Compressible Euler Flows on a

Convergent–Divergent Surface:

Steady Subsonic Flows

Hairong Yuan

Department of Mathematics, East China Normal University, Shanghai 200062,

China

Abstract

In this paper, we construct various special solutions on a convergent-divergentsurface for the steady compressible complete Euler system and established the sta-bility of the purely subsonic flows. For a given pressure p0 prescribed at the “entry”of the surface, as the pressure p1 at the “exit” decreases, the flow patterns on thesurface change continuously as those happen in a de Laval nozzle: there appear sub-sonic flow, subsonic–sonic flow, transonic flow and transonic shocks. This work mayhelp us understand subsonic flows in de Laval nozzles. Our results indicate that,to determine a subsonic flow in a two-dimensional de Laval nozzle, if the Bernoulliconstant is uniform in the flow field, then this constant should not be prescribedif the pressure, density at the entry and the pressure at the exit of the nozzle aregiven; if the Bernoulli constant and both the pressures at the entrance and the exitare given, then the average of the density at the entrance is totally determined.

Key words: Euler system, surfaces, special solutions, nozzles, subsonic flows,transonic flows, transonic shocks1991 MSC: 35M10, 58J, 76J25

1 Introduction

1. In recent years, there has been an increasing interest in studying varioussystems of conservations laws originating from mathematical physics on Rie-mannian manifold. Apart from the purely theoretical curiosity on what is the

Email address: hryuan@math.ecnu.edu.cn; hairongyuan0110@gmail.com

(Hairong Yuan).

Preprint submitted to Elsevier 12 November 2006

effect of the metric of a Riemannian manifold to the solvability and solutions,the motivation may be twofold. First, some physical problems can be natu-rally formulated on nontrivial Riemannian manifold. For example, the study ofshock wave solutions in general relativity [11], and the problem of supersonicflow past infinite cones of arbitrary cross section and arbitrary angle of attackin aerodynamics [12]. Second, some physical problems can be well “approxi-mated” by fixing them up into certain Riemannian manifolds, and the lattermay be more easier to carry out a rigorous analysis. This paper is devoted todemonstrate this in some detail. We begin with a physical problem.

2. It is well known that one of the challenging problems in aerodynamics isto rigorously analyze the flow fields in a de Laval nozzle for given both thepressures at the entrance and the exit. A de Laval nozzle is a duct consistingof a convergent part and a divergent part, while the narrowest part of it iscalled as the throat. De Laval nozzles have many important applications inbuilding wind tunnels, propulsion of jets and rocket engines etc. to transportfluid flows and control their motions [30]. Numerous physical experiments andnumerical simulations have shown that for steady flows, for a given pressurep0 at the entrance of the nozzle, the following steady flow patterns may occur(see [7,22,25,30]):

(1) (Subsonic flow) If the pressure p1 at the exit of the nozzle lies slightly belowp0, there will be a tender wind in the nozzle which flows from the entrance tothe exit, and the flow field is always subsonic;

(2) (Subsonic–sonic flow and supersonic bubbles) However, there is a pe < p0

such that if p1 = pe, the flow may accelerate to sonic at the throat, andthen decelerate to subsonic. Observing shows that there may also appear localsupersonic regions near the throat if p1 is close to pe, as that happens forhigher subsonic flow past a wing [13,14,21];

(3) (Supersonic flow) There is a pt < pe, such that the flow always acceleratesin the nozzle, which is sonic at the throat and becomes supersonic after passingit;

(4) (Transonic shock) For p1 ∈ (pj, pe) with a pj > pt, the flow is subsonic inconvergent part, sonic at the throat, and becomes supersonic after passing it,and then a normal shock front appears and the flow becomes subsonic behindit, with the pressure increases then to p1 at the exit. As p1 decreases to pj,the position of the shock front moves to the exit;

(5) (Various jets) If p1 < pj, various very complex jet flows may occur insideand outside of the nozzle.

For simplicity, we sometimes call these possible flows in a de Laval nozzle as“de Laval flows”.

2

3. So we see that, a rigorous analysis of the flow field in a de Laval nozzle byusing full compressible steady Euler system is a formidable task up to now.The difficulties one may encounter include:

(1) The Euler system for subsonic–transonic steady flows is of quasi-linearelliptic–hyperbolic composite–mixed type;

(2) To determine the transonic shock is a nonlinear free boundary problem;

(3) The nozzle walls are characteristic boundaries by posing the natural invis-cid and impenetrability conditions on them;

(4) The admissible boundary conditions on the entrance and exit of the nozzleare not clear.

4. Thus, aiming to fully understand the de Laval flows, a reasonable programis to construct appropriate approximate models to attack these difficultiesseparately.

For transonic shocks, some important works have been carried out by manyauthors. Based on a special transonic shock solution (“the first class” as calledin [28]), Canic, Keyfitz, Lieberman [3] initiated the study of stability of tran-sonic shocks by using the transonic small disturbance equation, then Chen,Feldman (see [4,5] etc.) and Xin, Yin [27] carried out their research on the sta-bility of transonic shocks in ducts for full potential equations, then Chen [6]firstly studied this problem in a straight duct for Euler system, and Yuan [29]established the instability of this first class of transonic shock for Euler flows.Thus it is shown that the first class of transonic shock is not physical. Thenbased on [7], in [28] Yuan established the second class of transonic shocks indivergent nozzles and in [20] Liu and Yuan showed that they are stable. Thisresult may partly explain the transonic shock phenomena in de Laval nozzles.

Various innovative ideas and methods are developed in these works, and manyinteresting phenomena are observed, such as a new type of nonlocal ellipticproblem [20].

The study of transonic shocks shows that various special solutions play a verysignificant role in understanding flows in de Laval nozzles. That is, the generalphilosophy of utilizing special solutions as building blocks to construct moregeneral solutions, such as using various self–similar solutions of Riemann prob-lem to study general Cauchy problems of hyperbolic systems of conservationlaws [2], also works here.

5. There has already lots of important works on de Laval flows based on thewell known quasi–one–dimensional model [22,25] reduced by supposing thenozzle is slowly–varying and the state of the flow is independent of the direc-

3

tions perpendicular to the axis of the nozzle. Although this model is somewhatrough at first glance, it works surprisingly well [25]. The reason might be that,as showed in [7,28], in certain circumstances, the special solutions obtainedby quasi–one–dimensional model are exact solutions of the Euler system! Ina series of paper [9,10,17,18], Liu and Glimm etc. have already studied thisquasi–one–dimensional model by applying theory of hyperbolic balance laws.See also [15]. Their results also conform well to the experiments, thus demon-strate validity of quasi–one–dimensional model and the importance of certainspecial solutions of Euler equations in studying nozzle flows.

In the elegant monographs [13,14] by Kuz’min, by assuming the existence ofcertain exact solutions, transonic flows are also studied by using von Karmanequation and Chaplygin equation. These monographs also provide many nu-merical results. Morawetz contributes greatly to the study of transonic flowsin many ways, see, for example, [21], where she also presents many outstand-ing open problems and important methods on transonic flows, some of which,such as compensated compactness, do not depend on special solutions.

Purely subsonic plane flows have been studied for a long time by using full po-tential equation. See [1,14,26] and references therein. Recently Liu [19] studiedinstability of subsonic flow in a straight duct. However, it seems that no workshave been done in analyzing subsonic de Laval flows by using Euler systembefore.

6. Now, a natural question is, can we construct special solutions which can de-scribe various flow patterns in a de Laval nozzle for stationary two–dimensionalcompressible complete Euler system?

This might be very difficult in physical spaces and no any result in physicalspace is known, at least to the author’s knowledge. However, on a convergent–divergent surface (i.e., a two–dimensional Riemannian manifold), the answeris positive and the construction of special solutions is indeed not so hard.

We note that in [23], L. M. Sibner and R. J. Sibner considered isentropic ir-rotational flows on a two–dimensional axially symmetric torus, and obtainedspecial solutions to the full potential equation, which exhibit similar phenom-ena as that in a de Laval nozzle. However, they considered only compact man-ifold without boundary; the admissible boundary conditions and the stabilityof these solutions are not studied.

Motivated by the discovery of Sibners, we consider in this paper compress-ible Euler flows on a convergent–divergent surface (for example, the wall of athree–dimensional de Laval nozzle, see also (9) for such a surface). We thenestablished special solutions as in [23] (see Lemma 2) and especially studiedstability of the uniform subsonic flows (see Theorem 5, 6 and 7) based onmethods developed in [29] and [19]. Further studies on other flow patterns,

4

such as transonic flows, will be reported in forthcoming papers.

7. The study of compressible Euler flows on a surface may help us to under-stand the flows in a two–dimensional de Laval nozzle in many ways.

(1) For a nozzle, it is clear that the boundary conditions on the walls should bethe slip condition: the velocity of the flow is perpendicular to the normal of thewall. (This implies that the wall is a characteristic boundary.) However, it isnot clear that, what should be the physically appropriate admissible boundaryconditions on the entrance and the exit of the nozzle even for potential flowequation. (One often gives potential there [14,26], but since potential is notphysical, such conditions are not quite well in understanding the physicalphenomena.) By considering the steady subsonic Euler system, which is ahyperbolic–elliptic composite system, the situation may be more perplex, sincethere is still no general theory on boundary value problems of such systems.

However, since the wall is characteristic, that is, it is not independent of theequations, so roughly speaking it might not influence the boundary conditionson the entrance and the exit. Then we may cancel the walls by consideringflows on a surface, while the convergent–divergent effect of the walls is replacedby the non–flat metric of the surface.

Our analysis shows that, for subsonic steady Euler flows, for given the pressure,density at the entrance, and the pressure at the exit of a de Laval nozzle, if weassume that the Bernoulli constant is the same in the whole flow field, thenthis constant should not be prescribed (see Theorem 5). Otherwise, there maybe no any solution. Similarly, if we are given Bernoulli constant and both thepressures at the entrance and the exit, then the density at the entrance shouldnot be totally given: it should contain an unknown constant to be solved, thatis, the average of the density at the entrance has already been determined bythe other given datum.

Such admissible boundary conditions at the entrance of the nozzle might beinstructive for later works on studying other de Laval flows (transonic flowsetc.) since all such flows are subsonic near the entrance.

(2) It is noticed for a long time by studying isentropic irrotational flows that,for given both the pressures at the entrance and the exit of a nozzle, theboundary value problem of full potential equation is not well–posed [13,14,26].As showed in [19] and in this paper, this is also true for Euler System. Oneneeds an integral condition to solve a first order elliptic system. It turns outthat, this is closely connected to the total mass flux in the nozzle. For givenpressure and density at the entrance, if the pressure at the exit is in certainrange, one may adjust the mass flux of the flow (or the Bernoulli constant)to satisfy this integral condition to realize a subsonic flow. Since the flow maybe “choked” and there is a maximum mass flux, such an integral condition is

5

naturally related to the bifurcation phenomena for de Laval flows.

8. Now there are several comments on the method we used to prove the stabil-ity of subsonic flows in this paper. It grew from the study of transonic shocks[20,29] and then used by Liu to study subsonic flow in a slowly–varying two–dimensional duct. The key point is that by introducing the Lagrangian trans-formation and characteristic decomposition one may reduce the Euler systemto a 2×2 system coupled with two algebraic equations (i.e., the Bernoulli’s lawand invariance of entropy along streamlines for C1 flows). Since the 2×2 systemis uniformly elliptic for subsonic flows and we are dealing with a small pertur-bation problem, we can solve the nonlinear problem by using solely Banachfixed point theorem. Compare to the study of the transonic shocks in [20,29],a difference one should pay special attention to is that, since the mass fluxη0 is unknown, then in Lagrangian coordinate, we encounter a “semi-fixed”boundary value problem as called in [20]. That is, we consider a boundaryvalue problem on the rectangular NL = [0, 1] × [0, η0] with η0 unknown.

9. This paper is organized as follows. In section 2 we first introduce steadycompressible Euler system on a Riemannian manifold and then write it explic-itly on a given convergent–divergent surface. By using conservation of masswe can employ Lagrangian coordinates and characteristic decomposition toreduce the Euler equations to a 2 × 2 system and the Bernoulli’s law andinvariance of entropy for C1 flows on streamlines. In section 3 we constructspecial solutions which behave like those flows in de Laval nozzles by solvingseveral algebraic equations. Section 4 to section 6 are devoted to study thestability of the constructed subsonic flows. In section 4 we state the main the-orems and the nonlinear boundary value problems to be solved. In section 6we solve the linearized problems. Finally, in section 6, by using Banach fixedpoint theorem to an appropriately constructed nonlinear mapping, we thenprove the three theorems stated in section 4.

2 Reduction of Euler System

1. Let M be a Riemannian manifold with a metric g. The equations governingthe motions of compressible steady perfect fluids on M are the followingconservation laws (Euler system)[24]:

∇ · (ρu) = 0, (conservation of mass) (1)

∇ · (ρu ⊗ u) + ∇p = 0, (conservation of momentum) (2)

∇ · (ρEu) = 0. (conservation of energy) (3)

Here u is a vector field on M represents the velocity of the fluid flows; p, ρ, E

6

are the scalar pressure, density of mass, and density of energy respectively. ∇·and ∇ are the divergence and gradient operator on (M, g). Equation (2) maybe simplified as

ρ∇uu + ∇p = 0, (4)

with ∇XV the covariant derivative on (M, g).

We consider in this paper polytropic gas; that is, the equation of state is

p = A(S)ργ , (5)

where γ > 1 is the adiabatic exponent, S is the entropy, and A(S) = exp(S/Cν)for some constant Cv. Then the speed of sound is

a =√

γp/ρ. (6)

For C1 flows, the equation (3) may be replaced by

∇uS = 0; (7)

or by the following Bernoulli’s law

1

2|u|2 +

a2

γ − 1= c1, (8)

which also holds along streamlines, i.e., the integral curves of the vector fieldu. Here c1 is a number which may be different on different streamline, butremains the same on one streamline even across a shock front.

We say a flow is subsonic (supersonic, sonic) at a point p ∈ M if |u(p)| < a(p)(|u(p)| > a(p), |u(p)| = a(p)). Flows that are everywhere subsonic (super-sonic) on M are called subsonic (supersonic) flows. If the flow is sonic some-where and subsonic (supersonic) at the rest points, then the flow is calledsubsonic–sonic (supersonic–sonic). Finally, if both the sets of points wherethe flow is subsonic, sonic, supersonic are non-empty, then the flow is calledtransonic.

2. Now we introduce a convergent–divergent surface as follows. Let n(r) be apositive smooth function on [0, 1]. We take M to be the surface

7

x1 = r r ∈ [0, 1],

x2 = n(r) cos θ θ ∈ [0, 2π),

x3 = n(r) sin θ

(9)

with the induced metric of R3. Then (r, θ) ∈ N := [0, 1]×[0, 2π) is a coordinate

system on M. Direct calculation shows that the metric is

ds2 = (1 + n′(r)2)dr2 + n(r)2dθ2; (10)

that is,

g =

1 + n′(r)2 0

0 n(r)2

. (11)

Then by setting u = (u, v) with u, v the velocity component in r− and θ−direction, the Euler system may be written explicitly as

∂r

(

n(r)√

1 + n′(r)2ρu)

+ ∂θ

(

n(r)√

1 + n′(r)2ρv)

= 0,

u∂ru + v∂θu + 11+n′(r)2

· 1ρ∂rp = n′(r)

1+n′(r)2

(

n(r)v2 − n′′(r)u2)

,

u∂rv + v∂θv + 1n(r)2

· 1ρ∂θp = −2n′(r)

n(r)uv,

12

(

(1 + n′(r)2)u2 + n(r)2v2)

+ a2

γ−1= c1.

(12)

3. Next we write the above Euler system in Lagrangian coordinates (ξ, η) byusing conservation of mass as follows.

Let

dθ(r,h)dr

= vu(r, θ(r, h)),

θ(0, h) = h.(13)

Then θ = θ(r, h) is a streamline. Set

η = η(r, h) =∫ θ(r,h)

θ(r,0)n(r)

√

1 + n′(r)2ρ(r, θ)u(r, θ) dθ. (14)

By the first equation in (12), one gets ∂η(r, h)/∂r ≡ 0. Thus η = η(h) :=η(0, h) and we see η(0) = 0. Direct computation also shows that

8

∂η

∂h= n(0)

√

1 + n′(0)2ρu(0, h). (15)

So if ρu(0, h) 6= 0 for all h ∈ [0, 2π), we get the inverse function h = h(η). Setθ(r, η) := θ(r, h(η)). Then (14) is the following identity:

η =∫ θ(r,η)

θ(r,0)n(r)

√

1 + n′(r)2ρu(r, s) ds. (16)

Differentiate this with respect to η, one gets ∂θ/∂η = 1/(n(r)√

1 + n′(r)2ρu),

while (13) indicates ∂θ/∂r = v/u.

Now we introduce the Lagrangian coordinates (ξ, η) by

r = ξ,

θ = θ(ξ, η).(17)

Then the above definitions and computations show that

∂(r, θ)

∂(ξ, η)=

1 0

vu

1

n(r)√

1+n′(r)2ρu

, (18)

so

∂(ξ, η)

∂(r, θ)=

1 0

−n(r)√

1 + n′(r)2ρv n(r)√

1 + n′(r)2ρu

, (19)

thus

∂r = ∂ξ − n(ξ)√

1 + n′(ξ)2ρv∂η,

∂θ = n(ξ)√

1 + n′(ξ)2ρu∂η.(20)

By a lengthy computation, (12) may be written in Lagrangian coordinates asa symmetric system:

A∂ξU + B∂ηU + D = 0, (21)

where

9

A =

u(1 + n′2) 0 1ρ

0 un2 0

1ρ

0 u(ρa)2

, B = n√

1 + n′2

0 0 −v

0 0 u

−v u 0

,

D =

n′(u2n′′ − v2n)

2uvnn′

(n√

1+n′2)′

n√

1+n′2

uρ

, U =

u

v

p

. (22)

By (14), the domain N in Lagrangian coordinates is NL := [0, 1]× [0, η0) with

η0 =∫ 2π

0n0

√

1 + n′(0)2ρ(0, θ)u(0, θ) dθ. (23)

Note that η = 0 and η = η0 are the same streamline passing the point (0, 0) ∈N , so the periodical boundary conditions should be posed on them. That is,U should be periodic with respect to η with period η0.

4. Characteristic Decomposition. Straightforward computations show that

det A =un2

ρ2a2

(

u2(1 + n′2) − a2)

. (24)

Furthermore, assuming the flow is subsonic, we may solve the generalizedeigenvalues of B respect to A by

det(λA − B) = 0 (25)

as follows:

λ0 = 0; (26)

λ± = λR ±√−1λI , (27)

λR =ρv1

√1 + n′2

u21

a2 − 1, λI =

√1 + n′2ρu1

√

a2 − |u|2

a(u21

a2 − 1). (28)

Here we have set

u1 =√

1 + n′2u, v1 = nv. (29)

The corresponding left eigenvectors l (i.e., λlA = lB) are

10

l0 = ( u, v, 0 ); (30)

l± = lR ±√−1lI , (31)

lR = (λR

ρ+√

1 + n′2v1, −u1

n(1 + n′2), −λRu(1 + n′2)), (32)

lI = (λI

ρ, 0, −λIu(1 + n′2)). (33)

So for non–stagnation subsonic flow, the Euler system is of hyperbolic–ellipticcomposite type since det A 6= 0 and there are both real and complex eigenval-ues.

By multiplying the left eigenvector l0 to (21), we have

l0A(∂ξ + λ0∂η) + l0D = 0, (34)

or, by written explicitly,

1

2∂ξ|u|2 +

1

ρ∂ξp = 0. (35)

This is exactly invariance of entropy along streamlines by using Bernoulli’slaw and the equation of state (5):

∂ξ

(

p

ργ

)

= 0. (36)

Similarly, by setting

w =v

u, (37)

then multiplying l± from left to (21), after a long calculations we obtain

nu1(a2 − u2

1)∂ξw + (1 + n′2)(a2 − |u|2)∂ηp

−n√

1 + n′2u1v1ρa2∂ηw

+√

1 + n′2

(

n′(n′′u2 − nv2)v1 − 2n′v(u21 − a2) − a2v1

(n√

1 + n′2)′

n√

1 + n′2

)

= 0; (38)(

1 − u21

a2

)

∂ξp −√

1 + n′2ρv1∂ηp − nρ2u31∂ηw

+ρn′(n′′u2 − nv2) − ρu21

(n√

1 + n′2)′

n√

1 + n′2= 0. (39)

11

We remark that if the flow is supersonic and u1 6= a, then both the eigenval-ues are real and the Euler system is hyperbolic, and the above characteristicdecomposition method leads to the same equations (36)(38)(39). We summa-rizing the above results as the following lemma.

Lemma 1 If the flow is non–sonic and u1 6= a, ρu 6= 0, then the compressiblesteady Euler system (12) is equivalent to (36)(38)(39) and the Bernoulli’s law.Furthermore, if |u| < a (|u| > a), then (38)(39) is elliptic (hyperbolic).

PROOF. One can easily check that if |u| 6= a, ρu 6= 0 and u1 6= a hold, thenthe three left eigenvectors are linearly independent. 2

3 Special Solutions

In this section we construct exact special solutions of (12) which depend onlyon r. To be specific, we assume the following hold for the profile of the surface.

(N): n(r) is a positive smooth function on [0, 1] and

n′′(r) > 0, n(0) > n(1); (40)

n′(r) < 0 on (0, s), n′(r) > 0 on (s, 1) for a fixed s ∈ (0, 1). (41)

We note that (41) implies that the surface is “convergent–divergent”.

Lemma 2 Suppose (N) holds, and the flow depends only on r, and v ≡ 0.Then for given p(0) > 0, ρ(0) > 0 (i.e., the pressure and density at theentrance r = 0 of M),

(1) [subsonic flow] there exists a pe < p(0) depending only on p(0) and n suchthat if p(1) ∈ (pe, p(0)), then there exists a unique subsonic flow on M withthe pressure at the exit r = 1 is p(1);

(2) [subsonic–sonic flow] there exists a unique flow which is subsonic on 0 <r < s ∪ s < r < 1 and sonic at r = s with the pressure at the exitr = 1 is pe;

(3) [transonic flow] there exists a unique flow which is subsonic on 0 < r < s,sonic at r = s, and supersonic on s < r < 1, with the pressure atthe exit r = 1 is pt. Here the number pt < pe depends only on n(r) andp(0), ρ(0);

(4) [transonic shock] there exists a unique flow which is subsonic on 0 < r <s, sonic at r = s, and supersonic on s < r < rs, and subsonic onrs < r < 1, with r = rs a shock, such that: the pressure at the exitr = 1 is p1. Here p1 ∈ [pj, pe] with pj > pt being a number depending onlyon n(r) and p(0), ρ(0).

12

PROOF. 1. We first write down the Euler system in this simplified case. Wewrite u1 = u1(r), p = p(r) etc. in the following. Now (12) is

ρu1u′1 + p′ = 0, (42)

(nρu1)′ = 0, (43)

1

2|u1|2 +

a2

γ − 1= c1. (44)

Equation (42) may be replaced by S ′ = 0 by using (44) and the equation ofstate (5) if we consider only continuous flows. So

S(r) ≡ S0 := A−1

(

p(0)

ρ(0)γ

)

. (45)

Thus to prove claim (1)–(3) of this lemma, we are led to the following twoalgebraic equations

nρu1 = c0,1

2|u1|2 +

a2

γ − 1= c1, (46)

with c0, c1 two positive numbers.

2. For simplicity, we denote p(r) = pr, ρ(r) = ρr, a(r) = ar, n(r) = nr forr ∈ [0, 1]. By substituting the first equation in (46) to the second one, we seec1 is a function of c2

0:

c1 =1

2

c20

n20ρ

20

+a2

0

γ − 1. (47)

Now define

g(ρ) = c1ρ2 − γ

γ − 1A(S0)ρ

γ+1, ρ > 0. (48)

We need to solve ρ(r) by

g(ρ(r)) =1

2

c20

n(r)2, (49)

thus determine p(r), u(r) on M.

3. To guarantee that the function g to be positive, we see that there shouldhold

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0 < ρ < ρ∗ :=

(

γ − 1

γ

c1

A(S0)

) 1

γ−1

. (50)

Direct calculation shows that

g′(ρ) = ρ

(

2c1 −γ + 1

γ − 1a2

)

(51)

= ρ(u21 − a2)

> 0 if 0 < ρ < ρ∗, i.e., the flow is supersonic,

< 0 if ρ > ρ∗, i.e., the flow is subsonic.

Here

ρ∗ :=

(

2

γ + 1

) 1

γ−1

ρ∗. (52)

Thus the range of g on (0, ρ∗) is (0, g(ρ∗)] with

g(ρ∗) =γ − 1

γ + 1c1ρ

2∗. (53)

4. So to solve (49), one requires that for any r ∈ [0, 1],

c20

2n(r)2≤ g(ρ∗), (54)

or equivalently,

G(t) := c2

(

t

2n20ρ

20

+a2

0

γ − 1

)γ+1

γ−1

− t ≥ 0, (55)

with

c2 = 2γ − 1

γ + 1

(

2(γ − 1)

γ(γ + 1)A(S0)

) 2

γ−1

n(s)2,

and t represents c20. Note that n(s) = minn(r) : r ∈ [0, 1] by assumption

(N).

Since G(0) > 0 and G′(t) < 0 for

14

0 ≤ t ≤ t∗ :=a2

0

γ − 1

(

γ + 1

2

(

n0

ns

)γ−1

− 1

)

, (56)

while G(t∗) < 0, there exists uniquely one t∗ := supt ∈ [0, t∗) : G(t) > 0with G(t∗) = 0. So we see that if c0 ∈ [ 0,

√t∗ ], then (49) is solvable. The

solution (p(r), ρ(r), u(r)) depends on c0 as it varies. In aerodynamics, whenc0 =

√t∗, a de Laval nozzle is then “chocked”.

5. One easily checks that

G(ρ20a

20n

20) = ρ2

0a20(n

2s − n2

0) < 0, (57)

so we have an estimate

t∗ < ρ20a

20n

20. (58)

This means that when c0 varies in (0,√

t∗), the flow at the entrance is alwayssubsonic. That is, ρ0 > ρs.

Now for any fixed c0 ∈ (0,√

t∗), by assumption (N) and the graph of thefunction g(ρ), we see ρ(r) decreases if r ∈ (0, s) and attains the minimumρmin > ρ∗ at r = s, then increases as r ∈ (s, 1). Thus ρ(r) > ρ∗ for r ∈ (0, 1)and the flow is purely subsonic. By assumption (N), n1 < n0 implies thatρ1 < ρ0, thus p1 < p0.

6. Now from (48)(49), we see that

c1(t)ρ21 −

γ

γ − 1A(S0)ρ

γ+11 =

t

2n21

. (59)

As seen in step 4, ρ1 is a function of t = c20. So by differentiating the above

equation with respect to t, one obtains that

dρ1(t)

dt=

1

2n20ρ1(u2

1 − a2)

(

n20

n21

− ρ21

ρ20

)

. (60)

By the result of step 5, we know u21−a2 < 0 and n0 > n1, ρ0 > ρ1, so ρ′

1(t) < 0for all t ∈ (0, t∗). We denote the minimum ρ1(t∗) as ρe. Thus we obtain pe inclaim (1) of Lemma 2.

For any p1 ∈ (pe, p0) (thus ρ1 is known), by the monotonicity of the functionρ1(t), we may uniquely obtain a t ∈ (0, t∗) with ρ1(t) = ρ1. Then as in step 4

we obtain uniquely one smooth subsonic flow. This proves claim (1).

15

7. Now taking c0 =√

t∗ to solve (49), we see ρ(r) monotonically deceases toρ∗ as r increases in (0, s). The flow becomes sonic at r = s. Then as r variesin (s, 1), n(r) increases and ρ(r) also increases to ρe. Thus we proved claim(2).

Another possibility is that, due to the “∧

”–like shape of the graph of thefunction g(ρ), ρ(r) may further decreases to the “left branch” for r ∈ (s, 1).In this case since ρ(r) < ρ∗ for r ∈ (s, 1), the flow becomes supersonic. Thenpt is the smaller root to g(ρ) = t∗/(2n2

1). This proves claim (3).

8. (4) can be proved by analysis similar to those in [28]. 2

The next lemma concerns regularity of the above constructed flows.

Lemma 3 Under the assumptions of Lemma 2, the subsonic flow is smooth,the transonic flow is C1, while the subsonic–sonic flow is smooth on [o, s) ∪(s, 1], with jumps of the derivatives of ρ, u1 at r = s.

PROOF. 1. From (46), we have the following ordinary differential equationssatisfied by ρ, u1:

ρ′ =n′

n

ρu21

a2 − u21

, (61)

u′1 = −n′

n

u1a2

a2 − u21

. (62)

So subsonic flow is everywhere smooth, and the subsonic–sonic flow and tran-sonic flow are smooth on [0, s) ∪ (s, 1], since u2

1 − a2 6= 0 there.

2. We may determine ρ′(s) by L’Hospital principle in calculus for subsonic–sonic flows and transonic flows since n′(s) = a(s)2 − u1(s)

2 = 0. By (61)(62),we have

ρ′(s)2 =ρu2

1

n

n′′(s)

(a2 − u21)ρ

=ρ2

s

γ + 1

n′′(s)

n(s)6= 0, (63)

u′1(s)

2 =u1(s)

γ + 1

n′′(s)

n(s)6= 0. (64)

For subsonic–sonic flow, whose density decreases on (0, s) and increases on

(s, 1), one thus sees that ρ′(s∓) = ∓ρs

√

1γ+1

n′′(s)n(s)

, so the derivative is not

continuous at r = s.

16

However, for transonic flow, since ρ(r) always decreases and u1 increases, we

see ρ′(s) = −ρs

√

1γ+1

n′′(s)n(s)

and u′1(s) =

√

u1(s)γ+1

n′′(s)n(s)

. This finishes the proof of

Lemma 3.

In this paper, we will focus on the stability of the subsonic flow constructed inLemma 2 under appropriate boundary conditions. Study of the transonic flowand other patterns of flows on M will be presented in forthcoming papers. Forconvenience of this paper, we introduce the following definition and notations.

Definition 4 We call the subsonic flow constructed in Lemma 2 as the background solution determined by (p0, ρ0, p1), and denoted as Ub = Ub(r) =(pb(r), ρb(r), ub(r), 0). The constant c1 in Bernoulli’s law will be simply de-noted as cb for background solution; The mass flux η0 defined by (23) forbackground solution will be denoted as ηb.

4 The Perturbed–Subsonic–Flow Problem

We first state the results on stability of subsonic flows which will be provedin this paper.

Theorem 5 Let Ub be a background solution determined by (pb(0), ρb(0), pb(1)).There are constants ε0 and C0 depending on n(r) and Ub such that if:

(1) The Bernoulli constant c does not depend on streamlines;(2) On r = 0, θ ∈ [0, 2π) the following hold for some α ∈ (0, 1):

‖p0(θ) − pb(0)‖C2,α[0,2π] ≤ ε ≤ ε0, (65)

‖ρ0(θ) − ρb(0)‖C2,α[0,2π] ≤ ε ≤ ε0; (66)

then there exists a unique subsonic flow U = (u, v, p, ρ) on M satisfying p =p0(θ), ρ = ρ0(θ) at the entry r = 0, p = pb(1) at the exit r = 1, and

∫ 2π

0

v(0, θ)

u(0, θ)dθ = 0 (67)

together with the following estimate

‖U − Ub‖C1,α(M;R4) ≤ C0ε. (68)

This theorem may be strengthened a little as:

17

Theorem 6 For given pb(0), ρb(0), let (pb)e be the number determined bypb(0), ρb(0) and n as in Lemma 2 (2), and [pc, pd] ⊂ ((pb)e, pb(1)). Then thereare constants ε0 and C0 depending on n(r), pb(0), ρb(0) and pc, pd such thatfor background solution Ub determined by (pb(0), ρb(0), pb(1)) with any fixedpb(1) ∈ [pc, pd], if

(1) The Bernoulli constant c does not depend on streamlines;(2) On r = 0, θ ∈ [0, 2π) the following hold for some α ∈ (0, 1):

‖p0(θ) − pb(0)‖C2,α[0,2π] ≤ ε ≤ ε0,

‖ρ0(θ) − ρb(0)‖C2,α[0,2π] ≤ ε ≤ ε0;

then there exists a unique subsonic flow U = (u, v, p, ρ) on M satisfying p =p0(θ), ρ = ρ0(θ) at the entry r = 0, p = pb(1) at the exit r = 1, and

∫ 2π

0

v(0, θ)

u(0, θ)dθ = 0

together with the following estimate

‖U − Ub‖C1,α(M;R4) ≤ C0ε.

Another result is the following theorem.

Theorem 7 Let Ub be a background solution determined by (pb(0), ρb(0), pb(1)).There are constants ε0 and C0 depending on n(r) and Ub such that if:

(1) The Bernoulli constant c is given and does not depend on streamlines;(2) On r = 0, θ ∈ [0, 2π) the following hold for some α ∈ (0, 1):

‖p0(θ) − pb(0)‖C2,α[0,2π] ≤ ε ≤ ε0,

‖ρ0(θ) − ρb(0)‖C2,α[0,2π] ≤ ε ≤ ε0,

|c − cb| ≤ ε ≤ ε0;

then there exists a unique subsonic flow U = (u, v, p, ρ) on M and a constante satisfying p = p0(θ), ρ = ρ0(θ) + e at the entry r = 0, p = pb(1) at the exitr = 1, and

∫ 2π

0

v(0, θ)

u(0, θ)dθ = 0

together with the following estimate

18

|e| + ‖U − Ub‖C1,α(M;R4) ≤ C0ε.

A strengthened version of Theorem 7 analogy to Theorem 6 also holds.

Remark 8 The differences between Theorem 7 and Theorem 5 are that, theBernoulli constant is given in the former, but unknown in the latter; andthe density is given with a constant difference (i.e., containing an unknownconstant e to be solved) in the former, while is totally given in the latter. Theoccurrence of e in Theorem 7 may be viewed as that the average of the densityat the entrance is not arbitrary.

As mentioned in section 1, it is perplex that what is a physically meaningfulboundary condition to study nozzle flows. Although there may be numerousmathematical ways to pose boundary conditions at the entrance and exit (suchas the two ways showed above and see also [19]), it seems that (from theauthor’s opinion) formulate boundary conditions as in Theorem 7 and 5 aremore physical, since the pressures at the entrance and exit should be totallygiven. The methods of this paper can also deal with the case if the Bernoulliconstant depends on streamlines (c.f. the proof of Theorem 7).

The condition (67) comes from the periodic assumption on θ (as well as η)coordinate and is used to guarantee the uniqueness. For a two–dimensional deLaval nozzle, it should be replaced by the slip condition on walls.

Remark 9 We will show that for given pressure at the entrance and the exitsimultaneously, the problem requires an integral condition to be solvable. Thusthere needs a constant c or e to adjust the flow so that it is realizable. Thisreflects the change of total mass flow through a nozzle as pressure varies andis then natural from physical point of view (c.f. the construction of specialsolutions).

Remark 10 We may also perturb the metric g of the surface M a little andobtain stability of subsonic flows with the above given boundary conditions.

We now begin to prove Theorem 5. Theorem 7 may be proved in a similar wayand we will point out the difference later. Theorem 6 follows from a simpleobservation on how the constant C0 and ε0 depending on the backgroundsolutions.

We first note that η0 is unknown; that is, we encounter a “semi-fixed” bound-ary value problem. We may use the following transformation Φ : (ξ, η) 7→(ξ, η) :

19

ξ = ξ,

η = ηη0

(69)

to normalize the domain NL to Ω = (ξ, η) ∈ [0, 1]× [0, 1]. Then we will writeout the boundary value problem to be studied in an easy–to–be–linearizedform.

4.1 Perturbed Equations

Let

w2 = p(ξ, η) − pb(ξ). (70)

We need to find the equations of w and w2.

1. We begin with (38). Let

e2(ξ) :=1 + n′(ξ)2

n(ξ)· a2 − |u|2

(a2 − u21)u1ηb

∣

∣

∣

∣

∣

U=Ub(ξ)

, (71)

d2(ξ) :=

(

n′

n

a2 − 2u21

a2 − u21

− n′n′′

1 + n′2

)∣

∣

∣

∣

∣

U=Ub(ξ)

, (72)

and

f2(U) :=

√1 + n′2v1ρa2

η0(a2 − u21)

∂ηw +

(

e2(ξ) −1 + n′(ξ)2

n(ξ)· a2 − |u|2(a2 − u2

1)u1η0

)

∂ηp

+n′v2v1

u(a2 − u21)

+

(

d2(ξ) −(

n′

n

a2 − 2u21

a2 − u21

− n′n′′

1 + n′2

))

w. (73)

Then (38) is

∂ξw + e2(ξ)∂ηw2 + d2(ξ)w = f2(U). (74)

By setting

D2(ξ) := exp∫ ξ

0d2(s) ds, (75)

f2(U) := D2(ξ)f2(U), (76)

the equation (74) may also be written in divergence form as

20

∂ξ(D2w) + ∂η(D2e2w2) = f2(U). (77)

Note that f2 consists of higher order terms.

2. Now we turn to (39). By (61) one sees that pb(ξ) solves

dp

dξ=

n′

n

ρu21a

2

a2 − u21

. (78)

Then (39) is equivalent to

∂ξw2 −nρ2a2u3

1

(a2 − u21)η0

∂ηw − n′

n(H(U) − H(Ub))

=

√1 + n′2ρa2

(a2 − u21)η0

v1∂ηp +n′nρa2

a2 − u21

v2, (79)

where

H(U) :=γpu2

1

a2 − u21

. (80)

Since

u21 =

2c − 2a2

γ−1

1 + n2

1+n′2 w2(81)

holds by Bernoulli’s law, and

a2 = γ

(

p0(η)

ρ0(η)γ

) 1

γ

pγ−1

γ (82)

holds by constancy of entropy, and note that c is also an unknown number, wesee H is actually an analytical function of p, w and c. So by Taylor expansions,we have

H(U) − H(Ub) = ∂pH(Ub)w2 + ∂wH(Ub)w + ∂cH(Ub)(c − cb) (83)

+O(|U − Ub|2 + |c − cb|2)= d1(ξ)w2 + d0(ξ)(c − cb) + O(|U − Ub|2 + |c − cb|2),

where

21

d1(ξ) :=a4

(a2 − u21)

2

(

γ(

u1

a

)4

+ (1 − γ)(

u1

a

)2

− 2

)∣

∣

∣

∣

∣

U=Ub(ξ)

, (84)

d0(ξ) :=2γpa2

(a2 − u21)

2

∣

∣

∣

∣

∣

U=Ub(ξ)

> 0. (85)

Let

d1(ξ) :=n′

nd1, d0(ξ) :=

n′

nd0,

e1(ξ) :=nρ2a2u3

1

(a2 − u21)ηb

∣

∣

∣

∣

∣

U=Ub(ξ)

,

f1(U, c) :=

√1 + n′2ρa2

(a2 − u21)η0

v1∂ηp +n′nρa2

a2 − u21

v2

+

(

nρ2a2u31

(a2 − u21)η0

− e1

)

∂ηw + O(|U − Ub|2 + |c − cb|2).

Then (79) is

∂ξw2 − e1∂ηw − d1(ξ)w2 − d0(ξ)(c − cb) = f1(U, c). (86)

By introducing an auxiliary function

D1(ξ) := exp

(

−∫ ξ

0d1(s) ds

)

, (87)

and setting

f1(U, c) = D1(ξ)f1(U, c), (88)

then (86) in divergence form is

∂ξ(D1w2) − ∂η(D1e1w) − D1d0 · (c − cb) = f1(U, c). (89)

Note that f1 also consists of higher order terms.

4.2 The Perturbed–Subsonic–Flow Problem

So far we see that to prove Theorem 5, we need only to solve c, w, w2 andu, v, ρ, η0 from the following nonlinear boundary value problems:

22

(NP1) :

(77), (89) in Ω,

w2 = p0(η) − pb(0) on ξ = 0,

w2 = 0 on ξ = 1,

Periodic conditions on η = 0, η = 1,∫ 10 w(0, s) ds = 0;

(NP2) :

ρ = ρ0(η)(

w2+pb

p0(η)

) 1

γ ,

u1 =

√

2c− 2a2

γ−1

1+ n2

1+n′2w2

,

v = wu1√1+n′2

;

(NP3) : η0 =n(0)

√

1 + n′(0)2∫ 2π0 ρ0(θ) dθ

∫ 10

dηu(0,η)

.

There are several remarks to explain the above three problems.

Periodic conditions in (NP1) mean that the functions are periodical with re-spect to η with period 1.

The expressions in (NP2) come from Bernoulli’s law, invariance of entropyand definition of w.

The formula in (NP3) on computing η0 comes from (17). We know that onr = ξ = ξ = 0, there should hold

η0dη

dθ= n(0)

√

1 + n′(0)2ρ0(θ)u(0, η) (90)

and

η(0) = 0, η(2π) = 1. (91)

This is a two–point boundary value problem of an ordinary differential equa-tion. From (90) we see

η0

∫ η

0

ds

u(0, s)= n(0)

√

1 + n′(0)2

∫ θ

0ρ0(s) ds. (92)

23

Thus if (NP3) holds, then this two–point boundary value problem is uniquelysolvable since u(0, s), ρ0(s) are positive and bounded away from zero.

We note that in (NP1) and (NP2),

ρ0(η) := ρ0(θ(0, ηη0)), p0(η) := p0(θ(0, ηη0))

may depend on the solution. But if (65)(66) hold and the solution U is nearthe background solution, that is, U ∈ Oδ with

Oδ :=

U = (u, v, p, ρ) : U is periodic with respect to η with period 1,

and ‖U − Ub‖C1,α(Ω;R4) ≤ δ,

|u|2/2 + a2/(γ − 1) = const ∈ R

, (93)

then the estimate

‖p0(η) − pb(0)‖C1,α[0,1] + ‖ρ0(η) − ρb(0)‖C1,α[0,1] ≤ Cε (94)

is true. Here C is a positive constant depending only on Ub and δ is chosen sosmall that U is still subsonic and ρu 6= 0. This fact follows from results likeProposition 2.2 in [29] (or Lemma 4.6 in [4]) concerning C1,α homeomorphismson different domains.

5 Analysis of Linearized Problems

To solve the three coupled nonlinear problems (NP1)(NP2)(NP3), we willapply Banach fixed point theorem to a nonlinear mapping constructed byappropriately linearized problems.

5.1 Statement of the Linearized Problems

We begin with (NP1). Let f1, f2 ∈ Cα(Ω), g ∈ C1,α[0, 1] be periodical withrespect to η with period 1. Problem (P1) solves w, w2 and c from the followinglinear boundary value problem of first order elliptic system:

24

(P1) :

∂ξ(D1w2) − ∂η(D1e1w) − D1d0 · (c − cb) = f1,

∂ξ(D2w) + ∂η(D2e2w2) = f2 in Ω;

w2 = g on ξ = 0,

w2 = 0 on ξ = 1,

Periodic conditions on η = 0, η = 1,∫ 10 w(0, s) ds = 0.

For given ρ0(η), p0(η) and w, w2, c obtained from problem (P1), problem (P2)solves ρ, u and v by

(P2) :

ρ = ρ0(η)(

w2+pb

p0(η)

) 1

γ ,

a2 = γ(w2 + pb)/ρ,

u = 1√1+n′(ξ)2

√

2c− 2a2

γ−1

1+ n2

1+n′2w2

,

v = wu.

Problem (P3) solves η0 by

(P3) : η0 =n(0)

√

1 + n′(0)2∫ 2π0 ρ0(θ) dθ

∫ 10

dηu(0,η)

with a given ρ0(θ) and u obtained from (P2).

5.2 Solving Problem (P1)

Since (P2)(P3) are simple algebraic equations, we focus on the solution ofproblem (P1).

Lemma 11 Let f1, f2 ∈ Cα(Ω), g ∈ C1,α[0, 1] be periodical with respect toη with period 1. Then problem (P1) has a unique solution w, w2 and c. Inaddition, the following estimate holds:

‖w2‖C1,α(Ω) + ‖w‖C1,α(Ω) + |c − cb|≤C

(

‖f1‖Cα(Ω) + ‖f2‖Cα(Ω) + ‖g‖C1,α[0,1]

)

. (95)

PROOF. 1. By linearity of (P1), we first separate it as the following twoproblems:

25

∂ξ(D1w(1)2 ) − ∂η(D1e1w

(1)) − D1d0 · (c − cb) = f1,

∂ξ(D2w(1)) + ∂η(D2e2w

12) = 0 in Ω;

w(1)2 = g on ξ = 0,

w(1)2 = 0 on ξ = 1,

Periodic conditions on η = 0, η = 1,∫ 10 w(1)(0, η) dη = 0;

(96)

∂ξ(D1w(2)2 ) − ∂η(D1e1w

(2)) = 0,

∂ξ(D2w(2)) + ∂η(D2e2w

(2)2 ) = f2 in Ω;

w(2)2 = 0 on ξ = 0,

w(2)2 = 0 on ξ = 1,

Periodic conditions on η = 0, η = 1,∫ 10 w(2)(0, η) dη = 0.

(97)

Then clearly

w = w(1) + w(2), w2 = w(1)2 + w

(2)2 , (98)

and c solve (P1).

2. By the second equation in (96), we may introduce a potential function Φ(1)

on the simply connected domain Ω = [0, 1] × [0, 1] such that

w(1) = −∂ηΦ(1)

D2, w

(1)2 =

∂ξΦ(1)

D2e2. (99)

If w(1) and w(1)2 satisfy the periodic conditions and

∫ 10 w(0, s) ds = 0, note that

D2(0) = 1, we see that Φ(1) also satisfies the periodic conditions. The reverseis also true. So we may formulate a Neumann problem:

∂ξ

(

D1

D2e2∂ξΦ

(1))

+ ∂η

(

D1e1

D2∂ηΦ

(1))

− D1d0 · (c − cb) = f1 in Ω;

∂ξΦ(1) = e2(0)g on ξ = 0,

∂ξΦ(1) = 0 on ξ = 1,

Periodic conditions on η = 0, η = 1,

Φ(1)(0, 0) = 0.

(100)

Let

26

c − cb = −∫

Ω f1(ξ, η) dξdη +∫ 10 g(η) dη

∫ 10 D1(ξ)d0(ξ) dξ

, (101)

and note that by (N) the denominator is nonzero; see step 3 below. Then itis well known that (100) is uniquely solvable and the estimate

∥

∥

∥Φ(1)∥

∥

∥

C2,α(Ω)≤ C

(

‖f1‖Cα(Ω) + ‖g‖C1,α[0,1]

)

(102)

holds [8].

3. We claim that if the background solution is subsonic and (u1)b 6= 0, then∫ 10 D1(ξ)d0(ξ) dξ < 0; so it is also bounded away from zero for a compact

family of background subsonic flows (as those in the assumption of Theorem6).

Indeed, if we set

h(ξ) :=(u1)

2b

a2b

∈ (0, 1),

then by Bernoulli’s law (44) and (78)(61), we have

a2b(h) =

cb

12h + 1

γ−1

, (103)

ρb(h) =

cb

γA(Sb)(12h + 1

γ−1)

1

γ−1

> 0, (104)

n′

ndξ =

−(1 − h)dh

((γ − 1)h + 2)h, (105)

d1(ξ) =n′

n

γh2 + (1 − γ)h − 2

(1 − h)2, (106)

d0(ξ) =2ρb(h)

(1 − h)2

n′

n. (107)

So, by noting that the change of variables (105) is valid if h ∈ (0, 1), then

−∫ ξ

0d1(ξ) dξ = G(h(ξ))

:=∫ h(ξ)

h(0)

γh2 + (1 − γ)h − 2

h(1 − h)((γ − 1)h + 2)dh ≤ 0 (108)

and

27

D1(ξ) = exp(G(h(ξ))) ∈ (0, 1]. (109)

Hence we get

∫ 1

0D1(ξ)d0(ξ) dξ

= −∫ h(1)

h(0)

2ρb(h) exp(G(h)))

h(1 − h)((γ − 1)h + 2)dh. (110)

Note that by (N) (n(1) < n(0)), there must hold h(1) > h(0), so the aboveintegral is negative.

4. For (97), similar to step 2, we may introduce a potential Φ(2) such that

w(2) =∂ξΦ

(2)

D1e1, w

(2)2 =

∂ηΦ(2)

D1. (111)

The condition w(2)2 = 0 on ξ = 0 and D2(0) = 1 guarantee that Φ(2) satisfies

the periodic conditions on η = 0, η = 1. We need only to solve the followingequi-valued surface problem to determine w(2), w

(2)2 :

∂ξ

(

D2

D1e1∂ξΦ

(2))

+ ∂η

(

D2e2

D1∂ηΦ

(2))

= f2 in Ω;

Φ(2) = 0 on ξ = 0,

Φ(2) = e on ξ = 1,

Periodic conditions on η = 0, η = 1,∫ 10 ∂ξΦ

(2)(0, η) dη = 0.

(112)

Here e is also a number to solve.

By standard theory of equi-valued surface problem for elliptic equations (see,for example, [16]), (112) is uniquely solvable, and the following estimate holds:

∥

∥

∥Φ(2)∥

∥

∥

C3,α(Ω)+ |e| ≤ C ‖f2‖Cα(Ω) . (113)

5. The estimate (95) is then easily obtained, and Lemma 11 is proved.

6 Existence of Perturbed Subsonic Flows

In this section we prove Theorem 5, 6 and 7 by Banach fixed point theorem.

28

6.1 Construction of a Nonlinear Mapping

Let Oδ be defined as in (93). For any U ∈ Oδ, we will construct a nonlinearmapping T on Oδ which maps U to U in the following several steps by usingproblem (P1)(P2)(P3), and one easily sees that the fixed point of this mappingT is exactly a solution to problems (NP1)(NP2)(NP3).

1. It is easy to verify that Oδ is a closed subset of the Banach space C1,α(Ω; R4).

2. For any fixed U ∈ Oδ, we denote

c = |u|2/2 + a2/(γ − 1). (114)

Then there holds

|c − cb| ≤ Cδ. (115)

By (P3) we obtain a constant η0, and with this η0, due to (92), we have astrictly monotonic C2,α function θ = θ(η) with θ(0) = 0, θ(1) = 2π. Let

p0(η) := p0(θ(η)), ρ0(η) := ρ0(θ(η)). (116)

Then (94) holds as explained in section 4.

3. Now, recalling (76)(88), we will take

g = p0(η) − pb(0), f1 = f1(U, c), f2 = f2(U) (117)

as nonhomogeneous terms in (P1). Direct calculation and (94) show that

‖f1‖Cα(Ω) ≤ Cδ2, (118)

‖f2‖Cα(Ω) ≤ Cδ2, (119)

‖g‖C1,α[0,1] ≤ Cε (120)

with a positive constant C depending only on the background solution. Thusby Lemma 11 we infer not only unique existence of (w, w2, c), but also theestimate

‖w‖C1,α(Ω) + ‖w2‖C1,α(Ω) + |c − cb| ≤ C(δ2 + ε). (121)

29

4. Now by (P2), we have u, v, ρ. Since these expressions are analytical, weeasily get the following estimates by differential mean value theorem and(94)(121):

‖ρ − ρb‖C1,α(Ω) + ‖u − ub‖C1,α(Ω) + ‖v‖C1,α(Ω) ≤ C(δ2 + ε). (122)

Therefore we have obtained U and it also satisfies

∥

∥

∥U − Ub

∥

∥

∥

C1,α(Ω;R4)≤ C(δ2 + ε). (123)

By choosing

δ = 2Cε, ε0 ≤ 1/(2C)2, (124)

then the mapping T : U 7→ U is well defined on Oδ and maps Oδ into Oδ.

6.2 Contraction of the Nonlinear Mapping

To apply Banach fixed point theorem, for any U (i) ∈ Oδ (i = 1, 2), we need toshow that

∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω)≤ 1

2

∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω). (125)

1. We denote c(i) to be the Bernoulli constant corresponding to U (i). SinceU (i) ∈ Oδ, one easily has

|c(1) − c(2)| ≤ C ′∥

∥

∥U (1) − U (2)∥

∥

∥

C(Ω). (126)

Let η(i)0 and θ(i) = θ(i)(η) be the number and function determined by U (i)

according to (P3) and (92) respectively. Then there holds

|η(1)0 − η

(2)0 | ≤ C ′

∥

∥

∥U (1) − U (2)∥

∥

∥

C(Ω). (127)

Since

dθ(i)(η)

dη=

η(i)0

n(0)√

1 + n′(0)2ρ0(θ)u(i)(0, η)(128)

30

and ρ0(θ) is bounded away from zero, there holds

∥

∥

∥θ(1)(η) − θ(2)(η)∥

∥

∥

C2,α[0,1]≤ C ′

∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω). (129)

So for p(i)0 (η) = p0(θ

(i)(η)), ρ(i)0 (η) = ρ0(θ

(i)(η)), and let

g(i) = p(i)0 (η) − pb(0), (130)

we have

∥

∥

∥g(1) − g(2)∥

∥

∥

C1,α[0,1](131)

=∥

∥

∥p(1)0 − p

(2)0

∥

∥

∥

C1,α[0,1]≤ C ′ε

∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω),

∥

∥

∥ρ(1)0 − ρ

(2)0

∥

∥

∥

C1,α[0,1]≤ C ′ε

∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω). (132)

2. Now let

f(i)1 = f1(U

(i), c(i)), f(i)2 = f2(U

(i)), i = 1, 2. (133)

Straightforward computation shows that

∥

∥

∥f(1)j − f

(2)j

∥

∥

∥

Cα(Ω)≤ C ′ε

∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω)j = 1, 2. (134)

By Lemma 11, we then see

∥

∥

∥w(1) − w(2)∥

∥

∥

C1,α(Ω)+∥

∥

∥w(1)2 − w

(2)2

∥

∥

∥

C1,α(Ω)+ |c(1) − c(2)|

≤C ′ε∥

∥

∥U (1) − U (2)∥

∥

∥

C1,α(Ω). (135)

3. Now using mean value theorem to (P2), one then easily gets the estimate(125) by choosing ε0 further small. This finishes the proof of Theorem 5.

Proof of Theorem 4.2. We only need to note that, more specifically, theconstant C, C ′ depend only on the distance of the background solution Ub

to the sonic surface

U : |u| = a

and the stagnation–vacuum surface

U :

ρu = 0

in the phase space to guarantee uniform ellipticity and positivenessof ρ, u. 2

31

6.3 Proof of Theorem 7

We now sketch out the proof of Theorem 7.

1. The first difference is the equation (89). Now, for the Bernoulli constantgiven, the function H(U) in (80) is actually depending on p, w and ρ0 and wehave

∂ρ0H(Ub) = d0(ξ) :=

2γcbpba2b

(a2b − (u1)2

b)2ρb(0)

6= 0. (136)

Then by setting d0 := n′d0/n, and using the boundary condition of ρ at ξ = 0,(39) is

∂ξ(D1w2) − ∂η(D1e1w) − D1d0 · (ρ(η) − ρb(0) + e) = f1(U, e),

or

∂ξ(D1w2) − ∂η(D1e1w) − D1d0e = f1(U, e) (137)

with f1 = f1 + D1d0 · (ρ(η) − ρb(0)).

We may then formulate an elliptic problem like (NP1) and solve its linearizedversion by adjusting e as done before.

2. To solve the nonlinear problem, the construction of a nonlinear mappingshould be modified as follows.

Now define

Oδ :=

U = (u, v, p, ρ)t : U is periodic with respect to η with period 1,

and ‖U − Ub‖C1,α(Ω;R4) ≤ δ,

(138)

and

Kδ = e ∈ R : |e| ≤ δ. (139)

For any U ∈ Oδ and e ∈ Kδ, we first use (P3) to solve η0:

η0 =n(0)

√

1 + n′(0)2∫ 2π0 (ρ0(θ) + e) dθ

∫ 10

dηu(0,η)

; (140)

32

then determine θ = θ(η) by using (90) with η0 there replaced by η0. So we getp0(η) and ρ0(η).

Now we can solve corresponding problem (P1) to obtain w, w2 and e.

Then in problem (P2), by setting

ρ = (ρ0(η) + e)

(

w2 + pb

p0(η)

) 1

γ

(141)

and substituting it in the other expressions in (P2), we may obtain further uand v.

One may show that the above procedure established a contractive mappingon Oδ ×Kδ for some δ = Cε if ε < ε0 is sufficiently small.

This finishes the proof of Theorem 7.

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