THE TRANSONIC SHOCK IN A NOZZLE FOR 2-D AND 3-D COMPLETE EULER SYSTEMS

Transcription

1 THE TRANSONIC SHOCK IN A NOZZLE FOR -D AND 3-D COMPLETE EULER SYSTEMS Zhouping Xin Department of Mathematics and IMS, CUHK, Shatin, N.T., Hong Kong Huicheng Yin Department of Mathematics and IMS, Nanjing University, Nanjing 0093, China The Institute of Mathematical Sciences, CUHK, Shatin, N.T., HongKong Abstract In this paper, we study a transonic shock problem for the Euler flows through a class of -D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging or converging part. If the supersonic incoming flow is symmetric near the divergent or convergent part of the nozzle, then, as indicated in [], there exist two constant pressures P and P with P <P such that for given constant exit pressure P e P,P, a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock is completely determined by P e. Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the Multi-dimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of a unsteady symmetric transonic shock under the exit pressure P e when the initial unsteady shock lies in the symmetric diverging part of the -D or 3-D nozzle. On the other hand, it is shown that a unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle. Keywords: Steady Euler equation, unsteady Euler equation, supersonic flow, subsonic flow, transonic shock, nozzle, Cauchy-Riemann equation Mathematical Subject Classification: 35L70, 35L65, 35L67, 76N5. Introduction and the main results This is a continuation of our studies on the transonic shock problem in a nozzle [8-30]. In [9-30], under the assumptions that the flow is steady, isentropic and irrotational, we use the potential equation to study the well-posedness or ill-posedness of a transonic shock to the steady flow through a general -D or 3-D slowly variable nozzle with a large exit pressure induced by the appropriate boundary condition on the exit. In [8], the ill-posedness results in [9-30] was extended to the -D complete Euler flow case when the nozzle is arbitrary but slightly curved. However, for a suitably curved -D nozzle with symmetric supersonic incoming flows, as indicated in Section 47 of [], it is shown in Theorem 5. of [8] that there exist two constant pressures P and P with P <P which depend only on the incoming flow and the shape of the nozzle, such that if the exit pressure P e P,P, then for the -D complete steady Euler system, a unique symmetric transonic * Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4040/06P, CUHK-408/04, and RGC Central Allocation Grant CA05/06.SC0. Yin is supported by the National Natural Science Foundation of China No and the National Basic Research Programm of China No.006CB Typeset by AMS-TEX

2 shock exists in the diverging or converging part of the nozzle. In this paper, we first study the well-posedness of the steady transonic shock problem when a steady symmetric supersonic incoming flow goes through a slightly curved -D or 3-D nozzle whose diverging or converging part is symmetric with an appropriately given constant exit pressure at the exit of the nozzle. Although the existence and uniqueness in the symmetric class can be established quite easily by theory for ordinary differential equations, the uniqueness of such a symmetric transonic shock in the multi-dimensional setting requires more delicate analysis. Next, we focus on the unsteady transonic shock problem. More precisely, for symmetric unsteady supersonic incoming flows through a symmetric De Laval nozzle with an appropriate constant pressure at the exit of the nozzle, we will establish the global existence, stability and the long time asymptotic behavior of a unsteady symmetric transonic shock in a nozzle when the initial shock lies in the diverging part. On the other hand, it is shown that a unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the converging part as observed in physical experiments and numerical computations. The m-dimensional complete compressible Euler system can be written as t ρ divρu =0, t ρudivρu u P =0, t ρ e u div ρ e u P u =0,. where u =u,,u m is the velocity, ρ, P, e and S represent the density, the pressure, the internal energy and the specific entropy respectively. Moreover, the equations of states, P = P ρ, S and e = eρ, S, are assumed to be smooth such that ρ P ρ, S > 0 and S eρ, S > 0 for ρ>0. For convenience, we sometimes write the equations of states as ρ = ρp, S and e = ep, S. In the case of the ideal polytropic gases, the equations of states read as P = Aρ γ e S cv, and e = P γ ρ, here A, c v and γ are positive constants, and <γ<. The sound speed c is given by c = ρ P ρ, S. In the case of steady flows, the system. is reduced to divρu =0, divρu u P =0, div ρ e u P u =0.. We now describe the classes of nozzles and supersonic incoming flows we are going to study. Let X 0 be any fixed positive constant. First, for -D case, it is assumed that the walls of the nozzle are given by two curves Γ and Γ, which are C 4 regular for r [X 0,X 0 ] with r = x x x. Furthermore, we assume that Γ i can be decomposed into two curves Π i and Π i such that Π and Π include the converging part of the nozzle while Π and Π form a two-dimensional angular section with its vertex at the origin 0, 0, more precisely, Π i is given by x = i x tgα 0 for r X 0 4,X 0,α 0 0, π,

3 so that Π and Π form a portion of the diverging part of the nozzle, see Figure. Figure Similarly, for the 3-Dimensional case, the wall of the nozzle, Γ, is assumed to be C 4 regular for r [X 0,X 0 ]r = x = x x x 3, such that Γ is a disjoint union of Π and Π with Π being part of a circular cone surface given by x x 3 = x tg α 0 for x > 0, r [ X 0 ] 4,X 0, where α 0 is a positive constant, α 0 0, Π. See Figure. Figure We will assume that the steady supersonic incoming flow, ρ 0,u 0,S 0 x, is C3 -smooth and symmetric near r = X 0, i.e., ρ 0 x =ρ 0 r, u 0 x =U 0 rx r, S 0 = constant near r = X 0. We now focus on the well-posedness of steady transonic shock solution. First, by an analyzing some systems of ordinary differential equations as in Section 47 of [], one can obtain the following existence and uniqueness of symmetric transonic shock solutions: 3

4 Theorem.. Existence Let m-dimensional nozzle and the steady supersonic incoming flow be given as above. Then there exist two constant pressures P and P with P <P, which are determined by the coming flow and the nozzle, such that if the end pressure P e P,P, then the system. has a unique symmetric transonic shock solution { P P, u, S = 0 r,u 0 x,s 0, for r<r 0, P 0 r,u 0 x,s 0, for r>r 0, here u 0 x =U 0 r x r, S 0 is a constant, and P 0 r,u 0 r is C3 smooth. Moreover, the position r = r 0 with r 0 X 0,X 0 and the strength of the shock are uniquely determined by P e. Remark.. Although the proof of Theorem. can be carried out as sketched for the -D case in [8], yet for completeness, we still give a detailed proof, which yields more useful estimates of the solutions that will be used in the later analysis. Next, we turn to the uniqueness of the symmetric transonic shock solution constructed in Theorem. in a large class which is not necessarily symmetric. Assume that the shock Σ is given by x = ξx with x = x,,x m, and the flow behind the shock is denoted by ρ,u,s x. The Rankine-Hugoniot conditions on Σ read: [, x ξx ρu] =0, [, x ξx ρuu], x ξx t [P ]=0, [, x ξx ρ e u P.3 u]=0, where ρ = ρp, S. Then entropy condition requires see [] P x >P x on Σ..4 At the exit of the nozzle, one poses the following end pressure condition P x =P e for x = r = X 0,.5 here the constant pressure P e is given as in Theorem.. A natural boundary condition on the wall of the nozzle, Γ, is the no-flow condition, which reads as u tg α 0, x x =0 on Π,.6 for 3-D, and u = f ix u on Π i,.7 where x = f i x i x tg α 0 for -D. Let Ω be the subsonic region, i.e., Ω = {x : ξx <x < X 0 x, x <x tg α 0 }, D is the projection of Σ onto x -plane, L = Σ Γ, and x 0 Π be a fixed point. Finally, we assume that X 0 is suitably large and α 0 is sufficiently small so that X 0 tg α 0 =, η <α 0 <η 0.8 hold, where η 0 is a small constant. We note that the condition.8 implies that Π is close to the cylinder x = for r [X 0 4,X 0 ]. We now can state our main uniqueness theorem. Theorem.. Uniqueness Let the assumptions in Theorem. and.8 hold. Then the steady transonic shock problem,.-.7, has no more than one pair of solution P x,u x,s x; ξx with the following properties: 4

5 i There exists positive constants δ 0 0, and ε such that ξ C 3,δ0 D, x 0 Π Σ, i.e., x 0 = ξx, and C ξx r0 x ε,.9 3,δ 0 D where r 0 is given as in Theorem., and ε depends only on η 0 and the incoming supersonic flow. ii P,u,S x C,δ0 Ω C 3 Ω such that P x ˆP 0 r,u x û 0 x,s x S 0 C,δ 0 Ω ε,.0 here û 0 x =Û 0 r x r, and ˆP 0 r, Û 0 r stands for the extension P 0 r,u 0 r in Ω. Remark.. For 3-D, the uniqueness holds with less regularity. Indeed, it suffices to assume that ξ C,δ0 D such that C ξx r0 x ε.9,δ 0 D and P,S x C,δ0 Ω C 3 Ω, u x C,δ0 Ω C Ω such that P x ˆP 0 r,s x S 0 C,δ 0 Ω u x û 0 x C,δ 0 Ω ε..0 This is explained in more details in Appendix B. Remark.3. It can be shown that the compatibility of the boundary conditions.3 and.6-.7 holds at the corner L = Σ Π see Lemma 4. in 4 and Lemma 6. in 6. Thus, the assumptions on the regularities of the solution P x,u x,s x; ξx in Theorem. are plausible. This follows from Remark. in [8] or one can see [-3], [9-0], and the references therein for -D. In the 3-D case, an explanation is given in the Appendix B. It is interesting that such a compatibility condition is satisfied naturally for any C Ω -regular solution in contrast to the general unsteady shocks [3-4]. Remark.4. It can be verified see that P 0 r,u 0 r in Theorem. can be the domain {x : X 0 4 r X 0, x x tg α 0 }, so that ˆP 0 r, Û 0 r in Theorem. is well-defined. Remark.5. Consider a general -D nozzle. Let the diverging part of the walls of nozzle, given by Γ i : x = f i x, i =,, be curved slightly and intersect the shock surface Σ at the point x i =x i,x i. Then a necessary condition for the existence of a weak transonic shock solution P x,u x,s x; ξx C Ω is that f i xi =0. This implies that, in general, one cannot expect the existence of a C Ω regular transonic shock solution in the diverging part of a -D De Laval nozzle. The proof of this fact is given in Appendix A. Remark.6. It follows from the proof of Theorem. that one can actually obtain a more general uniqueness result even if the supersonic coming flow is not symmetric and the nozzle walls are general but slightly curved for r 0 δ<r<x 0 with a fixed constant δ>0. Remark.7. In order to illustrate the validity of C,δ0 regularity of the solution to., we require that the function GM0 0, where GM 0 = γm 0 μ M0 M0 = U0 r 0 cρ r 0,S0 and μm 0 = U 0 r 0 U0 r 0 γ μ M 0 3μM 0 μm 0 with, see Lemma 6. of 6 for more details. It follows from this and Appendix B that the assumptions on the regularities of solution P x,u x,u x,u 3 x,s x; ξx,x 3 in Theorem. are plausible. Remark.8. For the unsteady multidimensional compressible Euler systems, A.Majda in [3-4] has shown the existence and stability of a multidimensional shock under the appropriate compatibility conditions on the discontinuous initial data along the initial shock curve. But for the steady transonic multidimensional Euler system., the compatibility condition will be satisfied naturally for any C Ω regular solution see Lemma 6. and Remark 6.. This is an interesting fact. 5

6 Analogously, we can study the well-posedness of steady transonic shock in a nozzle with a symmetric converging part. Indeed, consider a m-d m =, 3 nozzle whose wall contains a straight section given by x = x tg α 0, x < 0, α 0 0, π, for X 0 r x <X 0. In addition, we assume that the supersonic incoming flow is C 3 -smooth, isentropic, and symmetric near r = X 0 3 4, which can represented as P 0 x,u 0 x = P 0 r, U 0 r x r near r = X Then as a counter part of Theorem.-., we can show Theorem.3. Let the nozzle and the supersonic incoming flow be as described above. Assume further that the flow is isentropic. Then for suitably large X 0 > 0, there exist two constant pressures P and P with P <P, such that for P e p,p, the steady transonic shock problem.-.8 has a unique solution given by { P P, ux = 0 r,u 0 x, r > r 0, P 0 r,u 0 x, r < r 0, where U 0 x = U 0 r x r, r 0 X 0,X is uniquely determined by P e, and P 0 r,u 0 r is C3 -smooth. Remark.9. In Theorem.3, the uniqueness is in the class which can be described analogously as in Theorem.. Remark.0. The proof of Theorem.3 is similar to that of Theorem.-., for completeness, we give the sketch in Appendix C. Next, we turn to the problem of dynamical stability of a steady symmetric transonic shock, constructed in Theorem. and Theorem.4, under small generic unsteady symmetric perturbations for simplicity in presentation, we will only study the isentropic flows. We start with transonic shocks in a symmetric expanding nozzle. Thus suppose that the initial flow is a small perturbation of the steady symmetric transonic shock solution, ρ ± 0 r,u± 0 r for r [X 0 4,X 0 ], given in Theorem., i.e. ρ ± 0,r=ρ ± 0 rερ± r, U± 0,r=U ± 0 rεu ± r, r [X 0 4,X 0 ],. where ρ ± 0 r,u± 0 r is defined in Theorem. with ρ± 0 r =P ± 0 r A γ, and ρ r,u r C 0X 0 4,r 0 and ρ ± r,u± r C 0r 0,X 0, and ε>0 is a suitably small constant. We will impose the following unsteady boundary condition at the entry and the exit of the nozzle: ρ,u t, r = X 0 =ρ 0 4,U 0 X 0 ερ 4,U t,. and ρ t, r = X 0 =ρ e ερ t,.3 here ρ t,u t; ρ t C 00, and ρ e = Pe A γ with Pe as given in Theorem.. Let the unsteady shock front Σ be denoted by r = rt and the flow field before and behind the shock be given by ρ,u t, r and ρ,u t, r respectively. It then follows from. for isentropic flows for r rt, { t ρ ± r ρ ± U ± m r ρ ± U ± =0, t ρ ± U ± r ρ ± U ± P ± m r.4 ρ ± U ± =0. On the shock front Σ, the Rankine-Hugoniot conditions become { [ρ]r t [ρu] =0, [ρu]r t [ρu P ]=0. 6.5

7 In addition, ρ ±,U ± t, r should satisfy Lax s entropy condition [7]: λ ρ,u t, rt0<r t <λ ρ,u t, rt 0, r <λ ρ,u t, rt 0.6 with λ ρ, U =U cρ and λ ρ, U =U cρ. Then we have the following nonlinear stability result for a transonic shock in an expending nozzle: Theorem.4. Global Existence and Dynamical Stability Consider the problem of unsteady transonic shocks in an expanding symmetric nozzle as described above. Assume that X 0 > 0 is suitably large and the steady transonic shock ρ ± 0,U± 0 r, r [X 0 4,X 0] is weak in the sense that 0 < minu 0 ± r 0 cρ ± 0 r 0 maxu ± r 0 cρ ± 0 r 0 <δ 0 for a suitably small positive constant δ 0. Then there exists a positive constant ε 0 such that for ε ε 0, the initial-boundary value problem.-.6 has a unique global ρ ±,U ± ; rt with the property that rt C [0,, and ρ ±,U ± t, x is C -smooth for r rt. Furthermore, the location r = rt of the shock front and the flow field after the shock, ρ,u t, r tend to r = r 0 and ˆρ 0, Û 0 r respectively with a rate of decay as t, here ˆρ 0, Û 0 r denotes the extension of ρ 0,U 0 r for r [X 0 4,X 0 ]. Remark.. Theorem.4 shows that a weak steady symmetric transonic shock in an expanding symmetric nozzle globally in time nonlinear stable for generic unsteady symmetric perturbations with prescribed pressure condition at the exit of the nozzle. Furthermore, it is remarkable that the solution is globally in time piecewise smooth and there are no other discontinuities in the solution besides the main perturbed transonic shock, which are in sharp contrast to the theory if Cauchy problems in [-, ]. Remark.. The boundary condition. guarantees the global existence of a shock. Otherwise, other singularities may form see [6], [33] and so on. Remark.3. Since the isentropic compressible Euler systems.4 are used to describe the transonic flow, then it is plausible to require that the shock is weak in the sense that although U0 r 0 >cρ 0 r 0 and U 0 r 0 <cρ 0 r 0, U0 r 0 cρ 0 r 0 and cρ 0 r 0 U 0 r 0 are suitably small. Remark.4. The rate of decay to the steady transonic shock stated in Theorem.4 is not optimal. In fact, it follows from the proof of Theorem.4 that for any positive m, there exists a positive constant ε 0 depending only on m such that if ε<ε 0, then the solutions, ρ t, r,u t, r; rt in Theorem.4, tends to ˆρ 0 r, Û 0 r; r 0 as t approaches to infinity with a rate of order t m. Finally, we study the instability of a m-d steady symmetric transonic shock in a symmetric converging nozzle as given in Theorem.3 under generic unsteady small perturbations. For convenience, in this part of the presentation, we will use the variable r = r instead of Ω, and denote the states before and behind the shock by ρ, Ũ t, r and ρ, Ũ t, r respectively. As in Theorem.4, the initial data is assumed to be a small perturbation of the steady symmetric transonic flow ρ ± 0 r,u± 0 r for r [ X 0 3 4, X 0], i.e., ρ ±, Ũ ± 0, r =ρ ± 0,U± 0 rε ρ±, Ũ [ X ± r, r 0 3 ] 4, X 0,.7 where ε is small positive constant, ρ ± 0,U± 0 r is given in Theorem.3 with ρ± 0 r ± =P 0 r A γ, and ρ, Ũ C0 X 0 3 4, r 0 and ρ, Ũ C 0 r 0, X 0. In addition, the boundary conditions at the entrance and the exit of the nozzle are imposed as: ρ, Ũ t, X 0 4 =ρ 0,U 0 X ε ρ, Ũt.8 and ρ t, X 0 =ρ e ε ρ t.9 here ρ, Ũ ; ρ C 00, and ρ e = Pe A γ with Pe given in Theorem.3. Denote by r = rt the unsteady shock front Σ. Then it follows from. that { t ρ ± r ρ ± Ũ ± m r ρ ± Ũ ± =0, r rt, t ρ ± Ũ ± r ρ ± Ũ ± P ± m r.0 ρ ± Ũ ± =0, r rt. 7

8 Across the shock front Σ, the Rankine-Hugoniot conditions are { [ ρ] r t [ ρũ] =0, [ ρũ] r t [ ρũ P ]=0, and the Lax s geometrical entropy conditions become. λ ρ, Ũ t, rt0< r t <λ ρ, Ũ t, rt 0, r t <λ ρ, Ũ t, rt0,. with λ ρ, Ũ =Ũ c ρ and λ ρ, Ũ =Ũ c ρ. Then we have the following instability result: Theorem.5. Dynamical Instability Let ρ ± 0,U± 0 r denote a m-d symmetric steady transonic shock solution in a symmetric converging nozzle as described in Theorem.3. Assume that X 0 > 0 is sufficiently large and the strength of the transonic shock is suitably weak. Then there exist appropriately choosen perturbations ε ρ ±, Ũ ± r and ε ρ t, Ũ t; ρ t of the initial-boundary value such that the solution to the problem.7-. is asymptotically unstable in the sense that there is no uniform constant C 0 > 0 independent of ε such that ρ, Ũ t, ˆρ 0, Û C[ rt, X0] rtr 0 r t C 0 ε for all t 0..3 It should be noted that there have been many studies on m-dimensional steady transonic shock waves see [5-8], [-], [7], [5-6], [8-3], [35], and the references therein. In particular, for a flat nozzle of the form N,N 0,b;0,b in 3-D, the existence and uniqueness of a transonic shock for the steady compressible Euler are established under the assumptions that the shock front goes through a fixed point and the pressure condition is given with freedom one. However, as conjectured by Courant-Friedrich s in [], such transonic shock phenomena occur in a class of physically interesting nozzles, such as the De Laval nozzle whose wall cannot be flat, and physically relevant condition at the exit of a nozzle should be a given suitably large pressure. Furthermore, it is of great important to study the effects of geometry of the nozzle and boundary condition, in particular, how to determine the shape and location of the transonic shock front []. In [9-30], for -D and 3-D steady potential equation, we have established the uniqueness of the transonic shock wave pattern as conjectured by Courant-Friedrich s for general slightly curved finite nozzles with arbitrarily given large pressure at the exit of the nozzle, proved the existence of transonic shock wave solutions in such a nozzle for a class pressures induced by appropriate boundary conditions at the exit of the nozzle, and more surprisingly, the problem is ill-posed in general by showing no such piecewise smooth transonic shock wave pattern for a class of nozzles, which include both De Laval type nozzles and the flat nozzles, for arbitrarily given large pressure at the exit. The ill-posedness results for the potential in [9-30] were extended to the transonic shock problem for the full steady compressible Euler system. for flat nozzles or slightly curved nozzles with given pressure at the exit in [8]. In this paper, Theorem., Theorem. and Theorem.3 yield the existence and uniqueness of a steady transonic shock wave pattern for a special class of m-d nozzle with appriately pressure given at the exit of the nozzle. The studies on the unsteady transonic shocks began with the works of Liu [-], where he studied the dynamical stability of transonic shock in a duct by Glimm s method for a quasi-one dimensional model of the form: t ρ x ρu = a x ax ρu, t ρu x ρu P = a x ax ρu, t ρe x ρeu Pu= a x ρeu Pu, ax.4 where E = e u is the total energy and ax is the cross section of the duct. It is shown in [-] that flows along the expanding part of the nozzle are asymptotically stable, while flows with standing shock waves in a 8

9 contracting duct are dynamically unstable by studying weak solutions to Cauchy problems for.4 based on Glimm s random choice method. For some recent generalization of the results in [-], see []. However, these results are different from the results in Theorem.4-.5 in this paper due to the boundary conditions and the structures of the solutions. We now comment on the proofs of the main results. First, we note that the steady compressible Euler system. is hyperbolic-elliptic in the subsonic region, it is challenging to investigate even the fixed boundary value problem for such systems. Thus, to prove the m-d uniqueness of a free-boundary value problem, our main strategy is to decompose the m-d full system. into a second order elliptic equation on the pressure P with some mixed boundary conditions and m first order equations on u and S by using the Bernoulli s law. Based on this decomposition and the Rankine-Hugoniot relations, we are able to use the theory for second order elliptic equations and the characteristic method to estimate P x ˆP 0 r,u x û,0 x,s x S 0 in the subsonic region Ω in terms of u û,0,,u m û m,0 x, which can be estimated by its values on the shock surface Σ and the system. using the method of characteristics. It should be noted that on the shock surface Σ, u û,0,,u m û m,0 x is governed by the Cauchy-Riemann system with a natural boundary condition on L, the intersection of Σ with the wall of the nozzle, so its estimate on Σ can be obtained without much difficulties. These will lead to the proof of Theorem.. Next, we turn to the study on unsteady transonic shocks. In the case of the divergent duct, the keys to the asymptotic stability of the symmetric steady transonic shock are some global in time uniform decay estimates for ρ r, t ˆρ 0 r,u r, t Û 0 r,rt r 0 and its derivatives which can be established by making use of the properties of the background solution P 0 ± r,u± 0 r given in Theorem.. The strategy is similar to that in [8, 3-33]. While for converging nozzle, one of the crucial elements of the analysis for the dynamical instability of the transonic shock is that we can derive an ordinary differential equation on the shock position rt r 0 to show that rt increases rapidly in time, as motivated by the work in [], which can yield the unstable phenomena. The rest of the paper is organized as follows. In, we prove Theorem. and study some useful properties of the steady symmetric shock solutions. In 3, we reformulate the -D problem. with the boundary conditions.3-.7 by some useful decomposition of the 4 4 two dimensional full Euler system. In 4, we establish some a priori estimates on the difference P x ˆP 0 r,u x Û 0 r,u x,s x S 0 ; ξx r 0 x based on the decompositions in 3, which yields the proof of Theorem. for -D. The reformulation of the 3-D problem and the decomposition of the 5 5 full Euler system are given in 5. In 6, using the decompositions in 5, we derive some a priori estimates on P x ˆP 0 r,u x û x,u x û x,u 3 x û 3 x,s x S 0 ; ξx,x 3 r0 x x 3, which yields the proof of Theorem.3 for 3-D as in 4. In 7, we give a reformulation on the problem.4 with the boundary conditions.-.3 and Subsequently, we complete the proof on Theorem.4 in 8. Finally, we prove Theorem.5 in 9. In Appendix A, the stated fact in Remark.4 will be shown. In Appendix B, we will give a detailed explanation on the regularity assumption of solution P x,u x,u x,u 3 x,s x; ξx,x 3 in Theorem.. In Appendix C, we will give a proof on Theorem.3 In what follows, we will use the following convention: OY means that there exists a generic constant C such that OY CY, here C is independent of ε and η 0.. The existence of steady symmetric transonic shock solution In this section, we will sketch the proof of the existence of a steady symmetric transonic solution in Theorem., and list some important properties of such solutions which will be used later. Details of the analysis can be found in [, 0, 5, 34, 8]. The proof of Theorem.. Since we are looking for piecewise smooth solutions of. separated by a transonic shock. We may assume the entropy are piecewise constant S0 and S 0 before and after the shock. Due to the symmetric properties of the incoming flow and the nozzle, we can look for symmetric solutions of 9

10 the form ρ, u, Sx =ρ ± 0 r,u± 0 r x r,s± 0 for r r 0. Then the full steady Euler system is reduced to d dr rm ρ ± 0 U 0 ± =0, d dr U ± 0 hρ ± 0,S± 0 =0,. where hρ, S is the enthalpy such that ρ hρ, S = c ρ, S ρ and c ρ, S = ρ P ρ, S. Let the location of the shock be given by r = r 0 with r 0 [X 0,X 0]. The Rankine-Hugoniot conditions at r = r 0 are [ρ 0 U 0 ]=0, [ρ 0 U0 P 0 ]=0,. [ρ 0 U 0 e 0 U 0 P 0 U 0 ]=0. Now we divide the proof of Theorem.. II into four steps. Step. For the given supersonic state ρ 0 r 0,U0 r 0,S0, then it follows from. that there exists a unique subsonic state ρ 0 r 0,U 0 r 0,S 0 such that. holds. This is given in [, 7] so is omitted here. Step.. has a unique supersonic solution ρ 0 r,u 0 r,s 0 for r [X 0 4,X 0 ]. In fact, due to the radial symmetries of both the nozzle for r [X 0 4,X 0 ] and ρ 0,U 0,S 0 x at r = X 0, the unique smooth solution to. should be radial symmetric and satisfies the following relations: f ρ 0,U 0,r rm ρ 0 ru 0 r C 0 =0, f ρ 0,U 0,r U 0 r hρ 0 r,s 0 C =0 with C 0 =X 0 m ρ 0 X 0 U 0 X 0 and C = U 0 X 0 hρ 0 X 0,S 0. Since du 0 m C = 0 c ρ 0,S 0 dr r m ρ 0 U 0 c ρ 0,S 0, then one has du 0 c ρ 0,S 0 dr = m ρ P ρ 0,S 0 ρ 0 ρp ρ 0,S 0 U 0 r 3 U 0 c ρ 0,S 0, U 0 r c ρ 0 r,s 0 U 0 X 0 c ρ 0 X 0,S 0 > 0 for r X 0..3 This implies that one the interval of existence of ρ 0,U 0,S 0 r, U 0 r and U 0 r c ρ 0 r,s 0 are increasing in r, which, in return, implies that du 0 dr is bounded a priorily. This, together.3, yields that. has a unique supersonic solution ρ 0 r,u 0 r,s 0 for r [X 0,X 0 ]. Step 3.. has a unique subsonic solution ρ 0 r,u 0 r,s 0 for r [r 0 δ 0,X 0 ], here δ 0 > 0isa fixed and small constant. If the assumption.8 holds, then the subsonic solution ρ 0 r,u 0 r,s 0 of. exists uniquely for r [X 0 4,X 0 ]. This can be proved as in Step. Step 4. The end pressure P e = P 0 X 0 is a decreasing function of the shock position r = r 0 for r 0 [X 0,X 0 ]. Indeed, for r 0 [X 0,X 0 ], let ρ 0 r,u 0 r,s 0 r = ρ 0 r,u 0 r,s 0 r 0 for r [r 0,X 0 ]be the unique subsonic solution given in Step 3. It follows from. and. that r m ρ 0 ru 0 r C 0, U 0 r hρ 0 r,s 0 r C 0.4

11 for r [r 0,X 0 ], with hρ, S =eρ, S P ρ,s ρ, and C 0 and C are positive constants determined by the incoming supersonic flow. In particular, the end pressure P e P 0 X 0 is the unique solution of and Note that Hence F P e,s 0 r 0 C 0 X 0 m ρ 0 P e,s 0 r 0 hρ 0 P e,s 0 r 0,S 0 r 0 C = 0.5 F P = ρ 0 X 0 F U S = 0 X 0 ρ 0 X 0 dp e dr 0 γ γ U 0 X 0 C > 0,.6 X 0 P e ρ 0 X 0 ρ S X 0 > 0..7 F F ds = 0 r 0 < 0,.8 S P dr 0 provided that ds 0 r 0 > 0..9 dr 0 One need to verify.9. Since.4 holds at r = r 0 and C and C 0 are independent of r 0, one can get from direct computations that d dr 0 ρ ± 0 r 0U ± 0 r 0 = m r 0 ρ ± 0 r 0U ± 0 r 0, ρ 0 r 0U 0 r 0 d dr 0 U 0 r 0= ρ 0 r 0T 0 r 0 ds± 0 r 0 dr 0 dp 0 r 0 dr 0, with T>0 being the absolute temperature. Thus, ρ 0 r 0T 0 r 0 ds 0 r [ ] 0 du = ρ 0 U 0 r 0 dr 0 dr 0 [ dp0 dr 0.0 ] r 0. Since ds 0 r0 dr 0 = 0. On the other hand, it follows from. and.0 that [ ] [ ] m [ρ 0 U0 du 0 dp0 ]= ρ 0 U 0 r 0 r 0 r 0 dr 0 dr 0. Hence, one obtains from.-. that ds 0 r 0 = m dr 0 r 0 ρ 0 r 0T 0 r 0 [ρ 0U0 ]r 0 > 0.3 here one has used the entropy condition [P 0 ]r 0 > 0 and [ρ 0 U 0 P 0 ]r 0 = 0. Thus, we have shown that the end pressure P e is a strictly increasing function of the shock position r = r 0. We can now complete the proof of Theorem.. For r 0 [X 0,X 0 ], by Step, there exists a unique supersonic flow in [X 0,r 0]. Moreover, it follows from Step and Step 3 that there exist a unique shock at r 0 and a unique subsonic flow in [r 0,X 0 ]. Thus the function F r 0 =P 0 X 0 is well-defined for r 0 [X 0,X 0 ]. By Step 4, F r 0 is a strictly decreasing and continuous function on P 0 X 0. When r 0 = X 0 or r 0 = X 0, one can obtain two

12 different end pressures P and P with P <P. Therefore, by the monotonicity of F r 0, one can obtain a unique symmetric transonic shock for P e P 0 X 0 P,P. Hence, Theorem. is proved. Remark.. By the assumption.7 and the proof of Theorem., it can be checked easily that there exists a constant δη 0 > 0 with δη 0 0 as η 0 0 such that for r 0 r X 0 d k U 0 r dr k d k P 0 r dr k δη 0, k =,, 3. Remark.. It follows from the derivation in Step that one can get an extension ˆρ 0 r, Û 0 r of ρ 0 r,u 0 r for r X 0,X The reformulation of the -D problem To prove Theorem. in the -D case as in [8] and [35], we reformulate the nonlinear problem.-.7 so that one can obtain a second order elliptic equation on P and a system on the angular velocity U. First, due to the Bernoulli s law, for any C solution, the system. in Ω is equivalent to ρ u ρ u =0, u u u hρ,s =0, u u u u P 3. ρ =0, u S u S =0. Next we derive a second order equation on the pressure P from 3.. By the state equation of gas dynamics, we can assume ρ = ρp, S and e = ep, S. For simplicity, set D = u u. Then it follows from the first equation in 3. that D ρ ρ D u u Dρ ρ =0. Since D i u i = i Du i i u i u u, i =,, then combining these with. yields D ρ ρ P ρ P ρ Dρ ρ u u P u P =0. 3. Additionally, in terms of DS = 0 in 3., one can derive that Thus 3. becomes Dρ P,S = P ρ DP and D ρ = P ρ DP P ρ D P. P ρ D P ρ P ρ P ρ P ρ P ρ ρ DP u u P u P =0. 3.3

13 Furthermore, 3.3 can be rewritten as u c ρ,s P u u u c ρ,s P u c ρ,s P u u u c ρ,s c ρ,s DP ρ ρ P ρ ρ P P ρ P ρ ρ c ρ,s P DP u u P u P = Next, we derive a Dirichlet boundary condition for P on the shock Σ. It follows from.3 that ξ x = [ρu u ] [P ρu ], ξx =x 0. Substituting 3.5 into.3 yields on Σ G P,u,u,S [ρu u ][ρu ] [ρu ][P ρu ]=0, G P,u,u,S [ρu u ] [P ρu ][P ρu [ ] ]=0, G 3 P,u,u,S ρu u hρ, S [P ρu ] [ ] ρu u hρ, S [ρu u ]= To derive the relations between P,S and u,u on Σ, we use the polar coordinates { x = r cos θ, x = r sin θ 3.7 and the decomposition { u = U cos θ U sin θ, u = U sin θ U cos θ. Then,. takes the form ρu r ρu θ ρu =0, r r r ρu P r θρu U ρu U =0, r r ρu U r θp ρu r ρu U =0, r ρu u hρ, S θ ρu r u hρ, S U r In addition, for any C solution, 3.9 is equivalent to ρu r ρu θ r ρu r =0, U r U U r θ U rp ρ U r =0, U r U U r θ U r θp ρ U U r =0, U r S U r θ S =0. 3 ρ u hρ, S =

14 Denote the shock Σ by r = rθ in the polar coordinates. Then, the R-H conditions become [ρu ] r θ rθ [ρu ]=0, [ρu P ] r θ rθ [ρu U ]=0, [ρu U ] r θ rθ [P ρu ]=0, [ ] [ ] ρu u hρ, S r θ ρu rθ u hρ, S =0. Thus 3.6 is reduced to G P,U,U,S [ρu U ][ρu ] [ρu ][P ρu ]=0, G P,U,U,S [ρu U ] [P ρu ][P ρu ]=0, [ ] G 3 P,U,U,S ρu u hρ, S [P ρu ] [ ] ρu u hρ, S [ρu U ]= Due to the radial symmetry of the data and the nozzle, the incoming supersonic flow must be symmetric and P,U,U,S P0,U 0, 0,S 0. Then it follows from. and 3. and a direct computation that on r = rθ, ρ 0 r 0U U 0 r 0 P ρ 0 r 0U 0 r 0P P 0 r 0 S ρ 0 r 0U 0 r 0S S 0 =g, ρ 0 r 0U 0 r 0U U 0 r 0 P ρ 0 r 0U 0 r 0 P P 0 r 0 S ρ 0 r 0U 0 r 0 S S 0 =g, ρ 0 r 0e 0 r 0 3 ρ 0 r 0U 0 r 0 P 0 r 0 U U 0 r 0 P ρ 0 r 0U 0 r 0 P ρ 0 e 0 r 0 U 0 r 0P P 0 r 0 S ρ 0 e0 r 0 Sρ 0 r 0U 0 r 0 U 0 r 0S S 0 =g 3, 3.3 where g i = g i U, U U 0 r 0, P P 0 r 0, P P 0 r 0S S 0, S S 0, U U 0 r 0P P 0 r 0, U U 0 r 0S S 0,P 0 P 0 r 0,U0 U 0 r 0 i =,, 3 is smooth on its arguments and g i 0, 0, 0, 0, 0, 0, 0, 0, 0 = 0. Furthermore, it can be verified that the determinant Δ of coefficient matrix in 3.3 satisfies Δ 0. Indeed, for the polytropic gas, one has by a direct computation that ρ 0 P ρ 0 U 0 S ρ 0 U 0 Δ = det ρ 0 U 0 0 r 0 ρ 0 e 0 ρ 0 U 0 P 0 P ρ 0 e 0 U 0 S ρ 0 e 0 U 0 = S ρ 0 r 0U 0 r 0 0 det ρ 0 e 0 P 0 ρ 0 U 0 P ρ 0 e 0 P ρ 0 e 0 U 0 = S ρ 0 U 0 ρ0 e0 P 0 U 0 c ρ 0 r 0 > 0. by use of e = 4 r 0 P γ ρ, ρe = P ρ and Sρ<0

15 Thus, on Σ, it follows from the implicit function theorem that U U 0 r 0= g U,P0 P 0 r 0,U0 U 0 r 0, P P 0 r 0= g U,P0 P 0 r 0,U0 U 0 r 0, S S 0 = g 3U,P0 P 0 r 0,U0 U 0 r An important property of g i is g i = OU OP 0 P 0 r 0 OU 0 U 0 r 0. Roughly speaking, this implies, on the shock, the influence of U on U, P and S can be almost neglected. Next, we derive the boundary conditions of P on the fixed boundaries Γ i : θ = i θ 0. In fact, in terms of the polar coordinates, the boundary condition.7 is equivalent to Thus the third equation in 3.0 implies that U =0 on θ = ±θ n P θ P =0 on θ = ±θ 0, 3.6 here n represents the derivative along the outer normal direction of the nozzle wall. Consequently, P in Ω can be determined by the following boundary value problem u c ρ,s P u u u c ρ,s P u c ρ,s P u c ρ,s P u u c ρ,s c ρ,s DP ρ ρ P ρ ρ P P ρ P ρ ρ DP u u P u P =0, P P 0 r 0= g U,P0 P 0 r 0,U0 U 0 r 0 on r = rθ, n P =0 on θ = ±θ 0, P = P e on r = X Next, we derive an algebraic relation for P,U,U and S so that we can determine U in terms of P,U and S. It follows from the second equation in 3. and the boundary conditions.7 and 3.4 that U r U r θ U hρ,s =0, U = U 0 r 0 g U,P0 P 0 r 0,U0 U 0 r 0 on r = rθ, P = P 0 r 0 g U,P0 P 0 r 0,U0 U 0 r on r = rθ, S = S 0 g 3U,P0 P 0 r 0,U0 U 0 r 0, U =0 on θ = ±θ 0. Let θ = θr, β be the characteristics starting from the point rβ, β for the first order differential operator U r U r θ, that is, θr, β satisfies 5

16 dθr, β = dr U r r, θr, β, U θ rβ,β=β, β [ θ 0,θ 0 ]. 3.9 Integrating the first order equation in 3.8 along θ = θr, β and noting that θ = ±θ 0 is the characteristics of U r U r θ starting from the point r±θ 0, ±θ 0, then we have in Ω U hρ,s r, θr, β = G 0 rβ,β,u rβ,β 3.0 with G 0 rβ,β,u rβ,β = e P 0 r 0 g,s 0 g 3 rβ,β U 0 r 0 g rβ,β U P 0 rβ,β r 0 g ρ P 0 r 0 g,s 0 g rβ,β. 3 Here it is noted that U rβ,β has not been determined yet. Finally, we determine U. It follows from 3.9 that d dr β θ = r θ U U r, θr, β β θ, θ β rβ,β= r β rβ U U rβ,β, β [ θ 0,θ 0 ]. 3. By 3.0, θ U hρ,s r, θr, β θ β = d dβ G 0 rβ,β,u rβ,β 3. and r U hρ,s r, θr, β = = r θ U hρ,s U U θ P P S e P,S θ S r, θr, β θ ρ r r, θr, β β G 0 rβ,β,u rβ,β θβr, θ, 3.3 here βr, θ represents the inverse function of θ = θr, β. In addition, the first equation and the third equation in 3.0 can be rewritten as r U r θu = U r ρ U rρ U r θ ρ, U ru U r θ U = r θp ρ U U r. Combining 3.3 with 3.4 gives r U = h P,U,U,S, r P, θ P, r S, θ S, θ U = h P,U,U,S, r P, θ P, r S, θ S, U r 0, θ 0 =0,

17 here h = Δ Δ 0,h = Δ Δ 0 with Δ 0 = r U, Δ = U r ρ θp U U rρ U rρ U r θρ U r S e P ρ Δ = U 3 r U P,S r S U ru U ρ U rρ U P e P ρ P P e ρ d dβ G 0βr, θ θ βr, θ r θρ U r P,S r P P S e ρ, θ P ρ U U r P,S r S U ru P,S r P d dβ G 0βr, θ θ βr, θ Additionally, it follows from 3.0 and 3.4 that S satisfies the following equation { U r U r θ S =0, S rβ,β=s 0 r 0 g 3 U,P0 P 0 r 0,U0 U 0 r 0 rβ,β. 3.6 Furthermore, 3.5 can be rewritten as r θ = rθ[ρu U ] [P ρu ], r θ 0 =r In order to show Theorem., we need only to treat the uniqueness problem and This will be done in next section. 4. The Uniqueness in -D We now prove the uniqueness of solutions stated in Theorem. for -D. It will be more convenient to change the domain Ω with a free boundary Σ into a fixed domain Q = {y : X 0 <y <X 0, θ 0 <y <θ 0 }.To this end, set y = X 0 y = θ. r rθ X 0 rθ, 4. For simplicity, in Q, we still write P,U,U,S as the state of fluid behind the shock in the new coordinates y =y,y. Noting that r = X 0 ry y, θ = X 0 y r y ry X 0 y y. 7

18 Then the equation 3.7 can be changed as follows with u D c ρ,s D P u u c ρ,s D P D u u c ρ,s D P u c ρ,s D P u D u c ρ,s D DP c ρ,s D ρ ρ D P D ρ ρ D P P ρ P ρ ρ DP u D u D P D u D P =0, P P 0 r 0= g U,P0 P 0 r 0,U0 U 0 r 0 on y = X 0, θ P =0 on y = ±θ 0, P = P e on y = X ry = ry X 0 ry y X 0, u = U cos y U sin y, u = U sin y U cos y, cos y D = X 0 ry X 0 y r y sin y y sin y ryx 0 ry ry y, sin y D = X 0 ry X 0 y r y cos y y cos y ryx 0 ry ry y, U D = X 0 ry X 0 y r y U y U ryx 0 ry ry y. Additionally, it follows from the equation 3.8 that D U hρ,s = The characteristics y = y y,βof D starting from the point X 0,β of 4.3 is given by dy X = 0 ry U dy ryu X 0 y r y U y X 0,β=β., 4.4 Thus it follows from 4.3 that U hρ,s y,y y,β = G 0 rβ,β,u X 0,β. 4.5 As in the derivation of 3.4, one can obtain from 4.3, 4.4 and 3.4 that { yi U = H i P,U,U,S, y P, y P, y S, y S, i =,, U X 0, θ 0 =0, 8 4.6

19 here H i = detãi detã0 for i =,, the 4 4 matrix Ã0 =l,l,l 3,l 4 is defined as l = 0,U, X 0 y r y ry ry X 0, U X 0 ry X 0 y r y U T, ry ry X 0 l = U, 0, ry, U T, ry T l 3 = 0,U, X 0 ry, 0, T l 4 = U, 0, 0, 0 and Ãii =, denotes the 4 4 matrix which is obtained from Ã0 by replacing the i column in Ã0 with the vector l = l 0, l 0, l 03, l 04 T defined as l0 = d dβ G 0 rβ,β,u X 0,β y βy y hρ,s, l0 = d dβ G 0 rβ,β,u X 0,β y βy dy y,β l03 = U ry ρ l04 = θ P ry ρ U U ry, U rρ U ry θρ y hρ,s, dy, where β = βy is an inverse function of y = y y,β. In addition, S solves the following problem { DS =0, S X 0,y =S 0 g 3U,P0 P 0 r 0,U0 U 0 r 0X 0,y. 4.7 Finally, 3.7 can be rewritten as r y = ry [ρu U ] [P ρu, ] r θ 0 =r To validate the regularity assumption in Theorem., we now give two lemmas to ensure the compatibility relations of any C Ω solution at the corned points formed by the shock curve and the nozzle walls. Lemma 4.. Orthogonality Under the assumptions in Theorem., we have r ±θ 0 =0. Namely, the shock curve is perpendicular to the walls of the nozzle. Proof. This fact follows from the third equation in 3. and the boundary condition 3.5 directly since the jump of the pressure is non-zero. Lemma 4.. Compatibility If the assumptions in Theorem. hold, then θ P x i =0, i =,. 9

20 In particular, the first order compatibility condition of the problem 4. at the point x i =x i,x i is satisfied with x =r 0 cos θ 0, r 0 sin θ 0 and x = rθ 0 cos θ 0, rθ 0 sin θ 0. Proof. By Lemma 4., U ± r, ±θ 0 = 0 and 3., one has [ρu ] r±θ 0, ±θ 0 =0, [ρu P ] r±θ 0, ±θ 0 =0, 4.9 [ρeρ, S ρu P ρ, SU ] r±θ 0, ±θ 0 =0 and θ [ρu ] r±θ 0, ±θ 0 =0, θ [P ρ, SρU ] r±θ 0, ±θ 0 =0, θ [ρeρ, S ρu P ρ, SU ] r±θ 0, ±θ 0 =0. This implies at the points r±θ 0, ±θ 0 that ρ θ U U θρ =0, ρ U θu c ρ,s U θ ρ S P ρ,s θ S =0, ρ eρ,s 3 ρ U P ρ,s θ U U eρ,s ρ ρ eρ,s U ρ P ρ,s θ ρ U ρ S eρ,s S P ρ,s θ S = For the polytropic gas, the determinant Δ of coefficients in 4.0 satisfies Δ = ρ U Se c ρ U 0. Thus, θ ρ r±θ 0, ±θ 0 = θ U r±θ 0, ±θ 0 = θ S r±θ 0, ±θ 0 =0. Consequently, θ P r±θ 0, ±θ 0 = 0 and the compatibility condition holds. Now we are ready to prove Theorem. in the -D case. Suppose that the problem and has another solution P,U,U,S ; ry with the corresponding regularities in Theorem.. Set W y =P y ˆP 0 r0 X 0 r 0 y X 0,W y =U y Û 0 r0 X 0 r 0 y X 0, W 3 y =U y,w 4y =S y S 0, Ξy = ry r 0. By 4.8, Lemma 4., the Remark. in and the assumptions in Theorem., one obtains after a careful computation that 4 Ξ y =a 0 y Ξy a i y W i ry,y 4. i= Ξ θ 0 =0, with a 0 y C,δ0 [ θ 0,θ 0 ],a i y C,δ0 [ θ 0,θ 0 ] i 4 satisfying a 0 C,δ 0 a C,δ 0 a 3 C,δ 0 a 4 C,δ 0 Cε δη 0, a C,δ 0 C. It follows from the Granwall s inequality, Lemma 4. and 3.4 that Ξy Cε δη 0 W L Q W 3 L Q W 4 L Q C W L Q

21 and Thus, implies that Ξy C [ θ 0,θ 0] Cε δη 0 W L Q W 3 L Q W 4 L Q C W L Q Ξy C,δ 0 [ θ0,θ 0] Cεδη 0 W C,δ 0 Q W 3 C,δ 0 Q W 4 C,δ 0 QC W C,δ 0 Q, 4.4 here δη 0 > 0 is a generic constant with δη 0 0asη 0 0. Based on 4.4 and the assumptions in Theorem., one can estimate W by 4.. Indeed, 4. implies that here D u c ρ,s D W u u c ρ,s D W = F ry, r y, r y,p, P,U, U,U, U,S, S, W = g U,P0 P 0 r 0,U0 U 0 r 0 on y = X 0, θ W =0 on y = ±θ 0, W =0 on y = X 0, F = k=,;j=,,3,4 yk b k 0jyW j b 3 yξy b 3 yξ y k=, yk b k 05yΞ y D u u c ρ,s D W u c ρ,s D W k=,;j=,,3,4 b k jy yk W j 4 b j yw j with b l ij y,b ij C,δ0 Q and b l ij y C,δ 0 Q b ij y C,δ 0 Q Cε δη 0. Due to Lemma 4., it follows from the known regularity estimates on second order elliptic equations of divergence form with corned boundaries and mixed boundary conditions see [-3], [9-0] and so on that W C,δ 0 C g C,δ 0 b k 0jW j C δ 0 b k j yk W j C δ 0 k=, k=,;j=,,3,4 b k 05yΞ y C δ 0 j 4 k=,;j=,,3,4 j= b j W j C δ 0 b 3 yξy b 3 yξ y C δ 0 Cε δη 0 W C,δ 0 W C,δ 0 W 3 C,δ 0 W 4 C,δ 0 Ξy C,δ 0. Substituting 4.4 into 4.5 yields Next, we estimate W 3. By 4.4, we obtain 4.5 W C,δ 0 Cε δη 0 W C,δ 0 W 3 C,δ 0 W 4 C,δ y y,β β C,δ 0 [X0,X 0; θ 0,θ 0] C 4 W i C,δ 0 Ξy C,δ 0 C i= 4 W i C,δ i= It follows from 4.6 that W 3 satisfies { yi W 3 = H i y, i =,, W 3 0, 0 =

22 here H i y has such a form H i y =d i y y W d i y y W d i 3y y W 4 d i 4y y W 4 d i 0yΞy d i yξ y d i y β y y,β βd i 3y U X 0,β, 8 d i kyw k 4 d i 9yy y,β β with β = βy being the inverse function of y = y y,β, d i k y C,δ0 for k 3 and k=5 3 k=5 d i k C,δ 0 Cε δη 0. Thus, combining the equation 4.8 with the estimate 4.7 yields W 3 C,δ 0 C H C δ 0 H C δ 0 C W C,δ 0 W C,δ 0 W 4 C,δ 0 Cε δη 0 W 3 C,δ 0. For sufficiently small ε and η 0, one has Next, we derive the estimate on W. By 4.5 and the estimates above, we obtain W 3 C,δ 0 C W C,δ 0 W C,δ 0 W 4 C,δ W C,δ 0 C W C,δ 0 W 4 C,δ 0 Cε W 3 C,δ 0 y y,β β C,δ 0 Cε δη 0 W C,δ 0 W 3 C,δ 0 C W 4 C,δ Finally, it follows from the equation 4.7 that W 4 C,δ 0 Cε δη 0 W 3 C,δ 0 y y,β β C,δ 0 Cε δη 0 Combining 4.6 and yields 4 W k C,δ 0 Cε δη 0 k= Thus, for small ε and η 0 we arrive at 4 W k C,δ 0. k= W = W = W 3 = W 4 =0. 4 W k C,δ k= It follows from 4.3 that Ξy =0. Therefore, we can obtain P y = ˆP 0 r0 X 0 r 0 y X 0,U y =Û 0 r0 X 0 r 0 y X 0,U y =0,S y =S 0 and ry =r 0 immediately. This leads to the proof on Theorem. in -D case.

23 5. The reformulation of 3-D Problem As for the -dimension problem in 3, we will use the Bernoulli s law to reformulate the nonlinear problem. with the boundary conditions.3-.7 as a second order elliptic equation on P and four first order equations for u =u,u,u 3 and S. First, for any C solution to. in Ω, it holds that divu Dρ ρ =0, Du P ρ =0, D u hρ,s here D = u u u 3 3, and ρ = ρp,s. Without loss of generality, we consider only the polytropic gases. Then the last equation in 5. is equivalent to u Du γ DP γ ρ P ρ Dρ =0. 5. Combining 5. with the second, third and fourth equations in 5. yields By 5., the first equation in 5. can be rewritten as Thus it follows from 5.4 and 5. that P ρ DDP γp =0, 5. DP = γp ρ Dρ. 5.3 divu DP = γp 3 i u j ju i = It is easy to verify that the equation 5.5 on P is a second order elliptic equation for the subsonic flow. Note that the third term in 5.5 is of the order O u, which can be almost neglected. Next we derive a Dirichlet boundary condition for P on the shock Σ as in 3. In fact, it follows from the third and fourth equations in.8 that i,j= { i ξx,x 3 = Δi Δ 0, i =, 3, ξx 0,x 0 3=x with Δ =[ρu u ][P ρu 3] [ρu u 3 ][ρu u 3 ], Δ =[ρu u 3 ][P ρu ] [ρu u ][ρu u 3 ], Δ 0 =[P ρu ][P ρu 3] [ρu u 3 ], here x 0 =x 0,x 0,x 0 3 Γ is defined in Theorem.3. 3

24 Substituting 5.6 into the other equations in.8 yields on Σ G P,u,S [ρu ]Δ 0 [ρu ]Δ [ρu 3 ]Δ =0, G P,u,S [P ρu ]Δ 0 [ρu u ]Δ [ρu u 3 ]Δ =0, G 3 P,u,S [ρu u hρ, S Δ 0, Δ, Δ ] = 0. As in 3, it follows from a direct computation and the implicit function theorem that on Σ u u,0 x = g u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0, P P 0 r 0= g u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0, S S 0 = g 3u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0, here u i,0 = U 0 r 0 xi r 0 i =,, 3 and g j 0, 0, 0, 0 = 0. Thus, by the assumption.8 and the Remark., we can conclude that g i satisfies g i =OεCη 0 Ou û,0 Ou 3 û 3,0 Oξx,x 3 r 0 x x3, here the generic constant Cη 0 0asη 0 0. This fact also illustrates that on the shock, the influence of u û,0 and u 3 û 3,0 on u û,0, P ˆP 0 and S S 0 can be almost neglected. Next, we derive the boundary condition of P on the cone surface Γ : x x 3 = x tg α 0. To this end, it is convenient to use the standard spherical coordinates r, θ, α, and the corresponding velocity decomposition U = cos αu sin α cos θu sin α sin θu 3, U = sin θu cos θu 3, U 3 = sin αu cos α cos θu cos α sin θu 3, with 0 θ<π and 0 α α 0. Then the system 5. becomes r ρ U r sin α θρ U r αρ U U 3 ρ r ρ U 3 r ctgα =0, r ρ U P r sin α θρ U U r αρ U U U 3 ρ r ρ U U 3 r ρ U U 3 r ctgα =0, r ρ U U r sin α θρ U P r αρ U U U 3 3ρ U r ρ U U 3 r ctgα =0, r ρ U U 3 r sin α θρ U U 3 r αρ U 3 P 3ρ U U 3 r ρ U U 3 r ctgα =0, r U hρ,s ρ U r sin α θ U hρ,s ρ U r α U hρ,s ρ U 3 r ρ U U hρ,s r ctgαρ U 3 U hρ,s = Correspondingly, Γ becomes α = α 0 and the boundary condition.6 reduce to U 3 =0 on α = α

25 Thus, it follows from the fourth equation in 5.9 that n P α P = ρ U ctgα 0 on Γ, 5. here n represents the outer normal of the surface Γ. It follows from the analysis above that P should solve the following problem P ρ D DP γp 3 i u j ju i =0, i,j= P P 0 r 0= g u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0 on x = ξx,x 3, n P = ρ U ctgα 0 on Γ, P = P e on r = X In addition, by 5.,.6, and 5.8, we arrive at the following first order equations on u and S Du P ρ =0, u u,0 x = g u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0 on x = ξx,x 3, 5.3 u x tg α 0 u x u 3 x 3 =0 on x x 3 = x tgα 0 and DS =0, S S 0 = g 3u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0 on x = ξx,x 3, u x tg α 0 u x u 3 x 3 =0 on x x 3 = x tgα It remains to determine u u,0 and u 3 u 3,0. Once the values of u and u 3 on the shock are known, then we can solve the problems 5.3 and 5.4 by the characteristics method to estimate u u,0 and S S 0. Furthermore, by the third and fourth equation in 5., one can estimate u u,0 and u 3 u 3,0 in Ω as well. We now derive a system on u and u 3 on the shock. By , one has u 3u 3 = DP γp DP u ρ u Du u 3 Du 3 u u u u u 3 3u. 5.5 In addition, it follows from 5.6 that Δ 3 Δ ξx 0,x 3,x,x 3 = Δ Δ ξx 0,x 3,x,x 3. This implies 3 Δ Δ ξx,x 3,x,x 3 = This, together with a direct computation making use of 5., yields Δ 3 Δ 0 Δ Δ 0 Δ Δ Δ Δ ξx,x 3,x,x 3. Δ 0 Δ 0 Δ 0 Δ 0 3 u u 3 = F u u,u 3 u, P, S, P0, u 0, 5.6 here F 0, 0, 0, 0, 0, 0 = 0. 5

26 By the boundary conditions.6 and 5.8, we have that on the intersection line l = {x = ξx,x 3 } Γ x u x x u,0 x 3 u 3 x x 3 u 3,0 = g 0u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0, 3 here the function g 0 has the same property as in 5.8. Thus, on x = ξx,x 3, we have u 3u 3 = D P γp DP u ρ u Du u 3 Du 3 u u u u u 3 3u, 3 u u 3 = F u u,u 3 u, P, S, P0, u 0, x u x x u x3,0 u 3 x x 3 u 3,0 = g 0 u u,0,u 3 u 3,0,P 0 P 0 r 0, 3 u 0 u 0 r 0 on l, 5.7 here it should be noted that the position of the intersection l can be exactly estimated in terms of the C Ω regularity of P,u,S and the compatibility condition see Lemma 6. for details. By 5.7, we can obtain some useful estimates on u u,0 and u 3 u 3,0 on the shock see Lemma 6.. Then it follows from the third and fourth equation in 5. that u and u 3 can be determined by the following problems respectively, Du i ip ρ =0, u i = u i ξx,x 3,x,x 3 on x = ξx,x 3, x tg α 0 u x u x 3u 3 =0 on Γ, 5.8 here i =, 3. Therefore, in order to prove Theorem.3, one needs only to study the uniqueness problem of solutions to the equations 5.6, and This will be done in 6. Remark 5.. By the references [4] and so on, if the Cauchy-Riemann equation 5.7 has a C solution, then the solution is unique. Namely, the boundary condition in 5.7 is enough to give a priori estimate on u,u 3. Remark 5.. We can obtain a pressure boundary condition on the general curved nozzle wall Γ for the system.. Indeed, let U be any C -smooth solution to.. If Γ is represented by α = fr, θ with fr, θ C, then the boundary condition.6 can be written as U rf U θ f r sin α U 3 r =0 on Γ. 5.9 Then 5.9 implies that U ru rf U ru θ f r sin α U r U 3 r = h 0 α U, f, f 5.0 with h 0 α U, f, f= U α U 3 rf r U 3 r α U rf U r f α U It follows from the equations for the momentum in 5.9 and 5.0 that rf θ f r sin α U r θf r sin α. r f r P θf r sin α θp r αp = H 0 ρ,u,u,u 3, θ,αu, f, f on Γ. 5. 6

27 Moreover, for the small curved nozzle wall Γ i.e. β r,θ f α 0 is small for 0 β <, here α 0 > 0 is a small constant, 5. is a strictly oblique derivative boundary condition on P. Thus we can extend Theorem. to more general curved nozzles. 6. The Uniqueness in 3-D We now prove Theorem. for 3-D case. As in 4, we transform the domain Ω with a free boundary Σ into a fixed domain Q = {y : X 0 <y <X 0,y y 3 < } by the following transformation y = X 0 x ξx,x 3, X 0 x x 3 ξx,x 3 y i = x i x tgα, 0 i =, 3. For simplicity in presentation, in Q, we still denote by P,u,u,u 3,S and ζy the state of fluid behind the shock and the shock surface equation ξx y,x 3 y in the new coordinates y =y,y,y 3 respectively. With the notation 3 i xi = xi y j yj i =,, 3, the equation 5. can be rewritten as follows j= 3 i i P DP 3 D ρ γp i u j j u i =0, i= i,j= P P 0 r 0= g u u,0,u 3 u 3,0,P 0 P 0 r 0,u 0 u 0 r 0 on y = X 0, ñp = ρ U ctgα 0 on y y3 = y, P = P e on y = X with D = u u u 3 3 and ñ = tgα 0 y y 3 3. Additionally, 5.6 becomes { Δ y ζy = y x Δ Δ 0 y x 3 Δ 0, ζy 0 =x 0, 6.3 here x 0 x 0 3 y 0 =X 0, x 0 tgα, 0 x 0 tgα, 0 x i y =y i x ytgα 0, y x i y =y i tgα 0 y x y, i =, 3, x y = A 0y A 0 yb 0yy X 0 X 0 A 0 y, B 0 y A 0 y =X 0 y ζy, B 0 y =y X 0 y y3tg α 0, y x y = Ay = x yayb yδ 0 x yaybyx y ζyx yx 3 y Δ 0 Ayx y Ayx yδ x 3 yδ, X 0 x y x 3 y, By =Ay ζy with analogous expressions for yi x j yi, j =, 3. Roughly speaking, yi x j y =δ ij Oη 0 Oεi, j =,, 3 holds. 7