of E and E. Hence, the backward orbit mst approach E. The forward orbit of corse mst approach E + so we have a layer connection E E +. If ϕ = ϕ, then the backward orbit shold stay above the -nllcline and connect to E. This is a sbsonic to sbsonic layer soltion and has been discssed in [4] by sing the principle of Wazewski. In other words, as ϕ ϕ, the limit of the one-parameter family of E E + layers is the nion of two heteroclinic orbits: E E (wet shock) and E E +. Observe that not all of the sper to sbsonic layers are monotone in. The component of some soltions can go below + then retrns back to +. On the other hand as ϕ π, the limit of the one-parameter family of E E + layers is the niqe pair of two heteroclinic orbits: E E +, which is a spersonic to spersonic connection as stdied in [4], followed by the transonic layer soltion E + to E + (dry shock). Corollary 3.. Assmption of Theorem 3. can be weakened to (H) p < p eq for (, λ) = (, ) Proof. Note that Assmption still holds within the pentahedron W since p λ p(λ, m/) < 0 which follows from p ρ > 0 by Assmption. Complementing to the reslts of Theorem 3., we can show that > 0 and Theorem 3.. There does not exist any sb- to spersonic evaporation layer soltions connecting E S 0 to E + S. Proof. This can be seen from the fact that z < 0 for + < < +. An indirect proof of Theorem 3. is to se a reslt from [4] which shows that the branch of the one-dimensional stable manifold W s (E +) that stays in the pentahedron W will intersect with W (E ) and form a spersonic to spersonic layer soltion. Therefore W s (E +) will not intersect with W (E ). 5
(a) z = p ρ λ = z= 0 E_ E + E_ λ = 0 _ n = n _ + _ (b) (c) λ = λ _ E + _ λ = λ _ E + _ λ = 0 E _ E _ + λ = 0 E _ + E _ Figre 8. Figre (a) shows z as a fnction of (, λ) for a fixed n = n. The arrows in Figre (b) illstrate the signs of z in the (, λ) coordinates. A onedimensional family of layer soltions connecting E to E + are plotted in Figre (c). Those soltions are bonded by layer soltions E E +, E E + and the wet shock E E. 3.. Existence of sbsonic(spersonic) to sonic layer soltions. For n = n, there are three eqilibrim points {E, E +, E } on S 0 S marked by their coordinates < + <. Among them E = (, 0, 0) S 0 is sbsonic, E + = ( +,, 0) S is sonic, and E = (, 0, 0) S 0 is spersonic, F Theorem 3.3. Assme n = n and the Assmptions and 3 are satisfied with ū < < < ū. Then there exists a one-parameter family of non-monotone spersonic to sonic heteroclinic soltions to the layer problem (.4) connecting E to E + (see Figre 8). Every soltion in that family approaches E + by being tangent to the part of the -axis where < +. There is also a niqe sbsonic to sonic layer E E +. Together with the layer soltion E E along the line λ = 0 (wet shock), this pair of heteroclinic soltions is a bondary of the one-parameter family of E E + layer soltions. The other bondary of that one-parameter family of E E + layer soltions is a niqe monotone spersonic to sonic layer soltion E E + that approaches E + by being tangent to the part of W s (E + ) where λ <. Proof. Consider a pentahedron shaped solid W in (, λ, θ)-space based on the rectangle R(, ) as in Definition. Then by Lemma 3., for any point P W, the forward flow throgh P leaves W throgh the srfaces F b, F r, F s, F k. On the other hand the backward flow throgh P can leave W only throgh F f. It is also straightforward to show that the backward layer flow cannot leave W throgh the six edges of W that do not form the bondary of F f. 6
The -nllcline divides the base rectangle R(, ) into three components: the most left one R above the segment of the -nllcline that connects E and E +, the middle one R below the -nllcline, and the most right one R 3 above the segment of the -nllcline that connects E and E +. Note that V (R ) is backward invariant nder the flow of the layer problem. The eqilibrim point E = (, 0, 0) on the spersonic branch of S 0 is an nstable node. The eqilibrim point E = (, 0, 0) on the sbsonic branch of S 0 has a twodimensional nstable manifold W (E ). The corresponding nstable eigenspace is spanned by the eigenvectors {v, v 3 }. Only the eigenvector v corresponding to the weaker nstable eigenvale points into W (we have l < 0 < l < l 3 ); see the definitions of eigenvectors in (.3). This already garantees that part of the (local) manifold W (E ) lies within the pentahedron W. The point E + = ( +,, 0) on the sonic srface of S has a two-dimensional center-stable manifold W cs (E + ). The corresponding center-stable eigenspace is spanned by the eigenvectors {v, v } (we have l < l = 0 < l 3 ); see the definitions of eigenvectors in (.3). The eigenvector v points along an edge of W while the eigenvector v points into W. The signs of the components of v are (+,, +), therefore there is a branch of the stable manifold W s (E + ) that enters the region V (R ) which is invariant with respect to the backward flow. The backward flow of that branch of W s (E + ) will remain in V (R ) and its α-limit set is the point E. This proves the existence of a niqe layer connection E E + that approaches E + exponentially fast and is monotone in. The local center-stable manifold of W cs (E + ) can be expressed as W cs (E + ) = {(, λ, θ) θ = θ (, λ), (, λ) < δ 0 }. Note that W c (E + ) is semistable: it is stable from the side < + and is nstable from the side > +. The center-stable manifold W cs (E + ) is divided by W s (E + ) into two components, call it the stable and nstable parts of W cs (E + ). An orbit on W cs (E + )\W s (E + ) mst follow the stable fibers of the foliation of W cs (E + ) and, hence, approaches W c (E + ) exponentially. On the other hand, the motion of the projection of sch an orbit along the stable fibers on W c (E + ) is only algebraic (nonexponential). If the projection of sch an orbit along the stable fibers is on the stable part of W cs (E + ), then it will approach E + nonexponentially and tangent to the branch of W c (E + ) that is below +. If it is on the nstable part of W cs (E + ), then it will leave E + following the flow of the nstable branch of W c (E + ) where > +. To stdy these two possibilities, define the semi-circle on W cs (E + ) as in (3.3). Then P (ϕ) := {((ϕ), λ(ϕ), θ(ϕ)), 0 ϕ π} is a smooth arc on W cs (E + ). Its projection to the (, λ) plane intersects the sbsonic part of the -nllcline at ϕ = ϕ and the projection of W s (E + ) at ϕ = ϕ 3. Let ϕ be the infimm of all the angles ϕ sch that the backward orbit throgh P (ϕ) will enter the region V (R ). Note that 0 < ϕ < ϕ < π/ < ϕ 3 < π. If ϕ < ϕ < π, then the orbit passing throgh P (ϕ) approaches E in backward time de to the facts the region V (R ) is backward invariant and the (λ, ) components in V (R ) are monotone. Consider the ω-limit set of an orbit throgh P (ϕ). If ϕ < ϕ 3 then the ω-limit set of the orbit throgh P (ϕ) is E +, while if ϕ > ϕ 3, the ω-limit set does not exist since the orbit leaves W throgh the side F k and is nbonded. This shows if ϕ < ϕ < ϕ 3, the one-dimensional family of spersonic to sonic connection E E + exists. Each orbit in that family is not monotone in and does not approach E + exponentially. As ϕ ϕ 3, the 7
θ λ v 3 W (E ) v E + v W cs (E + ) v. E v E Figre 9. The niqe sbsonic to sonic heteroclinic connection E E + corresponds to a transverse intersection of the manifolds W cs (E + ) and W (E ) within the pentahedron W. limit of the one-parameter family of spersonic to sbsonic layer is the niqe E E + connection that passes throgh P (ϕ 3 ). It is monotone in and approaches E + exponentially along the stable manifold W s (E + ). We now show that the orbit throgh P (ϕ ) is a connection E E +. For any κ > 0, there exist ϕ ± sch that ϕ ± ϕ < κ, ϕ < ϕ < ϕ + and the backward orbit throgh P (ϕ ) leaves V (R ) throgh the srface F f and the backward orbit throgh P (ϕ + ) leaves V (R ) throgh the srface whose projection to the (, λ) plane is the -nllcline to V (R ). By the principle of Wazewski, there exists a ϕ 0 (ϕ, ϕ + ) sch that the orbit throgh P (ϕ 0 ) remains in V (R ) in backwards time. Hence it approaches E in backward time. Since ϕ 0 ϕ can be arbitrarily small, the orbit throgh P (ϕ ) shold also approach E in backward time. The ω-limit point of this orbit is E + de to the fact 0 < ϕ < ϕ 3. We have shown that the one-parameter family of spersonic to sonic layer E E + has a limit as ϕ ϕ. The limit consists of a pair of heteroclinic orbits, E E (wet shock) followed by a sbsonic to sonic layer soltion E E +. Finally, note that dim(w (E ) W cs (E + )) =, i.e. W cs (E + ) has a fll intersection with W (E ). Frthermore, dim(w (E ) W s (E )) =, i.e. W s (E ) has a fll intersection with W (E ). Since we showed that the pair of heteroclinic orbits, E E and E E +, is part of the bondary of the two-dimensional manifold W (E ) W cs (E + ), it follows that the tangent space of the two-dimensional manifold W cs (E + ) is locally spanned near E by its stable eigenvector v and the weak stable eigenvector v. Observe that the strong nstable eigenvector v 3 is transverse to the tangent space spanned by {v, v }. Ths T W (E ) + T W cs (E + ) is 3-dimensional near E. This shows that the intersection of these manifolds is transverse and the sbsonic to sonic layer E E + is niqe (see Figre 9). Similar to Corollary 3., we have the following Corollary 3.. The Assmption of Theorem 3.3 can be weakened to (H) p < p eq for (, λ) = (, ). 8
4. Analysis of the redced (oter) problem on slow manifolds The phase space of the redced system (.5) is the critical manifold S (.8). We only focs on the redced flow on S 0 and S since S e does not play a role in or analysis of evaporation waves (Assmption ). Proposition 4.. The redced vector field on S 0 and S points away from the sonic fold srface = p ρ. Proof. Since S 0 and S are given as graphs over (r, m, )-space, we stdy the redced flow in this single (r, m, )-chart. Hence, we differentiate n = N(m, ) with respect to the slow scale y to obtain the redced system (.5) in the (r, m, )-chart (4.) (4.) r y = m y = m r N y = m r + N m m r. Since n = N(m, ) = m p(λ(m, ), m ), we have N = m p λ Λ p ρ ρ and N m = p λ Λ m p ρ ρ m. The derivatives of the fnction Λ evalated on S 0 respectively S are Λ m = Λ = 0. Using ρ = m/ to evalate ρ and ρ m, we obtain the redced system on S 0 respectively S : r y = (4.3) m y = m r ( p ρ ) y = r p ρ. Note that this system is singlar along the sonic srface defined by = p ρ. We desinglarize the system by the rescaling dy = ( p ρ )dȳ which gives the desinglarised system rȳ = ( p ρ ) (4.4) mȳ = m r ( p ρ ) ȳ = r p ρ > 0. The phase portraits of the redced and desinglarised system are eqivalent p to a change of orientation in the sbsonic domain, i.e. < p ρ. Hence y < 0 in the sbsonic domain and y > 0 in the spersonic domain which implies that the redced flow moves away from the sonic fold srface = p ρ on S 0 respectively S. Note, this follows becase the sonic srface is given as a graph = U(m) for m > 0 independent of r (see also Proposition 5.). Since we assme > 0, this also implies that there exists no ordinary or folded singlarities on S 0 or S. Remark 4.. Observe that S 0 and S represent manifolds of eqilibria for the layer problem (.4). The part of S 0 and S that satisfies p ρ consists of hyperbolic eqilibrim points of the first three eqations of the layer problem. Ths, on S 0 or S, if p ρ, can be solved 9
as a fnction of (m, n, r) and it has two smooth branches. One is sbsonic and the other is spersonic according to the signs of p ρ. If we se d = p ρ to measre the distance to the sonic srface, it is shown in [4] that when restricted to the sper or sbsonic branch of S 0 S, (d ) y > 0 with respect to the redced problem (.5). Therefore, a soltion staying on the critical manifold S 0 (or S ) will not correspond to a transonic wave. 5. Existence of transonic evaporation waves for 0 < ε From Proposition 4. it follows that a transonic evaporation wave mst have an internal layer at some r = r i (r 0, r ) as shown in Figre. To form a singlar standing wave profile Γ for r [r 0, r ] sch an internal layer soltion γ i has to be concatenated with two (oter) soltions from the redced problem, one on the sbsonic (spersonic) branch of S 0 for r [r 0, r i ] denoted by γ 0 and the other on the spersonic (sbsonic) branch of S for r [r i, r ] denoted by γ. In the following, we show the existence of (non-degenerate) transonic evaporation waves for 0 < ε that exist near a singlar (ε = 0) wave profile Γ = γ 0 γ i γ (see Figre 3) in the spersonic to sbsonic case (section 5.) and the sbsonic to spersonic case (section 5.). The sb- and spersonic branches of S 0 (respectively S ) away from the sonic srface are normally hyperbolic, with three zero eigenvales whose eigenspace is tangent to S 0 (respectively S ). Fenichel s theory [6] implies that these normally hyperbolic manifolds pertrb smoothly to O(ε)-close slow manifolds S 0 (ϵ) (respectively S (ϵ)), and the slow flow on these manifolds is an O(ε) smooth pertrbation of the corresponding redced flow. In the same manner, the end points P 0 and P of Γ pertrb smoothly to P 0 (ϵ) and P (ϵ), respectively. Remark 5.. It is well known that the slow manifolds S 0 (ϵ) respectively S (ϵ) are not niqe bt they represent a family of slow manifolds that lie within O(exp( K/ε)) distance from each other for some K > 0. We make an arbitrary choice of S 0 (ϵ) respectively S (ϵ) and show that the reslts obtained in this section are independent of sch choice. 5.. Spersonic to sbsonic evaporation waves. Assme that an internal layer soltion γ i from Theorem 3. is given which connects the three-dimensional manifold W (E ) on the spersonic region of S 0, denoted by S sper 0, to the two-dimensional manifold W s (E+) on the sbsonic region of S, denoted by S sb (see Figre 7). To constrct the concatenated orbit Γ, we mst find γ as a forward orbit from E+ and γ 0 as a backward orbit from E. The existence of sch a soltion γ that stays on the sbsonic region S sb is garanteed since the redced flow of (4.3) moves away from the sonic fold. We assme that γ forward connects E+ to P at r = r. On the other hand, the backward soltion γ 0 on S sper 0 with initial condition given by E might reach the sonic fold for some r (r 0, r i ). To avoid this, the following assmption will be sed: Assmption 4. The distance r i r 0 is sfficiently small sch that the backward soltion γ 0 of the redced problem (4.3) starting at r = r i with initial condition E S sper 0 given by the internal layer γ i stays in the spersonic region S sper 0 for all r [r 0, r i ]. By Assmption 4, we assme γ 0 backward connects E to P 0 at r = r 0 following the redced flow on S sper 0. Theorem 5.. Under Assmptions - 4, let Γ = γ 0 γ i γ denote a singlar evaporation wave in the phase space of system (.3) which is a concatenation of an internal layer soltion 0
γ i of (.4) at r = r i connecting the spersonic branch S sper 0 with the sbsonic branch S sb, an oter soltion γ 0 of the redced problem (.5) on the spersonic branch S sper 0 for r [r 0, r i ] and another oter soltion γ of the redced problem (.5) on the sbsonic branch S sb for r [r i, r ]. If ε 0 > 0 is sfficiently small, then for all ε (0, ε 0 ) there exists a two-dimensional family of standing wave soltions Γ(ϵ) of system (.3) lying within O(ϵ) of Γ. Proof. The normally hyperbolic spersonic branch S sper 0 has an associated local nstable manifold W (S sper 0 ) = p 0 S sper W (p 0 ) (nstable layer fibration) and the normally hyperbolic sbsonic branch S sb has an associated local stable manifold W s (S sb ) = p S 0 W s (p sb ) (stable layer fibration). Fenichel s theory [6] implies that these local nstable and stable manifolds (fibrations) W (S sper 0 ) and W s (S sb ) pertrb smoothly to O(ε)-close local nstable and stable manifolds (fibrations) W (S sper 0 (ε)) and W s (S sb (ε)). Observe that the two projections of the oter soltions γ j, j = 0,, of Γ onto the slow variable space (m, n, r) intersect at the common n(r i ), which is the n-coordinate for E and E+. From the third eqation of (.5) it is easily proved that the two (projected) soltions intersect transversely. Ths for 0 < ε, the two (projected) soltions intersect transversely O(ε) nearby, i.e. at the common n-coordinate n ε = n(ri ϵ = r i + O(ε)) and we denote the corresponding common points by E (ϵ) and E+(ϵ). The oter trajectory γ 0 S sper 0, connecting E to P 0, pertrb O(ε) smoothly to the trajectory γ 0 (ε) S sper 0 (ε), connecting E (ε) to P 0 (ε), while γ S sb, connecting E+ to P, pertrb O(ε) smoothly to the trajectory γ (ε) S sb (ε), connecting E+(ε) to P (ε). In Theorem 3. we have shown that there exists a spersonic to sbsonic evaporation layer soltion γ i connecting base points E S sper 0 and E+ S sb. Since this layer fiber intersection of W (E ) W (S sper 0 ) and W s (E+) W s (S sb ) is transverse and the projection of γ 0 (ε) and γ (ε) onto the slow variable space is transverse at the common base points E (ϵ) and E+(ϵ), this fiber intersection will persist for 0 < ϵ, i.e. the nstable fibers W (E (ϵ)) transversely intersect with the stable fibers W s (E+(ϵ)). Ths W (E (ε)) W s (E+(ε)) is a two-dimensional sbmanifold. For each point ζ on the two-dimensional intersection srface W (E (ε)) W s (E+(ε)), we constrct an orbit Γ(ε) which exponentially approaches S sper 0 (ε) in backward time along its nstable fibers and exponentially approaches S sb (ε) in forward time along its stable fibers. Since the nstable fibers of S0 sb (ε) are backward invariant with respect to the flow, Γ(ε) is on the nstable fibers of P 0 (ε) at r = r 0. Also the stable fibers are forward invariant to the flow. Ths Γ(ε) is on the stable fibers of P (ε) at r = r. Let the soltion that passes throgh ζ at r = ri ϵ be denoted by q(r, ζ, ε). Then q(r 0, ζ, ε) = (m, n,, λ, θ)(r 0 ) is exponentially close to P 0 (ϵ), and is in the spersonic region with λ(r 0 ) 0. Also q(r, ζ, ε) = (m, n,, λ, θ)(r ) is exponentially close to P (ε), and is in the sbsonic region with λ(r ). Ths q(r, ζ, ε), whose orbit is Γ(ε), is a sper to sbsonic evaporation wave. Sch waves form a twodimensional family determined by the choice of ζ. As mentioned in Remark 5., the soltion Γ(ϵ) is niqe once the choices S sper 0 (ϵ) and S sb (ϵ) have been made, and the differences de to choices are only exponentially small. 5... Bondary conditions that determine a niqe sper to sbsonic wave. Theorem 5. proved the existence of a two-dimensional family of sper to sbsonic evaporation waves. Note, since y = r, and z = r/ε, we write soltions as fnctions of r.
B λ = 0 λ= B 0 0 B q(r ) P ( ε) P ( ) q(r 0 0 ) ε i B Figre 0. Only the fast variables are plotted. The forward image of the manifold B 0 is B 0 while the backward image of the manifold B is B. At r = r ε i, these manifolds intersect transversely. It does not address the niqeness of these sper to sbsonic evaporation waves. In the following we prescribe bondary conditions at r = r 0 and r = r to obtain a niqe transonic evaporation wave. The bondary condition shall be given by two bondary manifolds B j, j = 0, : (r, m, n,, λ, θ)(r j ) B j, j = 0,. For simplicity, we assme that B 0 and B are linear affine planes that are defined by splitting the conditions on slow and fast variables. First, we define the following slow sbmanifolds B s j, j = 0, : Assmption 5. For the slow variables (r, m, n) B s j, either n(r 0 ) = n 0 or n(r ) = n is given. Also, either m(r 0 ) = m 0 or m(r ) = m is given. The conditions given in Assmption 5 provide for choices of ptting constraints on the slow part Bj s of the bondary manifolds B j, j = 0,. There is an obvios constraint on the variable r, that is, at y = r 0, r(r 0 ) = r 0. Using, dr/dy =, at the other end of the bondary we will have r(r ) = r. All together we have introdced three conditions on the slow variables (m, n, r). For ϵ = 0, we assme an internal layer at r = r i. Ths for a given layer position r = r i and for bondary conditions on (m, n) given by Assmption 5, the end points {P 0, P, E, E+} of the segments of the singlar orbit Γ = γ 0 γ i γ defined in Theorem 5. are niqely determined. For the fast variables (, λ, θ), we define the following fast sbmanifolds B f j, j = 0, : Assmption 6. The manifold B f 0 is one-dimensional and is transverse to the the weakest two nstable eigenspaces of P 0, spanned by {v, v }. The manifold B f is two-dimensional and is transverse to the local nstable manifold of P. Definition 3 (Bondary Manifolds, sper to sb). Under Assmptions 5-6, let B 0 = B s 0 B f 0 and B = B s B f, be the prodct of slow and fast planes. The bondary manifolds are given by B 0 = B 0 + q(r 0, ζ, ε), ζ W (E (ε)) W s (E +(ε)), and B = B + Q where Q is any point near P.
Remark 5.. In a typical linear PDE, the bondary conditions are given first then the existence of soltions are proved. If we assign bondary conditions withot any pre-knowledge of the soltions of the nonlinear problem, the soltion of the BVP may not exist. So or bondary conditions are based on the knowledge of the singlar limit soltion of the transonic waves. Note, B 0 depends on the prescribed ϵ while B is independent of ϵ. If B happens to pass throgh q(r, ζ, ϵ) then q(r, ζ, ϵ) obviosly satisfies the bondary conditions defined by B 0 and B (and is the niqe sch soltion). However, we do not want to define the bondary manifolds to pass the end points of the standing wave, then claim that there exists a soltion. We want the bondary condition to have some room for error, and the soltion shold be robst with respect to pertrbations of the bondary conditions. From the application point of view, it is important that one can maintain physical conditions at both ends of the nozzle with some error and still have a soltion. This explains why we assme that Q is close to bt not eqal to P. Recall that for the spersonic branch S sper 0, the eigenvales for the fast system with (m, n, r) S 0 satisfy the condition 0 < l, 0 < l < l 3. The following assmption shall be sed in the next theorem. Assmption 7. For the singlar wave γ 0 on S sper 0 defined from r = r 0 to r = r i, the eigenvales for the fast system where (m, n, r) γ 0 satisfy l < l 3, i.e., l 3 is the strongest nstable eigenvale. Theorem 5.. Let q(r, ζ, ϵ) be a family of evaporation waves that passes the initial vale ζ W (E (ε)) W s (E +(ε)) at r = r ϵ i = r i + O(ϵ). Assme that the family of soltions are tangent to the two weaker nstable fibers based at P 0 (ϵ) and that Assmption 7 is satisfied. Then the bondary manifolds B 0 and B given by Definition 3 determine a niqe sper to sbsonic evaporation wave. Proof. The standing wave that satisfies the bondary conditions (Definition 3) can be expressed as (5.) U(r, ϵ) := Φ(r r 0, ε)b 0 Φ(r r, ε)b, r 0 r r. It sffices to show that for one particlar r, the intersection defined by the right hand side of (5.) is a niqe nonempty point. To this end, we shall se Fenichel theory on the persistence of hyperbolic part of the slow manifolds and their fibrations (or foliations). Fenichel theory implies for sfficiently small 0 < ε that B 0 transversely intersects the fast sbspace spanned by the two weakest nstable fiber directions of P 0 (ϵ) as well as B transversely intersects the nstable fiber of P (ϵ). First, we focs on the intersection of the slow components of B 0 and B projected onto the slow variable space. Under Assmption 5, the singlar limit segments γ j, j = 0,, of Γ are niqely determined. As shown in the proof of Theorem 5., the two projections of γ j, j = 0,, onto the slow variable space (m, n, r) intersect transversely at the common n(r i ). Ths for 0 < ε, the two (projected) soltions intersect transversely O(ε) nearby, i.e. at the common n-coordinate n ε = n(r ε i ) and, as before, we denote the corresponding common points by E (ϵ) and E +(ϵ). Next, we shall se the theorem of graph transformation to complete the proof. When the flow is near a saddle eqilibrim point, sch theorem is called the Lambda Lemma, or 3