{'fraction_non_alphanumeric': 0.07189264444272565, 'fraction_numerical': 0.03480547606156702, 'mean_word_length': 3.6143825838686654, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 11, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Viscous singular shock profiles for a system of conservation laws modeling two-phase flow

This paper is concerned with singular shocks for a system of conservation laws modeling incompressible two-phase fluid flow. We prove the existence of viscous profiles using the Geometric Singular Perturbation Theory. Weak convergence and growth rates of the unbounded family of solutions are also obtained.


Introduction
Keyfitz et al [KSS03,KSZ04] considered the system of conservation laws For Riemann problems with data in feasible regions, they constructed uniquely defined admissible solutions. It can be readily shown that this system is not everywhere hyperbolic, and hence standard methods does not apply (see e.g. [Smo83,Daf10]). To resolve this problem, along with rarefaction waves and regular shocks, the concept of singular shocks was adopted. A singular shock solution, roughly speaking, is a distribution which contains delta measures and is the weak limit of a sequence of approximate viscous solutions. For details of the definition, we refer to [Sev07,Key11]. The existence of singular shocks for (1.1) was proved in [KSZ04]. In that work, for certain Riemann data (1.3) (β, v)(x, 0) = (β L , v L ), x < 0 (β R , v R ), x > 0 approximate solutions of the regularized system via Dafermos regularization (1.4 ) β t + (vB 1 (β)) x = tβ xx v t + (v 2 B 2 (β)) x = tv xx were constructed. A family of exact solutions of (1.4 ) and (1.3), rather than approximate solutions, is called a viscous profile. In this paper, we prove existence of viscous profile, also we give descriptions of their limiting behavior including weak convergence and growth rates. The main tool in our study is the Geometric Singular Perturbation Theory (GSPT), which will be introduced in later sections. The use of this tool on singular shocks was first introduced in the pioneering work of Schecter [Sch04].
The system (1.1) is equivalent to a two-fluid model for incompressible two-phase flow [DP99,p.248] of the form where the drag terms F i are neglected and the pressure terms satisfy p 1 = p 2 . To reduce (1.5) to (1.1), in [KSS03] the volume fractions α 1 and α 2 = 1 − α 1 have been replaced by a density-weighted volume element β = ρ 2 α 1 + ρ 1 α 2 and the momentum equations replaced by a single equation for the momentum difference v = ρ 1 u 1 − ρ 2 u 2 − (ρ 1 − ρ 2 )K, where K = α 2 u 1 + α 2 u 2 is taken to be zero. This is a simple example of continuous model for two-phase flow, but it shares with other continuous models the property of changing type -that is, it is not hyperbolic for some (in this case, most) states. The purpose of this study is to shed light on the mathematical properties of the change-of-type system that appear in continuous models of two-phase flow. The original studies [KSS03,KSZ04] showed the existence of self-similar solutions with reasonable properties. Specifically, the singular shocks that appear can be considered to be propagating phase boundaries. In this paper, we focus on viscous profiles of singular shocks and unveil some of their limiting behavior.
In Section 2, we state our main result, and in Section 3 the validity of the assumptions of the theorem is discussed, with some proofs for the sufficient conditions postponed to Section 8. In Section 4, we recall and enhance some tools in GSPT, including Fenichel's Theorems and the Exchange Lemma. Section 5 is devoted to describing the structure of the system. The proof of the main theorem is completed in Section 6, and numerical simulations are shown in Section 7.

Main Result
In standard notation for conservation laws, we write (1.1) as where u = (β, v), and write Riemann data for Riemann problems in the form where H(x) is the step function taking value 0 if x < 0; 1 if x > 0.
Then there is a singular shock with Dafermos profile for the Riemann data (u L , u R ). That is, for each small > 0, there is a solutionũ (ξ) of (2.9 ) and (2.10), andũ (ξ) becomes unbounded as → 0. Indeed, Moreover, if we set u (x, t) =ũ (x/t), then u (x, t) is a solution of (2.8 ) and in the sense of distributions as → 0.
The trajectories in (H3) are illustrated in Fig 1. Remark 1. A similar result holds if v L < 0 and v R < 0. In that case, the assumption e 0 > 0 in (H2) is replaced by e 0 < 0, and +∞ in (H3) is replaced by −∞.   The notation tδ {x=st} in (2.21b) denotes, following [TZZ94,CL03], the functional on C ∞ c (R × R + ) defined by The weight √ 1 + s 2 in the integral is to normalize the functional so that it is independent of parametrization of the line {x = st}.
The estimate (2.20) confirms the asymptotic behavior conjectured in [KSS03]. In [KSZ04], some approximate solutions for the Dafermos regularization were constructed, but they were not exact solutions to (2.9 ). The results in the main theorem can also be compared to [Sch04] and [KT12], where Dafermos profiles were constructed for a system motivated by gas dynamics. Those authors obtained families of unbounded solutions to (2.9 ), but they did not give descriptions of asymptotic behaviors the of solutions.
This will be verified in the next subsection.
Proposition 3.2 says that (H2) holds whenever (H1) holds, so the Main Theorem requires only (H1) and (H3). The author believes that (H3) is also a consequence of (H1). This needs further work to be verified.
For the sample set of data (2.26), we have Since w 2R < w 2L , (H2) holds. From Proposition 3.1 and 3.3, (H1) and (H3) also hold. Hence the main theorem applies. Note that the conditions (H1)-(H3) persist under perturbation of the Riemann data (u L , u R ), so those assumptions still hold for any data close to (2.26).

Geometric Singular Perturbation Theory
Our main goal is to solve the boundary value problem (2.9 ) and (2.10). Note that (2.9 ) is a singularly perturbed equation since the perturbation d 2 dξ 2 u has a higher order derivative than the other terms in the equation. To deal with singularly perturbed equations, we will apply Geometric Singular Perturbation Theory (GSPT). The idea of GSPT is to first study a set of subsystems which forms a decomposition of a system, and then to use the information for the subsystems to conclude results for the original system. Prototypical examples include relaxation oscillations for forced Van der Pol Equations [DR96, KS01a,KS01b] and FitzHugh-Nagumo Equations [JKL91, KSS03,LVV06]. Surveys on this topic can be found in [Jon95,Kap99,KJ01,RT02].
In Section 4.1 and 4.2, we recall some fundamental theorems in GSPT. In Section 4.3 we state and give new proofs for a version of the Exchange Lemma. 4.1. Fenichel's Theory for Fast-Slow Systems. Note that (2.12 ) is a fast-slow system, which means that the system is of the form where (x, y) ∈ R n × R l , and is a parameter. In order to deal with fast-slow systems, Fenichel's Theory was developed in [Fen74,Fen77,Fen79]. Some expositions for that theory can be found in [Wig94,Jon95]. An important feature of a fast-slow system is that the system can be decomposed into two subsystems: the limiting fast system and the limiting slow system. The limiting fast system is obtained by taking = 0 in (4.30 ); that is, On the other hand, note that the system (4.30 ) can be converted to, after a rescaling of time, Note that the limiting slow system (4.33) describes dynamics on the set of critical points of the limiting fast system (4.31), so we will need to piece together the information of the limiting fast system and the limiting slow system in the vicinity of the set of critical points. To piece this information together, normal hyperbolicity defined below will be a crucial condition.
Definition 1. A critical manifold S 0 for (4.31) is an l-dimensional manifold consisting of critical points of (4.31). A critical manifold is normally hyperbolic if D x f (x, y, 0)| S0 is hyperbolic. That is, at any point (x 0 , y 0 ) ∈ S 0 , all eigenvalues of D x f (x, y, 0)| (x0,y0) have nonzero real part.
Now we turn to discussing normal hyperbolicity for general systems where z ∈ R N , N ≥ 1. A manifold S ⊂ R N is locally invariant if for any point p ∈ S \ ∂S, there exist t 1 < 0 < t 2 such that p · (t 1 , t 2 ) ∈ S, where · denotes the flow for (4.34). In the vicinity of a locally invariant manifold, under certain conditions the system can be decomposed into lower-dimensional subsystems. For instance, when S = {p 0 } is an isolated hyperbolic equilibrium for (4.34), the stable and unstable manifolds W s (p 0 ) and W u (p 0 ) exist according to the Hartman-Grobman Theorem [Har64], and the union of their tangent spaces at p 0 spans R N .
A locally invariant C r manifold S ⊂ R N , r ≥ 1, is normally hyperbolic for the system (4.34) if the growth rate of vectors transverse to the manifold dominates the growth rate of vectors tangent to the manifold. (Note that this is consistent with Definition 1.) In this case, from the standard theory for normally hyperbolic manifolds (see, for example, [HPS77,VvG87,CL88]) it is assured that stable and unstable manifolds W s (S) and W u (S) are defined.
For a locally invariant manifold Λ ⊂ R N for (4.34) which is not necessarily normally hyperbolic, a center manifold is a normally hyperbolic locally invariant manifold, with the smallest possible dimension, containing Λ. In classical cases, Λ = {p 0 } is an isolated non-hyperbolic equilibrium, and a center manifold for p 0 has dimension equal to the number of generalized eigenvalues of DF (p 0 ) with zero real part. For instance, the planar systemẋ = x 3 ,ẏ = y, has a non-hyperbolic isolated equilibrium p 0 = (0, 0), and the x-axis is a center manifold for p 0 . For general invariant sets Λ, we refer to [CLY00a,CLY00b]. Fenichel's Theory is a center manifold theory for fast-slow systems. For a normally hyperbolic critical manifold S 0 for (4.31), the stable and unstable manifolds W s (S 0 ) and W u (S 0 ) can be defined in the natural way. We denote them by W s 0 (S 0 ) and W u 0 (S 0 ) to indicate their invariance under (4.30 ) with = 0. Fenichel's Theory assures that the hyperbolic structure of S 0 persists under perturbation (4.30 ). Below we state three fundamental theorems of Fenichel's Theory following [Jon95].
Theorem 4.1 (Fenichel's Theorem 1). Consider the system (4.30 ), where (x, y) ∈ R n × R l , and f , g are C r for some r ≥ 2. Let S 0 be a compact normally hyperbolic manifold for (4.31). Then for any small ≥ 0 there exist locally invariant C r manifolds, denoted by S , W s (S ) and W u (S ), which are C 1 O( )-close to S 0 , W s 0 (S 0 ) and W u 0 (S 0 ), respectively. Moreover, for any continuous families of compact sets where · denotes the flow for (4.30 ).

Proof. See [Jon95, Theorem 3].
Remark 2. If S 0 is locally invariant under (4.30 ) for each , then the S can be chosen to be S 0 because of the construction in the proof of [Jon95, Theorem 3].
Note that W u (S ) and W s (S ) can be interpreted as a decomposition in a neighborhood of S 0 in (x, y)space. The following theorem asserts that this induces a change of coordinates (a, b, c) such that W u (S ) and W s (S ) correspond to (a, c)-space and (b, c)-space, respectively.
Theorem 4.2 (Fenichel's Theorem 2). Suppose the assumptions in Theorem 4.1 hold. Then under a C r -dependent coordinate change (x, y) → (a, b, c), the system (4.30 ) can be brought to the form in a neighborhood of S , where the coefficients are C r−2 functions satisfying for some ν > 0 and The family of trajectories for (4.33) forms a foliation of S 0 . The following theorem says that this induces a foliation of W u (S ) and W s (S ).
constants C and ν such that (4.35) holds with S replaced by Λ . Suppose in addition that S 0 is invariant under (4.30 ) for each . Then Λ can be chosen to be Λ 0 .
Proof. Using Fenichel's coordinates (a, b, c) in Theorem 4.2 for the splitting of S 0 , we can take W u (Λ ) and W s (Λ ) to be the pre-images of the sets for each , then from the remark after Theorem 4.1, we can take S = S 0 and hence Λ = Λ 0 The system (4.36 ) is called a Fenichel normal form for (4.30 ), and the variables (a, b, c) are called Fenichel coordinates.

4.2.
Silnikov Boundary Value Problem. We have seen in Section 4.1 that fast-slow systems (4.30 ) can locally be converted into normal forms (4.36 ), where A u and A s satisfy the gap condition (4.37), and E is a small term satisfying (4.38). If we append the system with the equation˙ = 0 and then replace c bỹ c = (c, ), we obtain a system of the form for which (4.37) and (4.38) are satisfied with E replaced byẼ. For convenience, we will drop the tilde notation in (4.39) in the remaining discussion.
A Silnikov problem is the system (4.39) along with boundary data of the form where T ≥ 0. This boundary value problem was posed in [Sil67] to study homoclinic bifurcation. A heuristic reason for the existence of solutions of a Silnikov problem is illustrated in Fig 3. Consider the simple casė a = a,ḃ = −b andċ = 0. There are infinitely many trajectories contained in the box We may parametrize the set of trajectories in T ≥ 0 by b(0) = b 0 and a(T ) = a 1 . On the a-axis and b-axis, the trajectories tend to the origin in backward and forward time, respectively. This suggests that trajectories near the axes can stay for an arbitrarily long time in the box, which implies that for any large T there exists a trajectory satisfying b(0) = b 0 and a(T ) = a 1 . When T grows to infinity, the trajectories approach the axes. In the general caseȧ = A u a andḃ = A s b in arbitrary dimension, both a-and b-spaces consist of solutions tending to the origin in forward or backward time, so we have the same conclusion. The critical manifold for (4.39) is {a = 0, b = 0}, on which the system is governed by the limiting slow system For a solution (a(t), b(t), c(t)) to the Silnikov boundary value problem (4.39) and (4.40), from conditions (4.37) and (4.38), it is natural to expect that a(t) and b(t) decay to 0 in backward time and forward time, respectively, and that c(t) is approximately the solution of (4.41). A theorem from [Sch08b] asserts that this is the case: Theorem 4.4 (Generalized Deng's Lemma [Sch08b]). Consider the system (4.39) satisfying (4.37) and (4.38) with C r coefficients, r ≥ 1, defined on the closure of a bounded open set is the solution of (4.41) with initial value c 0 . Let ν > 0 be the number in (4.37). Suppose there exists β > 0 such thatν := ν − rβ > 0 and Moreover, there is a number K > 0 such that for all (t, T, a 1 , b 0 , c 0 ) as above and for all multi-indices i with |i| ≤ r, Sketch of Proof. Here we sketch the proof in [Sch08a]. Write (4.39) aṡ is a solution of Silnikov problem (4.39) and (4.40) if and only if c(t) = φ(t, c 0 ) + z(t) and η(t) = (a(t), b(t), z(t)) satisfies (4.43) Define an linear operator L by the right-hand side of (4.43) for functions η(t) = (a(t), b(t), z(t)). It can be shown that the restriction of L on a neighborhood of 0 in the space of functions η(t) = (a(t), b(t), z(t)) equipped with the norm is a contraction mapping. Hence the existence of solution of (4.39) and (4.40) follows from the standard Banach fixed point theorem.
Remark 3. Theorem 4.4 is a generalization of the Strong λ-Lemma in Deng [Den90], and C r -Inclination Theorem in Brunovsky [Bru99]. In Deng's work, the boundary data lie near an equilibrium that may nonhyperbolic. In Brunovsky's work, the boundary data lie near a solution of a rectifiable slow flow on a ] to study singularly perturbed systems near a normally hyperbolic, locally invariant manifold. Some generalizations of the Exchange Lemma for a broader class of systems were given by W. Liu [Liu00] and Schecter [Sch08b]. Another generalization, given by Tin [Tin94], is the (k + σ)-Exchange Lemma, 1 ≤ σ ≤ l, which tracks the (k + σ)-manifold I * which evolves from a (k + σ − 1)-manifold I = {b = b 0 , c 0 ∈ Λ}, where Λ is a (σ − 1)-manifold. A major difference between the (k + σ)-Exchange Lemma and the general Exchange Lemma in [Sch08b] is that the estimates (4.42) for the derivatives in slow variables were not considered in [Sch08b].
We analyze Silnikov problems for fast-slow systems in normal form (4.36 ) in Lemma 4.1, and then, in Theorem 4.5, return to (4.30 ) to present a version of the (k + σ)-Exchange Lemma.
for some positive constantsC andν. See Fig 4. Sketch of Proof. Existence of solutions follows directly from Theorem 4.4, so it remains to prove (4.46). Write p = (a in , b 0 , c 0 ) and q = (a 1 ,b ,ĉ ), then (4.46) is equivalent to The estimate of the derivatives in (a 1 , c 0 ) ∈ A × Λ in (4.47) follows directly from (4.42). To prove the estimate of the derivatives in τ ∈ J, note that from (4.43) we have (4.48) As in the proof of Theorem 4.4, it can be shown that the derivatives of the integrands in (4.48) are exponentially small, so we obtain (4.47).
The following theorem is a modification of the (k + σ)-Exchange Lemma. The main difference is that in this version we assert the existence of certain trajectories, while in the original version those trajectories were assumed to exist. The proof of the original theorem [Tin94] is based on tracking tangent spaces to an invariant manifold using linearized differential equations in terms of differential forms, while the approach we present below relies on estimates for solution operators, following closely to the proof of the general Exchange Lemma in [Sch08a].

Singular Configuration
The fast-slow system (2.12 ) has multiple limiting subsystems corresponding to different time scales. In this section we will find trajectories, called singular trajectories, for those subsystems such that the union of those trajectories joins the end states u L and u R . The union of those singular trajectories is called a singular configuration. In later sections we will show that there are solutions of (2.12 ) close to the singular configuration.
5.1. End States U L and U R . The system (2.15) has a normally hyperbolic critical manifold where λ ± (u) are the eigenvalues of Df (u), defined in (2.16). The limiting slow system for (2.14 ) is From (H1) we have s < Re(λ ± (u L )), so (u L , w L , s) ∈ S 0 . Choose δ > 0 so that s + 2δ < Re(λ ± (u L )), and set (5.60) denotes the flow for (5.59). It is clear that U L ⊂ S 0 is normally hyperbolic with respect to (2.15), and is locally invariant with respect to (2.12 ). Note that each point in U L is a hyperbolic equilibrium for the 2-dimensional system (2.13), and the unstable manifold W u 0 (U L ) is naturally defined. Proof. This follows from Theorem 4.3 by taking U L to be U 0 . Although U L is not compact, it is uniformly normally hyperbolic since ξ − Re(λ ± (u L )) < −δ on U L , and the proof of Theorem 4.3 in [Jon95, Theorem 4] is still valid.
From (H1) we also have, by decreasing δ if necessary, s − 2δ > Re(λ ± (u R )), and hence a similar result holds for the set U R defined by (5.62) Proposition 5.2. Assume (H1). Let U R be defined by (5.62). Fix any k ≥ 1. There exists a family of invariant manifolds W s (U R ) which are C k O( )-close to W s 0 (U R ) such that for any continuous family {J } ∈[0, 0 ] of compact sets J ⊂ W s (U R ), for some positive constants C and µ.

5.2.
Intermediate States P L and P R . Consider the system (2.14 ). In order to study the dynamics at {v = +∞}, we set r = 1/v and κ = log(1/r). Then (2.14 ) is converted, after multiplying the equations by r, to Note that the time variable in (5.64 ) is different from that of (2.14 ). We use the same dot symbol to denote derivatives, but there should be no ambiguity since the different time scales can be distinguished by comparing the termξ. The limiting fast system for (5.64 ) is The obvious equilibria for (5.65), besides (β L , r L , w 1L , w 2L , s) and (β R , r R , w 1L , w 2L , s), where r L = 1/v L and r R = 1/v R , are The limiting slow system on P L is and on P R is The Fenichel coordinates near P L can be described as follows.
Proof. The linearization of (5.65) at P L corresponds to the matrix which has one positive and one negative eigenvalue. Note that P L is invariant under (5.64 ) for each . From Theorem 4.1 and the remark following it, W s (P L ) and W u (P L ) are well defined and both have dimension 1, and we may take W u (P L ) = {r = 0}. Note that {β = ρ 1 } is transverse to W s (P L ), so we can choose Fenichel coordinates (a, b, c) corresponding to this splitting with b = r and a = β − ρ 1 + φ(w 1 , w 2 , ξ, κ, , r)r for some C k function φ. Letβ = a + ρ 1 . Then the desired result follows.
An analogous result holds for P R . We omit it here.
Let γ 0 be the trajectory defined in Proposition 5.4. We set Then (5.92) π u P L (q 0 ) = π s P L (p in 0 ) • (5.68) τ 10 , π s P R (q 0 ) = π u P R (p out 0 ) •  Figure 6. The singular configuration γ 1 ∪ σ 1 ∪ γ 0 ∪ σ 2 ∪ γ 2 connects U L and U R . and (5.94) in (β, r, w 1 , w 2 , ξ, κ)-coordinates. Then we obtain the singular configuration connecting U L and U R . See Fig 6. 6. Completing the Proof of the Main Theorem We split the proof of the main theorem into two parts. In the first subsection we prove the existence of solutions of the boundary value problem (2.9 ), (2.10), and show that (2.20) holds. In the second subsection we derive the weak limit (2.21).
Proof. We will apply the Exchange Lemma (Theorem 4.5) with (k, m, l, σ) = (1, 1, 3, 1). From (5.77) and (5.71), we know that the (k + σ)-manifold I 0 is transverse to the (m + l)-manifold W s 0 (P L ) at p in 0 , and the image of the projection π s P L I 0 ∩ W s 0 (P L ) = Λ L is σ-dimensional, so (T1) in the Exchange Lemma holds. The limiting slow system on P L is governed by (5.68), and by the parametrization (5.79) of Λ L it follows that (T2) holds. Also it is clear that (T3) holds with τ 0 = τ 10 , where τ 10 is defined in (5.85). Theorem 4.5 implies that there exists a neighborhood V 0 of q 0 such that (6.100) . From Proposition 5.4, it follows that the projections of I * and J * in the 5dimensional space {r = 0} intersect transversally at a unique point in Γ near q 0 . For the relation r = exp(−κ/ ), we then recover a unique intersection point q ∈ I * ∩ J * ∩ Γ in (β, r, w 1 , w 2 , ξ, κ)-space. By construction we have (6.96) and (6.97). The estimates (6.98) follows from (4.53). Note that max where σ 1 , σ 2 and γ 0 are defined in Section 5.3, so we obtain (6.99).
6.2. Convergence of Trajectories. Based on the results in Proposition 6.1, we first derive some estimates for the self-similar solutionũ (ξ).
Proposition 6.3. Letũ = (β ,ṽ ) be the solution of (2.9 ) and (2.10) in Proposition 6.2. Let p in and p out be defined in Proposition 6.1. Then as → 0, where ξ in and ξ out are ξ-coordinates of p in and p out , respectively.
From the estimates in Proposition 6.3, we can derive the weak convergence ofũ(ξ) as follows.
Proof. Let ψ ∈ C ∞ c (R) be a smooth function with compact support. From (6.105) we have A similar inequality holds for ∞ ξ out ψu dξ, so (6.110) On the other hand, from (6.104) and (6.106) we have Combining (6.110) and (6.111) we obtain This holds for all ψ, so we have (6.109).
From (2.22), this means (2.21) holds. Now Propositions 6.2 and 6.5 complete the proof of the Main Theorem.

Numerical Simulations
Some numerical solutions for (1.1) using the Lax-Friedrichs scheme are shown in Figure 7. The solutions appear to grow unboundedly as the number of steps increases.
Also some numerical approximations for (2.8 ) are shown in Fig 8. The algorithm was a shooting method following the descriptions in [KSS03]. Since w 1 and ξ are essentially constant near the shock, we project the trajectories in the (β, r, w 2 ) space. Note that w 2 (ξ) does not converge as ξ → ±∞ while x 2 = w 2 + (ξ − s)v converges, we replace w 2 by x 2 (again, following [KSS03]). Note that x 2 is a mild modification of w 2 near the shock since within the -neighborhood of ξ = s the difference between x 2 and w 2 is of order o(1). As decreases, the minimal value of r-coordinate on the trajectories in Fig 8 tends to zero. This means the maximum of v tends to infinity. Also observe that the change of the value of x 2 concentrates in the vicinity of β = ρ 1 and β = ρ 2 . This is consistent with our proof for the main theorem.

Proof of Proposition 3.3
If the section we prove the sufficiency of the conditions in Proposition 3.3 for (H3), which says that there is a trajectory for (2.15) connecting (u L , w L , s) and {β = ρ 1 , v = +∞}, and a trajectory connecting (u R , w R , s) and {β = ρ 2 , v = +∞}. We will focus on finding the first trajectory while finding the second one is similar.
We will switch back and forth between (β, v)-and (β, r)-coordinates, where r = 1/v. The system (2.15) is converted to (5.65) in (β, r)-coordinates. It suffices to find trajectories connecting u L = (β L , r L ) and p L ≡ (ρ 1 , 0) for (5.65) with (w, ξ) = (w L , s). From (H1) we know that u L is a source, and we will also see that p L is a saddle. Our strategy is to construct a negatively invariant region in which every trajectory goes backward to u L , and one of those trajectories goes forward to p L . See Fig 9. To construct a such region, we first study the flow on the boundary of the feasible region {ρ 2 ≤ β ≤ ρ 1 }. The equation ofβ in (2.15) with (w, ξ) = (w L , s) iṡ β = −sρ 1 − w 1L on {β = ρ 1 }.
Then V forms a backward invariant region. See Fig 9. We claim that u L attracts every point in V in backward time. Note that ∂ ∂β (vB 1 (β) − sβ − w 10 ) + ∂ ∂v (v 2 B 2 (β) − sv − w 2L ) = vB 1 (β) − s + 2vB 2 (β) − s = 4vB 2 (β) − 2s, which is positive when β ∈ ( √ ρ 1 ρ 2 , ρ 1 ) and s is small. In the last equality we used B 1 (β) = 2B 2 (β). By Bendixson's negative criterion, the system has no periodic orbit inside V . Since V is backward invariant and u L is the only equilibrium on the closure of V , it follows from the Poincaré-Bendixson Theorem that every trajectory in V tends to u L in backward time.