Larson–Penston Self-similar Gravitational Collapse

Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=k\varrho $$\end{document}p=kϱ, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>0$$\end{document}k>0, and subject to Newtonian gravity. We rigorously prove the existence of such a Larson–Penston solution.


Isothermal Euler-Poisson system
The classical model of a self-gravitating Newtonian star is given by the gravitational Euler-Poisson system. We work in three spatial dimensions and assume radial symmetry. The unknowns are the gas density (t, r ), the pressure p(t, r ), and the radial velocity u(t, r ), where t is the time coordinate and r = |x| the radial coordinate in R 3 . The equations take the form Here √ k is the speed of sound and it is constant throughout the star. We are interested in the existence of self-similar solutions to (1.1)-(1.4) describing finite time gravitational collapse. The only invariant scaling for (1.1)-(1.4) is given through the transformation . (1.5) Motivated by this invariance, we seek a self-similar solution of (1.1)-(1.4) of the form: (t, r ) = ( √ 2π t) −2ρ (y), u(t, r ) = √ kũ(y), (1.6) where It is convenient to introduce the relative velocitỹ ω :=ũ (y) + y y . (1.8) Applying the above change of variables, the Euler-Poisson system (1.1)-(1.4) becomes (1.10) where the derivative notation is short for ∂ y . A simple Taylor expansion at the origin y = 0 and the asymptotic infinity y → +∞ shows that in order for a solution (ρ,ω) to (1.9)-(1.10) to be smooth and asymptotically flat, we must havẽ  For a solution to be smooth through the sonic point y * , it has to be the case that the sonic point is a removable singularity. Assuming smoothness, we can formally compute the Taylor coefficients of (ρ,ω) around y * . Two possibilities emerge (see e.g. [2])either ρ(y) = 1 y * − 1 y 2 ω(y) = 1 y * + 1 y * (1 − 2 y * )(y − y * ) + −5y 2 * + 19y * − 17 2y 3 * (2y * − 3) (y − y * ) 2 + O(|y − y * | 3 ), (1.15) orρ 16) . (1.17) Using numerics, in 1969 in their seminal works Penston [22] and Larson [17] independently discovered an asymptotically flat smooth solution to (1.9)-(1.10) which satisfies the boundary conditions (1.11)- (1.12). Their solution passes through a single sonic point y * and conforms to the expansion of the type (1.14)- (1.15). In the literature, this solution is commonly referred to as the Larson-Penston (LP) collapsing solution.
There have been numerous studies of self-similar collapse for isothermal stars in the astrophysics literature and here we only provide a brief overview. In 1977 Hunter [14] numerically discovered a further (discrete) family of smooth self-similar solutions, commonly referred to as Hunter solutions, see also the important work of Shu [24]. The Taylor expansion of the Hunter solutions around the sonic point is of the form (1.16)- (1.17). A thorough analysis of the various types of self-similar solutions is given by Whitworth and Summers [25]. In 1988 Ori and Piran [23] gave numerical evidence that the LP collapse is the only stable self-similar solution in the above family of solutions, and therefore physically the most relevant. Brenner and Witelski [2], Maeda and Harada [18] reached the same conclusion performing careful numerical analysis of the collapse. The LP-solutions also play an important role in the study of so-called critical phenomena [11] and are of central importance in astrophysics, see e.g. [12]. The central result of this work is the proof of existence of an LP-solution. where we recall (1.8).

Remark 1.3
There are two known explicit solutions to (1.9)-(1.12). One of them is the Friedman solutionρ F (y) =ω F (y) ≡ 1 3 (1.20) and the other one is the far-field solutioñ ρ ∞ (y) = 1 y 2 ,ω ∞ (y) ≡ 1. (1.21) The Friedman solution (1.20) is the Newtonian analogue of the classical cosmological Friedman solution-it satisfies the boundary condition (1.11), but is not asymptotically flat. On the other hand, the far-field solution (1.21) is asymptotically flat, but blows up at the origin y = 0.
If the linear equation of state (1.4) is replaced by the polytropic law p = γ , γ > 1, it is well known that there cannot exist any collapsing solutions with finite mass and energy in the regime γ > 4 3 , see [4]. When γ = 4 3 there exists a special class of self-similar collapsing and expanding solutions [4,5,7,19]. The nonlinear stability in the expanding case was shown in [13]. When 1 < γ < 4 3 the authors in [9] showed the existence of an infinite-dimensional class of collapsing solutions to the gravitational Euler-Poisson system. When one considers the Euler-Poisson system with an electric (instead of gravitational) force field, the dispersive nature of the problem becomes dominant. A lot of progress has been made in recent decades, we refer the reader to [6,8,10,15] and references therein.
To prove Theorem 1.2, it is natural to consider the following change of variables The unknown sonic point y * is mapped to z = 1. The system (1.9)-(1.10) takes the form (1.24) We shall work with this formulation for the rest of the paper and often, by abuse of terminology, refer to the point z = 1 as the sonic point. It is now obvious from the LP sonic point expansion (1.14)-(1.15) that we formally assume that locally around the sonic point In this notation ω 0 = ρ 0 = 1 y * and (1.14)-(1.15) gives us For any y * > 0 we shall say that a solution of (1.23)-(1.24) is of Larson-Penston (LP)type if the conditions (1.25) and (1.28) hold. We shall prove in Theorem 2.10 that for any y * > 3 2 the LP-type conditions (1.25) and (1.28) uniquely specify a real analytic solution in some small neighbourhood of z = 1. We denote this flow by (ρ(·; y * ), ω(·; y * )).

Methodology.
The sonic point in the original (t, r )-variables corresponds to the backward cone emanating from the singularity (0, 0) and it takes the form r −t = −u(t, r ) ± √ k, t < 0. More details on the geometric meaning of the sonic point in this context can be found for example in [2]. Sonic points appear naturally in selfsimilar formulation of equations of fluid mechanics (see [1,3,12,16,20] and references therein). They present a fundamental difficulty in our proof of Theorem 1.2, as we cannot use any standard ODE theory to construct a real analytic (or a C ∞ ) solution. This is well illustrated in a recent pioneering study of sonic points for the compressible Euler system [20], where the authors use the equation of state p = γ , γ > 1. Using delicate arguments the authors [20] systematically develop the existence theory for C ∞ self-similar solutions of the Euler flow and such a smoothness is crucial in the proof of their nonlinear stability [21].
The self-similar problem associated with the Euler-Poisson system leads to an ODEsystem which is not autonomous [see (1.23)-(1.24)]. We also emphasise that the presence of gravity fixes exactly one invariant scale in the problem, see (1.5). Our proof uses in essential way dynamic invariances specific to the flow (1.23)-(1.24). The sonic point separates the positive semi-axis z ≥ 0 into an inner region [0, 1] and an outer region [1, ∞) (i.e. [0, y * ] and [y * , ∞) in the y-variable). The first and the easier step is to construct an LP-type solution in the outer region satisfying the boundary condition (1.12). This can be done for any value of y * ∈ [2,3]. The remaining key step is to find a value of y * ∈ [2,3] such that the associated LP-type solution connects z = 1 with the singular point z = 0 in the inner region and satisfies the boundary condition (1.11). More specifically, our goal is to choose y * ∈ [2,3] so that the local LP-type solution extends to the left all the way to z = 0 and satisfies lim z→0 ω(z; y * ) = 1 3 . This motivates us to consider The curve (ρ, ω) ≡ ( 1 3 , 1 3 ) corresponds to the Friedman curve, see Remark 1.3. We will show that the solution curve ω(·; y * ) crosses the Friedman curve strictly inside the interior region and stays trapped below it for y * sufficiently close to 3. The idea is to lower the value of y * to the infimum of the set Y -we setȳ * := inf Y . The idea is that ω(·;ȳ * ) will achieve the value 1 3 exactly at z = 0 and this will lead to an LP-solution. Using the minimality ofȳ * it is indeed possible to show that the solution exists on (0, 1] and satisfies lim inf z→ ω(z;ȳ * ) ≥ 1 3 . To prove that lim z→0 ω(z;ȳ * ) = 1 3 we use a contradiction argument in conjunction with a continuity argument. To explain this, it is necessary to consider the solution of the initial value problem (1.23)-(1.24) starting from z = 0 to the right with the initial values Just like we did in the vicinity of the sonic point, we resort to Taylor expansion around z = 0 to prove that (Theorem 2.12) the initial condtions (1.30) specify a unique solution to (1.23)-(1.24) locally to the right of z = 0. We denote this solution by (ρ − (·; ρ 0 ), ω − (·; ρ 0 )).
By way of contradiction we assume lim z→0 ω(z;ȳ * ) > 1 3 . The strategy is then the following.
• Step 3: Intersection at z = z 0 and contradiction. With considerable technical care and the crucial proof of strict monotonicity of the map ρ 0 → ρ − (z; ρ 0 ) in a region 0 < z 0 z 1 (see Lemma 4.18), we show that for any y * ∈ [ȳ * , y * * ] there is a continuous map y * → ρ 0 (y * ) such that The Intermediate Value Theorem, Steps 1 and 2 show that there exists y * ∈ (ȳ * , y * * ) ⊂ Y such that (ρ(·; y * ), ω(·; y * )) is a solution to (1.23)-(1.24) such that inf z∈(0,1] ω(z; y * ) ≥ 1 3 , which is a contradiction to the Definition (1.29) of the set Y . Our work provides a general strategy to construct a solution connecting a sonic point and a singular point, such as the origin z = 0 in this case. The crucial feature of the problem that allows us to find the solution is the contrast between the dynamics of the "right" solution (ρ(·;ȳ * ), ω(·;ȳ * )) and the "left" solution (ρ − (·; ρ 0 ), ω − (·; ρ 0 )) in the region 0 < z 1. This is fundamentally caused by the presence of the singular denominator 1 z on the right-hand side of (1.24), which is a generic feature of the 3dimensionality of the problem and radial symmetry.
Plan of the paper. Section 2 is devoted to the proof of the local existence, uniqueness, and regularity theorems for LP-type solutions locally around the sonic point (Theorem 2.10) and around the centre z = 0 (Theorem 2.12). In Sect. 3 we analyse the solution in the outer region z > 1. The main statement is Proposition 3.3 which states that for any y * ∈ [2,3] there exists a global forward-in-z solution to our problem starting from the sonic point z = 1 (i.e. y = y * ). The most difficult part of our work is the analysis of the inner region z ∈ [0, 1) and it is contained in Sect. 4. In Sect. 4.1 we obtain various continuity results for the LP-type flows, including most importantly the upper semicontinuity of the so-called sonic time, see Proposition 4.5. In Section 4.2 we introduce the crucial set Y and show that the LP-type flow associated withȳ * = inf Y starting from z = 1 exists all the way to z = 0, see

Local Well-Posedness Near the Sonic Point and the Origin
2.1. Existence, uniqueness, and regularity near the sonic point. Recalling (1.27), our goal is to compute a recursive relation that expresses the vector (ρ N , ω N ) in terms of ρ 0 , . . . , ρ N −1 , ω 0 , . . . , ω N −1 . For a given function f we shall write ( f ) M to mean the M-th Taylor coefficient in the expansion of f around the sonic point z = 1. In particular, where the summation implicitly runs over all non-negative integers satisfying the indicated constraint.
To compute the Taylor coefficients in (1.27), we first multiply (1.
where we have written z in the form 1 + δz.
To prove (2.34) we first note that (2.37) We plug in (1.27) into (2.31) and obtain Equating the coefficients and using (2.37), we conclude that for any non-negative N we have which is precisely (2.34).

Lemma 2.2
The coefficients (ρ i , ω i ), i = 0, 1 satisfy the following formulas: This is of course consistent with the 0-th order sonic point condition (1.25). We now let N = 1 in (2.33)-(2.34) and obtain respectively In this case ω 1 = 1 − 2ω 0 (which corresponds to the Larson-Penston solution) or ω 1 = −ω 0 . We disregard the latter possibility as it corresponds to a trivial solution that appears due to multiplication of (1.23)-(1.24) by 1 − y 2 * z 2 ω 2 . If on the other hand By a careful tracking of top-order terms in Lemma 2.1 we will next express (ρ N , ω N ) as a function of the Taylor coefficients with indices less or equal to N − 1.

Lemma 2.3 Let N ≥ 2. Then the following identity holds:
and are nonlinear polynomials of the first N − 1 Taylor coefficients given explicitly by the formulas (2.42) and (2.44) below.
Proof We first isolate all the coefficients in (2.36) that contain contributions from vectors where we have used We now isolate all the coefficients in (2.34) that contain contributions from vectors

associated with an LP-type solution is singular if and only if
As a consequence, the matrix A L P N is invertible for any y * > 3 2 In particular

Remark 2.5 (Hunter solutions) In the case of Hunter
In particular It follows that the matrix A H N is singular if and only if y * = N + 1 for some N ≥ 2.
For any y * > 3 2 consider the formal series (1.27) of LP-type, i.e. with conditions (1.25) and (1.28) satisfied. By Lemmas 2.3 and 2.4 we have the explicit relations The assumption ω 0 < 2 3 [recall ω 0 = 1 y * by (1.25)] implies that there exists a universal constant α > 0 such that Our goal is to show that the formal power series ∞ To that end we need some simple technical bounds which will be important in establishing convergence later on.

Lemma 2.6
There exists a constant c > 0 such that for all N ∈ N the following bounds hold and this proves (2.51). Next where we have used the already established bound (2.51) in each of the last two lines above. This proves (2.52). Finally, and this completes the proof of (2.53).
Let M ≥ 1 be such that (2.54) Lemma 2.7 Let y * > 3 2 and α ∈ (1, 2). Assume that for some universal constantC. It is now clear, that the estimates for (ωρ) and (ρ 2 ) , ≥ 0 follow in the same way, as the only estimates we have used are (2.54) and the inductive assumptions (2.60)-(2.61), which both depend only on the index, and are symmetric with respect to ρ and ω. Finally, from (2.58) it is clear that 2). Then there exists a constant C * > 0 such that if C > C * and for any N ≥ 3 the following assumptions hold: then there exists a constantc =c(D) such that (2.63) Proof By the assumption (2.54) and the bound D ≥ M we trivially have where D is the constant from Lemma 2.7. In the following, the constant c is a generic constant which depends on D, but not on N , and may change from line to line. Throughout the proof we use the convention that any summation of the form b k=a with a > b is zero.
We start with the estimate on the first term in the definition (2.42) of F N .
where we have used Lemma 2.7 to estimate |(ω 2 ) 2 | and |(ω 2 ) |, ≥ 3, and the inductive assumption (2.61) in the second line. To obtain the bound in the third line we used the estimate (2.53), and the bound 1 < C 2−α 2 2 which holds for C sufficiently large since α < 2.
In a similar way where we have used the Lemma 2.7, the inductive assumptions (2.60)-(2.61), and the bound (2.51). In an entirely analogous way we obtain the bound Finally, using the bound (2.64) and the inductive assumptions (2.60)-(2.61), we obtain m+n=N 0<m<N where we have used (2.51), bound (2.64), and the inductive assumptions. From the definition (2.42) of F N and bounds (2.65)-(2.70) we conclude that (2.62) holds. We now turn our attention to the source term G N . Note that the first line of (2.44) can be transformed into the first line of (2.42) by formally replacing some of the ρ k -s with ω k -s. Similarly, the negative of the second line of (2.44) is formally equal to the second line of (2.42) after formally replacing some of the ρ k -s and ω k -s. Since our estimates only depend on the bounds (2.64) and the inductive assumptions (2.60)-(2.61)-which only depend on the index of ρ and ω and are therefore invariant under the formal exchange of ρ and ω-the estimates analogous to (2.65)-(2.70) imply that the first two lines of (2.44) are bounded by (2.71) Clearly where we note that the last estimate follows from N ≤ cC N −1− N 2 , for some constant c > 0 and all N ≥ 3, and C > 1 independent of N . By the proof of (2.72) we have for N ≥ 3 and C sufficiently large, but independent of N .
To bound the quadratic nonlinearities in the 4-th line of (2.44) we note the bound The last term in the fourth line of (2.44) has already been estimated in the proof of (2.70) and we obtain k+n=N 0<k<N It remains to bound the cubic expressions in the last two lines of (2.44). We start with (2.77) Here we have used the bound (2.64), Lemma 2.7, and the inductive assumption (2.61). For any m ≤ N − 5 we have where we have used the bound for some constant c and all N ≥ 3, and (2.73) in the last line. By the same proof we obtain We next estimate On the other hand, Finally, the last remaining term to estimate is for all N ≥ 2. Moreover, for any closed interval K ⊂ (0, 2 3 ) we can choose the same constant C for all ω 0 ∈ K . Proof The proof proceeds by induction. Recall that c is a constant which may change from line to line and depends on y * , α, but not on N . When N = 2 it is clear that there exists a C 0 = C 0 (y * , α) > 1 such that for any C > C 0 the bound Similarly, bounds (2.50) and (2.63) give It is now clear that we can choose C > C 0 large enough so that (2.86)-(2.87) is true, since α > 1. Therefore, for a sufficiently large choice of C the inductive claim follows. The uniformity statement with respect to a closed subinterval K ⊂ (0, 2 3 ) is clear, as the constant c varies continuously as a function of ω 0 . Theorem 2.10 Let K ⊂ (0, 2 3 ) be a closed interval. Let {ρ k , ω k } k∈N be the coefficients associated with an LP-type solution. There exists an 1 > r = r K > 0 such that for any ω 0 = 1 y * ∈ K the formal power series converge for all z such that |δz| = |z − 1| < r . In particular, functions ρ(z) and ω(z) are real analytic inside the ball |z −1| < r . We can differentiate the infinite sums term by term and (ρ(z), ω(z)) is an LP-type solution of (1.23)-(1.24) for |z − 1| < r . Moreover, the density ρ(·; y * ) is strictly positive for |z − 1| < r.
Proof Fix an α ∈ (0, 2). By Lemma 2.9 there exists a C = C(K , α) > 0 such that when |δz| < 1 C =: r . The claim follows by the comparison test. The real analyticity and differentiability statements are clear. Since it follows that for r > 0 sufficiently small, the function 1 − y 2 * z 2 ω 2 = 0 for all |z − 1| < r and z = 1. Functions ρ(z) and ω(z) are indeed the solutions, as can be seen by plugging the infinite series (2.89) into the left-hand sides of (2.31)-(2.32); all the functions appearing on the left-hand side are analytic for |z − 1| < r .

Theorem 2.12
Let ρ 0 > 0 be given. There exists an 0 <r < 1 such that the formal power series converge for all z such that 0 ≤ z <r . In particular, functions ρ − (·; ρ 0 ) and ω − (·; ρ 0 ) are real analytic on [0,r ). We can differentiate the infinite sums term by term and the functions ρ − (·; ρ 0 ) and ω − (·; ρ 0 ) solve (1.23)-(1.24) with the initial conditions ω − (0; ρ 0 ) = 1 3 and ρ − (0; ρ 0 ) = ρ 0 . Proof By analogy to the previous section, we must Taylor-expand the solution at the origin z = 0 in order to prove a local existence theorem starting from the left. An immediate consistency condition follows from the presence of 1−3ω z on the right-hand side of (1.24). Namely, in order to have a well-posed problem with initial data prescribed at z = 0 we must have ω(0) = 1 3 . Assume that locally around z = 0 Our starting point are the equations We plug in (2.95) into (2.97) and obtain where, by definitionρ k =ω k = 0 for k < 0. Equating the coefficients above, we conclude that for any non-negative N we have which is precisely (2.33). Similarly, after plugging in (2.95) into (2.96) we obtain Therefore for any N ≥ 0 we have It is clear from (2.98) that ρ 0 = ρ(0) is a free parameter. Identities (2.98)-(2.99) give the recursive relationshipsρ The rest of the proof is now entirely analogous to the proof of Theorem 2.10 and we leave out the details.

The Outer Region z > 1
.
In this section we show that for any value of y * ∈ [2,3] there exists a unique LP-type solution in the exterior region. Such a statement is true because the flow "moves" in a stable direction as z → ∞. This should be contrasted to the more delicate analysis of the flow in the inner region. Our first preparatory lemma lists a number of simple properties in the vicinity of z = 1, which follow by continuity and careful use of the LP condition (1.28).
Claim (3.109) follows since ω(1) = 1 y * and ω is by (3.106) locally strictly increasing and 1 y * z is clearly strictly decreasing. To prove (3.110), consider Note that F(1) = F (1) = 0 and it is therefore necessary to evaluate the second derivative of F. A direct calculation shows that F (1) = 2 y * − 2ρ 2 which is strictly positive by (2.112). Thus F is strictly increasing on (1, 1 + δ) for a sufficiently small δ.
The next lemma shows the crucial dynamic trapping property.
Finally, combining the previous two lemmas we obtain the desired forward global existence result in the outer region z ≥ 1.
Proof Let T be defined as in Lemma 3.2 and assume that T < ∞. Notice that by the bounds in Lemma 3.2, both ω and ρ remain bounded and away from the sonic point singularity for z ∈ (1, T ). At T we must have either 1 Since (ρy * z) = ρy * (1 + ρ z ρ ) < 0 by (2.117) for all z ∈ (1, T ), we must have ρ(T )y * T < ρ(1)y * = 1, a contradiction.

The Inner Region 0 ≤ z < 1
By Theorem 2.10 and Lemma 2.11 there exists an r > 0 such that for any y * ∈ [2,3] there is unique LP-type solution (ρ(z; y * ), ω(z; y * )) on (1 − r, 1 + r ), which is analyticin-z and uniformly continuous with respect to y * . The next lemma records the obvious statement that one can extend the existence interval as long as we are away from the sonic line. Proof The proof is a standard consequence of the local well-posedness theory for ordinary differential equations.
To every y * ∈ [2, 3] (i.e. ω 0 = ω(1) ∈ [ 1 3 , 1 2 ]) we associate the sonic time of existence to the left s(y * ) := inf z ∈ (0, 1) solution exists on(z, 1] and ω 2 (z; y * )z 2 y 2 * < 1 . (4.126) Clearly 0 ≤ s(y * ) < 1 − r . By Lemma 4.1 the solution can be continued to the left starting at z = 1 − r for a short time and the maximal time of existence to the left is smaller or equal to s(y * ). Sonic time is of central importance in our analysis and our first goal is to show that there exists a y * ∈ [2, 3] such that s(y * ) = 0. To that end we prove a number of preparatory lemmas. We next collect important a priori bounds that hold on (s(y * ), 1).
In order to prove the upper bound for ρ, we consider Using (1.23) it is checked that By the LP-type sonic conditions (1.25) and (1.28) it is easy to see that f (1) = f (1) = 0.
To determine the sign of f near z = 1 it is therefore necessary to compute the second derivative of f . Since we conclude that f > 0 locally around 1 as the above expression is strictly positive for y * ∈ [2,3]. In fact, by choosing a possibly smaller r , we may assume f (z) > 0 for all z ∈ [1 − r, 1). We letz 1] it follows that the right-hand side of (4.131) is negative for any z ∈ (z, 1]. Integrating (4.131) for any z ∈ [z, 1 − r ] we get By an analogous argument as above, we conclude f (z) > 0 and thereforez =z. Therefore f (z) > 0 on (s(y * ), 1) as claimed. From (4.131) we then conclude f (z) < 0 which is equivalent to (4.129).
The following lemma shows that solutions which are a finite distance η away from the sonic line and defined for all z ≥z > 0, can be extended to the left by a finite time depending only on η andz.

Lemma 4.3
Let y * ∈ [2, 3] be given and consider the unique LP-type solution (ρ(·; y * ), ω(·; y * )) to the left of z = 1, given by Theorem 2.10. Assume that for somez ∈ (0, 1 −r ) and η > 0 we havez > s(y * ) and the conditions hold. Then there exists a t = t (η,z) > 0 such that the solution can be continued to the interval [z − t, 1) so that Proof By Lemma 4.2 the following a priori bounds hold: Formally, for any 0 < z ≤z we write the Eqs. (1.23)-(1.24) in their integral form For an M > 1 sufficiently large and t = t (z, η) <¯z 2 sufficiently small (both to be specified below), we make the inductive assumptions Here we choose to start the iteration with constant functions (ρ 0 (z), ω 0 (z)) ≡ (ρ(z), ω(z)), z ∈ [z − t,z] so that it satisfies the inductive assumptions. From (4.139), we easily conclude and therefore, for t sufficiently small and a sufficiently large M (but from now on fixed), we obtain the bound (4.142) for k = n. From (4.140) we easily conclude and therefore, for t = t (η,z) sufficiently small we obtain the bound (4.141) for k = n using (4.133). We also observe that for any z ∈ [z − t,z] Subtracting two iterates (ω n , ρ n ) and (ω n−1 , ρ n−1 ) we conclude −F(y * , ρ n−2 , ω n−2 )) dτ, Allowing the constants C to change from line to line, but to possibly depend onz, η, we plug (4.148)-(4.149) back into (4.146)-(4.147) and usingz − t ≥¯z 2 we obtain for k = 1, 2, . . . n Choosing t = t (η,z) sufficiently small we conclude that there exists a constant 0 < c < 1 such that u k ≤ cu k−1 for all k = 1, 2, . . . , n. By (4.144) and (4.145) for t = t (η,z) and therefore c sufficiently small. Since we can choose t so small that t <¯z 2 bound (4.150) gives us (4.143) with k = n. By the standard arguments we pass to a limit as n → ∞ and obtain the unique LP-type solution on the interval [z − t, 1].
Clearly both solutions are well-defined on (Z , 1 − r ].. Our goal is to show that Z =z if δ is sufficiently small. To that end, assume the opposite, i.e. Z = s(ỹ * ). In the rest of the proof the constant C may change from line to line, but may depend only onz and y * .

The set Y and the minimality property.
We now partition the interval [2,3] in the the sets X , Y, Z that will play an important role in our analysis. We let X := y * ∈ [2,3] inf z∈(s(y * ), 1)  The next statement shows that sets Y and X are not empty.

4.3.
Properties of the solution from the origin to the right. In order to complete the intersection argument in Sect. 4.4 we must better understand the solutions emanating from z = 0 to the right. Recall that (ρ − (·; ρ 0 ), ω − (·; ρ 0 )) is the unique local solution to (1. We then have the following a priori bounds on (ρ − , ω − ).
Proof It is clear that lim inf z→0 ω(z;ȳ * ) ≥ 1 3 as otherwise we would haveȳ * ∈ Y , a contradiction to the definition (4.170) ofȳ * and the openness of Y . We distinguish three cases. Case 1.