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Variational Structure and Two Dimensional Jet Flows for Compressible Euler System with Non-zero Vorticity

In this paper, we investigate the well-posedness theory of compressible jet flows for two dimensional steady Euler system with non-zero vorticity. One of the key observations is that the stream function formulation for two dimensional compressible steady Euler system with non-zero vorticity enjoys a variational structure, so that the jet problem can be reformulated as a domain variation problem. This allows us to adapt the framework developed by Alt, Caffarelli and Friedman for the one-phase free boundary problems to obtain the existence and uniqueness of smooth solutions to the subsonic jet problem with non-zero vorticity. We also show that there is a critical mass flux, such that as long as the incoming mass flux does not exceed the critical value, the well-posedness theory holds true.


Introduction and main results
1.1. Background and motivation. One of the most important problems in fluid dynamics is the study of flows through nozzles and the associated free boundary problems, such as shocks, jets and cavities [28,10,12]. Historically lots of works are done in the irrotataional (zero vorticity) case. The problem of finding a steady irrotational subsonic flows in a two dimensional infinitely long fixed nozzle was first posed by Bers in [12]. For this problem, properties of smooth solutions were studied by Gilbarg [42] via comparison principles, and the well-posedness theory was first established in [62] when the flux of the flows is less than a critical number, cf. [63,36,46] for further generalizations. Compared with the above nozzle problem, the jet problem, where the flow is bounded partially by fixed nozzles and partially by free boundaries, is more challenging. The early investigation of the jet problem relied on hodograph method and complex analysis techniques, which mainly deal with two dimensional incompressible jet flows with restrictive assumptions on the nozzles, cf. [13,11,43] and references therein. A major breakthrough for the jet problem was made by Alt, Caffarelli and Friedman in 1980's: they developed a systematic regularity theory for free boundary problems which are equivalent to domain variation problems [1,5,39]. With the aid of this general theory, they established the well-posedness for two dimensional and three dimensional axisymmetric jet and cavity problems for steady irrotational incompressible and compressible subsonic flows [4,9]. Recently, the subsonic jet problem for irrotational flows with different boundary conditions was investigated in [26] by the method of Alt, Caffarelli, and Friedman. An important progress for the existence of subsonic-sonic jet flows and jet flows in bounded domains has been made in [59,58].
The vorticity plays an important role in understanding fluid dynamics not only mathematically but also physically [54,10]. For example, when the vorticity appears, the steady water waves (free boundary of the incompressible flows) have many new features comparing with the irrotational flows, see [57] and references therein. A major motivation for us to study compressible jet flows with non-zero vorticity comes from the investigation for the flow pattern with a transonic shock inside and a jet issued from the nozzle [28]. In the past two decades significant progress has been made on the stability of transonic shocks in a nozzle. It was proved in [65,66] that the transonic shock problem for isentropic irrotational flows in a finite nozzle is in general ill-posed with the prescribed exit pressure. However, when the vorticity is not zero in the downstream subsonic region, transonic shocks under the perturbation of the exit pressure were shown to be stable, see [49,23,52,61] in different settings. In order to study the flow patterns with both transonic shock and jet, it should be necessary to study the compressible jet flows with nonzero vorticity.
For rotational fluids (non-zero vorticity) the main difficulty in the study of the steady subsonic flow is that the associated Euler system is a hyperbolic-elliptic coupled system. However, for the nozzle problem in terms of the stream function the system can be reduced to a single second order quasilinear elliptic equation with memory [64]. This observation and a careful study for the associated quasilinear elliptic equation allow to establish the well-posedness theory for subsonic flows with non-zero small vorticity in an infinitely long fixed nozzle [64], see also [15,31,32,19,37,34,35,22,14] and references therein for various generalizations.
There are only very few analytical results concerning the effect of vorticity on jet flows. The well-posedness of the steady incompressible jet flows with small non-zero vorticity was established by Friedman in [40]. In this work he crucially relied on the fact that the steady incompressible Euler system can be reduced to a semilinear elliptic equation for the stream function which has a variational structure, hence the jet problem can be formulated as a domain variation problem, see [25] for recent progress in this direction for incompressible jet flows. As far as the jet problems for compressible flows with non-zero vorticity are concerned, to the authors' knowledge, there are no rigorous analytical results available. A major obstacle is that it is unknown whether such problem enjoys a variational structure. In this paper, we give an affirmative answer to this question and prove the existence and uniqueness of the steady compressible jet flows with non-zero vorticity.
1.2. Set-up and main results. Two-dimensional steady isentropic compressible ideal flows are governed by the following Euler system (1) ∇ · (ρu) = 0, ρu · ∇u + ∇p = 0, where u = (u 1 , u 2 ) denotes the flow velocity, ρ is the density, and p = p(ρ) is the pressure of the flow. Suppose that the flow is a polytropic gas, after nondimensionalization, the equation of state can be written as p = ρ γ γ , where the constant γ > 1 is called the adiabatic exponent. The local sound speed and the Mach number of the flow are defined as (2) c(ρ) = p (ρ) = ρ Consider a nozzle in R 2 bounded by two solid boundaries. For simplicity we assume that the nozzle is symmetric about x 1 -axis, but the results and the proofs in this paper work for asymmetric nozzles as well. Let i.e. the mouth of the nozzle is at A := (0, 1) and the nozzle is asymptotically horizontal with heightH at upstream x 1 → −∞ (cf. Figure 1). The main goal of this paper is to study the following jet problem.
Moreover,ρ,ū are uniquely determined by the Bernoulli function B and the flux Q; at downstreamρ = (γp e ) 1/γ and the downstream asymptotics ρ,ū,H are uniquely determined by B, Q and p e . (ii) (Uniqueness) the Euler flow which satisfies all properties of part (i) is unique; (iii) (Critical mass flux) Q c is the upper critical mass flux for the existence of subsonic jet flow in the following sense: either or there is no σ > 0 such that for all Q ∈ (Q c , Q c + σ), there are Euler flows satisfying all properties in part (i) and Remark 1.4. The jet flows obtained in this paper can be regarded as a first step for the study on the whole transonic flow pattern with a transonic shock inside a De Laval nozzle and a subsonic jet issued from the nozzle.
Here we give the main ideas for the proof of Theorem 1. First, if the streamlines have simple topological structure, one can reformulate the Euler system in terms of the Bernoulli function and the vorticity (cf. Proposition 2.1). With this equivalent reformulation, one can reduce the Euler system into a single second order quasilinear equation for the stream function with a complicated memory term coming from the vorticity. Moreover, the equation is elliptic if and only if the flow is subsonic (cf. Lemma 2.4). One of our key observations is that this quasilinear elliptic equation with a memory term also enjoys a variational structure as in the irrotational case (cf. Lemma 3.1). As a consequence, we can formulate the subsonic jet problem with non-zero vorticity as a domain variation problem, so that the framework developed by Alt, Caffarelli, and Friedman can be adapted to prove the Lipschitz regularity of the free boundary. The higher regularity is obtained with the aid of the regularity theory developed for the free boundary problem to inhomogeneous elliptic equations by De Silva, Ferrari and Salsa [29,30]. To solve the jet problem, a crucial step is to show that the solution obtained from the domain variation problem satisfies a monotonicity property, which guarantees the equivalence of the Euler system and the domain variation formulation in terms of the stream function. This is done via comparison principles inspired by the proof in [9]. As long as one has existence and uniqueness of the solution for the jet problem for sufficiently small incoming mass flux, the existence of the critical mass flux can be established by the compactness arguments adapted from [64].
The rest of the paper is organized as follows. In Section 2, the stream function formulations for both the Euler system and the jet problem are established. In Section 3, we give the variational formulation for the jet problem after domain and subsonic truncations. Section 4 is devoted to the study for regularity of the truncated free boundary problem. The monotonicity property of the solution is established in Section 5 , which is crucial for the equivalence of the stream function formulation and the Euler system. In order to remove the domain truncations later, we also prove some uniform estimates for the truncated problem in Section 5. In Section 6, the continuous fit, smooth fit and the far field asymptotic behavior of the solution are established. In Section 7 we remove the subsonic truncation and complete the proof for the existence of solutions to the jet problem with the small flux. The uniqueness of the subsonic jet problem is proved in Section 8. Finally, the existence of a critical mass flux is established in Section 9.
In this paper, for convention the repeated indices mean the summation.

Stream function formulation and subsonic truncation
In this section, we introduce the stream function formulation to reduce the Euler system into a single second order quasilinear equation, which is elliptic in the subsonic region and becomes singular elliptic at the sonic state. In order to deal with the possible degeneracy of the equation near the sonic state, a subsonic truncation is introduced so that the modified equation is always uniformly elliptic.
2.1. The equation for the stream function. First, motivated by the analysis in [64], one has the following equivalent formulation for the compressible Euler system. Proposition 2.1. LetΩ ⊂ R 2 be the domain bounded by two streamlines S 0 and Suppose u satisfies the slip boundary condition u · n = 0 on ∂Ω, and (12) u 2 < 0 inΩ, and the following asymptotic behavior (13) u 1 , ρ, and ∂ x 2 u 2 are bounded, while u 2 , ∂ x 1 u 2 , and ∂ x 2 ρ → 0, as x 1 → −∞.
Then (ρ, u) satisfies the system (1) inΩ if and only if (ρ, u) satisfies are the Bernoulli function, the vorticity, and the enthalpy of the flow, respectively.
Proof. It is easy to see that smooth solutions of the Euler system (1) satisfy (14).
On the other hand, it follows from the first and the third equations in (14) that Therefore, there exists a function Φ such that Thus the Bernoulli's law (the second equation in (14)) is equivalent to This implies that Φ is a constant along each streamline, in particular along S 0 andS 1 . Note that (13) implies ∂ x 2 Φ → 0 as x 1 → −∞. Hence one has On the other hand, it follows from (12) that through each point inΩ, there is one and only one streamline which satisfies and can be defined globally in the nozzle. Furthermore, it follows from the continuity equation (the first equation in (14)) that any streamline through some point inΩ cannot touch the nozzle wall. Indeed, suppose not, then there exists a streamline through (x 0 1 , x 0 2 ) inΩ which passes through (Θ(x 2 ),x 2 ). Due to the continuity equation and the slip boundary condition along each streamline, one has 0 =ˆΘ This contradicts (12). Therefore, one can conclude from (16) that Φ ≡ C in the whole domainΩ. This implies that ∂ x 1 Φ = ∂ x 2 Φ ≡ 0 inΩ, i.e., hold globally inΩ. The above two equations together with the continuity equation are exactly the original Euler system (1).
The Bernoulli's law shows that B(ρ, u) is conserved along each streamline. Hence the Bernoulli function and the stream function are functional dependent. Therefore, there exists a function B such that In order to determine the exact form of B, we first study the flow density and velocity at upstream where the relation between the stream function and Bernoulli function can be determined easily.
Let us digress for the study on the states with the given Bernoulli constant. For the state with given Bernoulli constant s, the density ρ and speed q satisfy Therefore, the speed q satisfies be the critical density and maximum density respectively, and be the critical speed and square of the critical momentum respectively for the states with given Bernoulli constant s. A straightforward computation shows that for given Bernoulli constant s, one has q(s, c (s)) = q c (s) = c( c (s)), where c is the sound speed defined in (2). Furthermore, the flow is subsonic if and only if ρ > c (s), or equivalently q(s, ρ) < q c (s). Let (20) B * := min Proposition 2.2. Let B ∈ C 1,1 ([0,H]) and Q > 0 be the incoming data at upstream in the sense of (5) and (6). Suppose that the flow satisfies the upstream asymptotic behavior (10). Suppose that B * > 0 and B satisfies (  Proof. Suppose that the flow satisfies asymptotic behavior (10), then one has Note thatρ → Q(ρ) is strictly monotone decreasing in ( c (B * ), * (B * )), as Thus as long as Q ∈ (Q( * (B * )), Q * ), where Q * := Q( c (B * )), there exists a unique solutionρ ∈ ( c (B * ), * (B * )) to (24) andū is uniquely determined through (23). Furthermore, if Q → Q * , thenρ → c (B * ).

A straightforward computation gives
Therefore, with the aid of (21) and (22), one has (29) κ Lemma 2.3. Let ρ and ψ satisfy the Bernoulli's law (17). Then ρ can be expressed as a function of |∇ψ| 2 and ψ, where c and * are functions defined in (18), and g : R + × R → R is a function which is smooth in its first component and C 2 in its second component.
Proof. Set The straightforward computations give Now, it is easy to see that with c and * defined in (18) one has (a) → F( , z) achieves its maximum at c (B(z)); Thus by the inverse function theorem, if t = F( , z), then there exists a function g, which is smooth in t and C 2 in z, such that where t c is defined in (19). This together with the Bernoulli's law completes the proof of the lemma.
Proof. Let X(s; x) be the streamlines satisfying It follows from the third equation in (14) that ω/ρ is a constant along each streamline.
Hence ω/ρ can be determined by the associated data in the upstream as long as the streamlines of the flows have simple topological structure, which is guaranteed by the assumption u 2 < 0. If x ∈ {ψ = z}, then Expressing the vorticity ω in terms of the stream function ψ and using (28) one has In view of (30) the above equation can be rewritten into (34).
Thus β 0 has uniform upper and lower bounds depending only on γ and B * , and β 1 has a uniform lower bound but blows up when |∇ψ| 2 approaches t c (B(ψ)). Therefore, the equation (34) is elliptic as long as |∇ψ| 2 < t c (B(ψ)), and is singular when |∇ψ| 2 = t c (B(ψ)). This completes the proof of the lemma.
Remark 2.5. In fact, the condition (8) implies that the vorticity at the upstream of the flow is small. Thus it follows from (29) that the right hand side of the equation in (39) is small.

2.3.
Subsonic truncation. One of the major difficulties to solve the equation (34) is that (34) becomes degenerate as the flows approach the sonic state. As in [9] our strategy here is to use a subsonic truncation so that after the truncation the equation (34) is always elliptic.
Let : R → [0, 1] be a smooth nonincreasing function such that For ∈ (0, 1/4), let (s) := (s/ ). We define where t c (B(z)) is defined in (19) and g * is the upper bound for g in (35). The properties of g are summarized in the following lemma.
Lemma 2.6. Let g be the function given in Lemma 2.3 and let g be the subsonic truncation of g defined in (40). Then the function g (t, z) is smooth in t and C 2 in z. Furthermore, it satisfies the following properties: (i) There exist C * , C * > 0 depending on B * and γ such that for all t > 0, (iii) There exist T * and C > 0 depending only on B * and γ such that and Proof. It follows from Lemma 2.3 and the definition of that g is smooth with respect to t and C 2 with respect to z.
To show (42) we note that if 0 < t ≤ t c (B(z)) − the equation (36) and that Then (42) follows from the definition for g (t, z) and (41). To show (43) a straightforward computation gives Since the second and the third terms are nonnegative, then the lower bound in (43) follows from the lower bound for g in (41). Combining (41), (42) and the definition of gives the upper bound in (43).
(ii). If t ∈ (0, t c (B(z)) − ), it follows from the Bernoulli law (17) that A differentiation of (47) with respect to t yields Furthermore, if one differentiates (47) with respect to z and uses (28), then one gets Combining (48) and (49) gives This together with (30) completes the proof for (44).
Let T * := sup z≥0 t c (B(z)). It follows from the definition of t c (B(z)) in (19) and (27) that if the incoming data is subsonic, then T * depends only on γ and B * . If t > T * , then by the definition of , ∂ z g = 0. If 0 ≤ t ≤ T * , using (44) and the explicit computation on t c one obtains (45).

Variational formulation for the free boundary problem
One of the key observations in this paper is that the equation (34) can be written as an Euler-Lagrange equation for a Lagrangian functional, and the jet problem is equivalent to a domain variation problem.
From now on, let Ω be the domain bounded by S 0 and S 1 ∪ ([0, ∞) × {1}) . Noticing that Ω is unbounded, we thus make a further approximation by considering the problems in a series of truncated domains Ω µ,R := Ω ∩ {−µ < x 1 < R}, where µ and R are two large positive numbers. To define the energy functional, we set

Straightforward computations show that
is positive by the ellipticity condition (43).
The existence of minimizers for (52) is proved in Lemma 4.2 for a slightly more general functional. Assuming the existence of minimizers, we now derive the Euler-Lagrange equation for the variational problem (52) in {ψ < Q}: Proof. Let η be a smooth function compactly supported in Ω µ,R ∩ {ψ < Q}. Direct computations give By the definition of G in (50) minimizers of (52) satisfy the equation (55).
Next, by the definition of G in (50) one has This together with (44) gives A substitution of (56) into the expression of ∂ z G (t, z) gives This completes the proof for the lemma.
Next we show that local minimizers of (52) satisfy the free boundary condition in the following domain variation sense.
Lemma 3.2. Let ψ be a local minimizer of the problem (52). Then where H 1 is the one dimensional Hausdorff measure.

Remark 3.3 (Relation between λ and Λ).
If the free boundary Γ ψ := ∂{ψ < Q}∩Ω µ,R is smooth and ψ is smooth near Γ ψ , then it follows from the monotonicity of t → Φ (t, z) for each z and the definition of λ in (54) that Moreover, there is a constant C > 0 depending only on B * and γ such that Indeed, by the definition of G and Φ in (50) and (51), respectively, one has Proof of Lemma 3.2. Let η ∈ C ∞ 0 (Ω µ,R ; R 2 ) and τ ϑ (x) := x+ϑη(x). If |ϑ| is sufficiently small, then τ ϑ is a diffeomorphism of Ω µ,R . Let ψ ϑ (y) := ψ(τ −1 ϑ (y)). Since ψ ϑ ∈ K ψ and ψ is a minimizer, then Dividing ϑ on both sides of (57) and passing to the limit ϑ → 0 yield Using the divergence theorem for (58) we have where we have used the equation of ψ in (55) in the open set {ψ < Q} to get the last equality. This finishes the proof of the lemma.

The existence and regularity for the free boundary problem
In this section we study the existence and regularity of the minimizer, as well as the regularity of the free boundary away from the nozzle.
over the admissible set Here ψ is given and assumed to be continuous with 0 ≤ ψ ≤ Q on ∂D.
Properties of G is summarized in the following proposition.
Proposition 4.1. Let G be defined in (50) . Then the following properties hold: (ii) One has Proof. (i). In view of the definition of G in (50) a straightforward computation yields Thus from (41) and (43)  (ii). To show (63) we note that This, together with the estimate for |∂ z g | in (44) and (45), , which follows from (26) and our extension, then (63) holds true.
In view of (50), one has G(0, Q) = 0. Thus as long as the estimates (65) and (63) hold, one immediately has (64) and (66). This completes the proof of the proposition.
With Proposition 4.1 at hand, the existence of minimizers for the variational problem (52) follows from standard theory for calculus of variations. (62) and (66). Then the minimization problem (59) over the admissible set (60) has a minimizer.
Then it follows from the Poincaré inequality that ψ k − ψ are bounded in L 2 (D). Therefore, there exists a subsequence (not relabeled) satisfying Since G is convex with respect to p, using Fatou's lemma one obtains This means that ψ ∈ K ψ is a minimizer.
4.2. L ∞ estimate and Hölder regularity. The main goal of this subsection is to establish the L ∞ bounds and the Hölder regularity for the local minimizers. Firstly, one can show that the minimizers are supersolutions of the elliptic equation Let ψ be a minimizer for (59). Then ψ satisfies Proof. Since ψ is a minimizer, for any ϑ > 0, where the second inequality follows from the simple fact {ψ + ϑζ < Q} ⊂ {ψ < Q}. Thus taking ϑ → 0 yields (68).
Next, one has the following L ∞ estimate for the minimizer.
The proof for the lower bound is similar. Setψ ϑ := ψ − ϑ min{0, ψ} with ϑ > 0. It is straightforward to check that for ϑ ∈ (0, 1), It follows from (63) and (61) that the measure of the set {ψ < 0} is zero. Hence ψ ≥ 0. This finishes the proof of the lemma. Now we are in position to prove the Hölder regularity for the local minimizers. The proof follows from standard Morrey type estimates.
Lemma 4.5. Let ψ be a minimizer for (59) over the admissible set K ψ . Then ψ ∈ C 0,α loc (D) for any α ∈ (0, 1). Moreover, Proof. Given any B r ⊂ D, let φ ∈ H 1 (B r ) be the solution to Such a solution can be obtained by minimizing the functional´B r G(∇φ, Q)dx among φ ∈ H 1 (B r ) with φ = ψ on ∂B r . It follows from the regularity theory for elliptic equations (cf. for example [45]) that Furthermore, applying the maximum principle one has 0 ≤ φ ≤ Q in B r .
Since ψ is a minimizer, one haŝ It follows from (62) and (65) that Multiplying the equation in (69) with ψ − φ and integrating by parts we obtain Combining the above estimates together one has The desired Hölder regularity then follows from the standard Morrey type estimates (cf. [41, the proof for Theorem 2.2 in Chap. III]).

4.3.
Lipschitz regularity and nondegeneracy. In this subsection we establish the (optimal) Lipschitz regularity of the solution and the so-called nondegeneracy property, which play an important role to get the measure theoretic properties of the free boundary. We start with the following comparison principle for the associated elliptic equation.
Lemma 4.6. Let ψ ∈ H 1 (D) be a supersolution of the equation (67) in the sense of (68) and φ ∈ H 1 (D) be a solution of (67) in the following sense Assume that φ ≤ ψ on ∂D. Then φ ≤ ψ in D as long as δ in Proposition 4.1 is sufficiently small depending on b * andH.
Applying Cauchy-Schwarz and Poincaré inequalities to η(x 1 , ·) for each x 1 (notice that where C > 0 is a universal constant. If δ is sufficiently small depending on b * andH, then necessarily ∇η = 0 in D, which implies that η = 0 in D. This completes the proof of the lemma.
Therefore, solutions to the rescaled equation (73) enjoys the comparison principle as well, provided δ is sufficiently small depending on a, r, b * , andH.
where Dx ,r := x−x r x ∈ D and By the scaling property, without loss of generality, we assume that r = 1 andx = 0 after the translation of the coordinates. The aim is to show that there exists a constant C depending only on b * such that m 0 ≤ Cλ. The proof is divided into four steps.
Step 2. Gradient estimate. We claim if δ = δ(b * ) is sufficiently small, then Indeed, the minimality of ψ * yieldŝ Noting that φ > 0 in B 2 (y) and arguing similarly as in Lemma 4.5 one has By Hölder and Poincaré inequalities, the first term on the right-hand side of (80) can be absorbed by the left-hand side of (80) when δ = δ(b * ) is sufficiently small. Thus one has (79).
Step 3. Poincaré type estimate. We claim the following Poincaré type estimate The proof is similar as [1, Lemma 3.2] or [5,Lemma 2.2]. For the completeness, we provide the details here. For w ∈ B 1 (y), consider a transformation A w from B 2 (y) to itself which fixes ∂B 2 (y) and maps w to y, for instance, On the other hand, by (78), one has Combining (82) and (83) together gives An integration of ξ over S 1 yields A further integration over w ∈ B 1 (y) yields This is the desired estimate (81).
The next lemma investigates the nondegeneracy of the free boundary, whose proof is similar to that for Lemma 4.7 and is based on the minimality and suitable barrier functions.
Lemma 4.8. Let ψ be a minimizer for (59) and let ψ * := Q−ψ. Assume that δ = δ(b * ) is sufficiently small. Then for any p > 1 and any 0 < r < 1, there exists a constant c r > 0 such that for any Proof. As in Lemma 4.7, we assume thatx = 0 and R = 1. By the L ∞ estimate for the subsolution of (72) (cf. [44,Theorem 8.17] with f = g = 0), one has for any r ∈ (0, 1), where C = C(p, b * ). Thus it suffices to show that if M r /λ is sufficiently small, then ψ * = 0 in B r . This is proved in three steps.
Step 1. Upper bound of the energy. We claim that for someC =C(b * ) > 0, Similar as the proof for Lemma 4.7, for µ = µ(b * ) sufficiently large and δ = δ(b * ) small, there is Since ψ * is a local energy minimizer, one has By the convexity of p →G (p, z) one haŝ where the last inequality follows from the bound for ∂ zG in (64) and 0 ≤ φ+s(ψ * −φ) ≤ M r √ r for s ∈ [0, 1]. Multiplying (85) by (ψ * − Φ 0 ) + and integrating by parts (noting This, together with (86) and (87), yields that if δ = δ(b * ) is sufficiently small, then In view of the expression forG and |∇Φ 0 | on ∂B r as well as (61) Combining the above two estimates gives the desired estimate (84).
Step 2. Lower bound of the energy. We claim that there is a constant C = C(b * , r) > 0 such that First, by the trace estimate there exists C = C(r) > 0 such that The Cauchy-Schwarz inequality and the fact 0 ≤ ψ * ≤ M r in B r yield It follows from the above two estimates that The estimate (66), together with Poincaré inequality, gives that if δ is sufficiently small depending on b * , thenˆB Combining all the above estimates together yields (88).
Step 3. Conclusion. From (84) and (88) there is a constant C = C(b * , r) such that Taking M r /λ sufficient small giveŝ This implies ψ * ≡ 0 in B r . Hence the proof of the lemma is complete.  Note that ψ x, * satisfies (73) in B 1 (with r = d(x) and a = 1). By virtue of (62), ∂ i ψ x, * solves a uniformly elliptic equation of divergence form and thus is C 0,α by the De Giorgi-Nash-Moser estimate ( [44]). Hence one has Here the upper bound for |∂ zG | in (74) has been used to get the last inequality.
(ii). If K D is connected and K contains a free boundary point x, then it follows from the Harnack inequality (Theorem 8.17 in [44]) and the connectedness of K that Q − ψ ≤ Cλ in K for some C depending on b * , δ, K and D. Given x ∈ K, either x is closer to the free boundary or x is closer to ∂K. Similar arguments as in (i) give that |∇ψ(x)| ≤ Cλ, where C depends on b * , b * , δ, K, and D.
Remark 4.10. By the boundary estimate for the elliptic equations, ψ is Lipschitz up to the C 1,α portion Σ ⊂ ∂D as long as the boundary data ψ ∈ C 0,1 (D ∪ Σ). Moreover, if K is a subset of D with K ∩ ∂D being C 1,α , K ∩ D is connected, and K contains a free boundary point, then |∇ψ| ≤ Cλ in K.

4.4.
Measure theoretic property and higher regularity of the free boundary. From the Lipschitz regularity (Proposition 4.9) and nondegeneracy of the solutions (Lemma 4.8), one has the following measure theoretic properties of the free boundary.
Proposition 4.11. Let ψ be a minimizer for J (ψ) in (59). Assume that δ = δ(b * ) is sufficiently small. Then one has (2) There is a Borel measure ς ψ such that (3) For any K D, there exist positive constants C ♦ , C ♦ such that for every ball B r (x) ⊂ K with x ∈ Γ ψ ∩ K, The proof is the same as that for [5, Theorem 3.2], so we omit the details here.
In order to get the regularity of the free boundary, the following strengthened estimate for |∇ψ| is needed. Lemma 4.12. Let ψ be a local minimizer. Then for K D, there exist α ∈ (0, 1) and where Λ is defined in (38).
An immediate consequence of Proposition 4.13 is the following corollary.
Corollary 4.14. Let ψ be a local minimizer to (59). Then every blow up limit of ψ at x ∈ Γ ψ ∩ D is a half plane solution with slope λ in a neighborhood of the origin.
The proof is the same as in [5,Corollary 4.4], which we do not repeat here. With Corollary 4.14 at hand, one can make use of the improvement of flatness arguments, which is rather standard now, to show that the free boundary is locally a C 1,α graph for some α ∈ (0, 1). For the proof we refer to [5,Section 5] for the case of minimizers of the energy´D G(∇ψ), and [29,30] for a different argument which does not rely on energy minimality and also allows to deal with the inhomogeneous terms.
Higher regularity of the free boundary follows from [47,Theorem 2]. For the completeness, we state the result here and omit the proof.
Proposition 4.15. The free boundary Γ ψ is locally C k+1,α if G(p, z) is C k,α in its components, and it is locally real analytic if G(p, z) is real analytic.
Remark 4.16. The results in this section hold for general variational problem as long as G satisfies the structure conditions in Proposition 4.1.

Fine Properties for the free boundary problem
The major property of the solutions obtained in this section is that streamlines, in particular the free boundary, are graphs. This is one of the key ingredients to justify the equivalence between Problem 1 and Problem 2.
The key step towards establishing the graph property for the streamlines is to show that the stream function is monotone in the x 1 variable. For that we take a specific boundary value on ∂Ω µ,R for the truncated problem. It should be noted that due to the uniqueness of the problem after removing the truncation (cf. Section 8), the solution to the subsonic jet problem does not depend on the boundary data which one takes for the truncated problems.
In order to state the dependence of solutions on the parameters clearly and make use of the special structure of the equation (55), we resume to use the notations introduced in Section 3. To describe the boundary datum on ∂Ω µ,R , some notations are needed.
The next lemma shows that the minimizers for (52) in K ψ µ,R must stay between ψ µ,R (−µ, ·) and ψ µ,R (R, ·). This in particular implies that for a minimizer ψ, ∂ x 1 ψ is nonnegative at the two sides x 1 = −µ and x 1 = R of Ω µ,R .
Since Ψ (τ * ) ≡ Q + τ for x 2 ≥ 1, then the equality cannot be attained at x 1 = −µ or the nozzle boundary S 1 either. Now one needs to consider the following two cases depending on the heightH: Case 1.H ≥ 1. This means that the free boundary is empty. Since one has Ψ (τ * ) > Q along x 2 = 1, the graph of Ψ (τ * ) has to touch the graph of ψ at an interior point. This contradicts with the strong comparison principle in [55]. Case 2.H < 1. It follows from the strong comparison principle that the graphs of Ψ (τ * ) and ψ touch at a free boundary pointx = (x 1 ,x 2 ). Note that necessarilyx 2 < 1 since Ψ (τ * ) > Q along x 2 = 1. Then by the boundary Hopf lemma (cf. [55]) which is a contradiction.
The next proposition concerns the uniqueness of the minimizer and the monotonicity of the minimizer with respect to x 1 . Proposition 5.3. Under the same assumptions as in Lemma 5.2, ψ is the unique minimizer to (52) An important consequence of the property ∂ x 1 ψ ≥ 0 obtained from Proposition 5.3 is that, the free boundary (if not empty) is an x 2 -graph, which is proved in Proposition 5.6 below. More precisely, Let denote those free boundary points which lie strictly below {x 2 = 1}. Note that if H ≥ 1, then Γ µ,R,Λ = ∅.
Before we prove that the free boundary is a graph, let us digress for the proof of the following non-oscillation property.
The following result on the uniqueness of the Cauchy problem for the elliptic equations, plays a crucial role in the proof for the graph property of the free boundary as well as the study of the various blowup limits.
Proof. The proof relies on the unique continuation property for the uniformly elliptic equations with Lipschitz coefficients. First, it follows from the boundary regularity for the elliptic equation ([5, Lemma 1.7]) that ψ,ψ ∈ C 1,α (D ∪ Σ). Let ϕ :=ψ − ψ. Then ϕ solves with the Cauchy data ϕ = ∂ ν ϕ = 0 on Σ. The Cauchy data allow to extend ϕ into a solution in a neighborhood U ⊃ Σ by setting ϕ = 0 outside of U ∩ D. It follows from the assumptions on G and W that b i ,b i and W ∈ L ∞ , and A(x) is uniformly elliptic and Lipschitz continuous. Thus by the unique continuation property, cf. for example [48,Theorem 1], one has ϕ ≡ 0 in U ∪ D. This means that ψ =ψ.
Now we are ready to show that the free boundary is a graph.
Proposition 5.6. Let Γ µ,R,Λ be as in (93). If Γ µ,R,Λ = ∅, then it can be represented as a graph of a function of x 2 , i.e., there exists a function Υ µ,R,Λ and l µ,R,Λ ≥H such that Proof. It follows from Proposition 5.3 that ∂ x 1 ψ ≥ 0. Hence there is a function Υ µ,R,Λ (x 2 ) with values in [−µ, R] such that for any x 2 ∈ (0, 1), It follows from Lemma 5.2 that one has This implies that there is no free boundary points in the strip R × [0,H). Thus there is a constant l µ,R,Λ ≥H such that the free boundary is a graph x 1 = Υ µ,R,Λ (x 2 ) for x 2 ∈ (l µ,R,Λ , 1).

Continuous fit, smooth fit and the far-field behavior
In the first part of this section, we show the continuous fit property: given a nozzle and incoming data (ρ,ū) at upstream, there exists a momentum Λ > 0 along the free boundary, such that the associated free boundary is nonempty and it matches the outlet of the nozzle. Subsequently, the free boundary join the nozzle wall in a continuous differentiable fashion via a blow-up argument. Finally, we remove the truncation of the domain by letting µ, R → +∞ and study the asymptotic behavior of the jet at downstream x 1 → ∞. The downstream asymptotic behavior plays an important role in proving the equivalence between Problem 1 and Problem 2.
Proof. The proof consists of two steps.
Step 3. Proof for (96). It follows from Lemma 5.4 that there exists σ 0 = σ 0 (ι) > 0, such that for any open subset U D ι,σ 0 one has ψ Λn < Q in U for sufficiently large n. Thus as n → ∞ the limit ψ Λ satisfies the equation in (96) in D ι,σ 0 . The Dirichlet boundary condition ψ Λ = Q follows from ∂ x 1 ψ Λ ≥ 0 and the uniform convergence of ψ Λn to ψ Λ in D ι,σ 0 . Thus one needs only to show that the Neumann condition holds.
(b). The arguments for lim x 2 →1− ∂ x 2 ψ ≤ Λ on D ι,σ 0 ∩ {x 2 = 1} are similar as those in [39,Theorem 6.1 in Chapter 3]. We only outline the proof here for completeness. For each x (0) ∈ D ι,σ 0 ∩ {x 2 = 1} and r > 0 sufficiently small, by the uniform convergence of ψ n to ψ as well as the convergence of the free boundaries, one can find a sequence of domains {K n } such that: (i) ψ n < Q in K n ∩ B r (x (0) ), and ∂D n ∩ B r (x (0) ) is given by a C 1,α graph, which converges to the free boundary ∂{ψ < Q} ∩ B r (x (0) ) = {x 2 = 1}∩B r (x (0) ) in C 1,α as n → ∞; (ii) there is a sequence of points {x (n) } ⊂ ∂K n ∩B r (x (0) ), such that ψ n (x (n) ) = Q for each n and x (n) → x (0) as n → ∞. Then let φ n be the solution to the Dirichlet problem φ n = Q on ∂D n ∩ B r (x (0) ), and φ n = ψ n on ∂B r (x (0) ) ∩ ∂K n .
The continuous dependence on the parameter Λ together with the nondegeneracy property of the solutions yields the following lemma: (i) If Λ is large, then the free boundary Γ Λ is nonempty and satisfies Proof. The proof is inspired by [9,Lemma 5.2].

≥CΛ.
HereC > 0 depends only on γ, r, and B * defined in (20). Since Q and r are fixed, there is a contradiction if Λ is large.
(ii). Suppose that Υ Λ (1) ≤ 0, we will prove that Λ cannot be too small. Similar to that in [9], for ϑ, σ > 0, denote One can increase ϑ till the first time ϑ 0 such that the graph of x 2 = η ϑ 0 (x 1 ) touches the free boundary at some It follows from Lemma 4.6 that φ ≥ ψ Λ in K. Hence On the other side, using the linear asymptotics of φ at x (0) , which can be obtained by solving the problem for the blow up limit at x (0) , one has for some C > 0 independent of Λ. Combining the above two inequalities together gives Q/Λ ≤ 1/C. This leads to a contradiction if Λ is small.
As a consequence of Lemmas 6.1 and 6.2, one has the following continuous fit property for the truncated problem: for positive constant C depending on B * , γ, andH, but independent of µ and R.
Note that (98) is the same as the governing equation for the irrotational flows. Thus with this scaling property at hand, one can use the compactness arguments in [9, Theorem 6.1] together with the unique continuation property for the limiting equation (98) to conclude Proposition 6.4. 6.3. Asymptotic behaviour of the jet. The main goal of this subsection is to remove the truncations of the domain by letting µ, R → ∞, which is a consequence of Corollary 6.3 and the uniform estimates (with respect to µ and R) of the solutions in Section 4 and Section 6.1. Hence one gets a solution in Ω, which is the domain bounded by S 0 and S 1 ∪ ([0, ∞) × {1}). Moreover, the solution inherits the properties of solutions in the truncated domain. The convergence and the properties of the limit solution are summarized in the following proposition. with ψ µ,R defined in (90). Then for any µ j , R j → ∞, there is a subsequence (still labeled by µ j and R j ) such that Λ j := Λ(µ j , R j ) → Λ ∞ for some Λ ∞ ∈ (0, ∞) and ψ µ j ,R j ,Λ j → ψ ∞ in C 0,α loc (Ω) for any α ∈ (0, 1). Furthermore, the following properties hold: i.e. J (ψ ∞ ) ≤ J (ψ) for allψ = ψ ∞ outside any bounded domain D.
The free boundary Γ ψ∞ := ∂{ψ ∞ < Q} \ S 1 is given by the graph for some function Υ, where Υ is C 1,α as long as it is finite.
Proof. The proof for (i)-(iii) is based on standard compactness arguments (cf. step 1 in the proof for Lemma 6.1), the regularity of the solution in Section 4 as well as the properties of ψ n in Section 5 and Section 6.1. We will not repeat it here. (iv) The continuous fit property follows from Proposition 5.6, Lemma 6.1 and Corollary 6.3.
Indeed, if Υ(l) < ∞, there must be anx 2 < l such that Υ(x 2 ) < ∞. Otherwise, ψ would solve the following Cauchy problem It follows from Lemma 5.5 that ψ = ψ 1d is a one dimensional solution and thus the free boundary is the horizontal line R × {l}. This is a contradiction. The fact that Υ(x 2 ) < ∞ implies that Υ(x 2 ) < ∞ for all x 2 ∈ (x 2 , 1) including l. Thus l is an interior point of I, which is a contradiction. Hence Υ(x 2 ) < ∞ for all x 2 ∈ (l, 1) and Υ(x 2 ) = ∞ for all x 2 ∈ (0, l]. Since lim x 2 →l+ Υ(x 2 ) = ∞, the flatness condition (cf. [5]) is satisfied when x 1 >R for sufficiently largeR. This implies that the free boundary is an x 1 -graph x 2 = f (x 1 ) for x 1 >R. In addition, one has f (x 1 ) → 0 as x 1 → +∞.
The next proposition is about the asymptotic behavior of the solution ψ ∞ at x 1 = ±∞.
Proposition 6.6. Let ψ ∞ be the solution obtained in Proposition 6.5. Then as is the stream function associated with the incoming flowρ,ū; as where ψ is the unique one dimensional solution to the free boundary problem with the constant momentum Λ ∞ . In particular, the asymptotic height at downstream is uniquely determined by Λ ∞ ,ρ,ū andH.
Proof. (i). Downstream asymptotic behavior. Let ψ (−n) (x 1 , x 2 ) := ψ ∞ (x 1 + n, x 2 ), n ∈ N. By the C 1,α regularity of the free boundary and the boundary regularity for elliptic equations, up to a subsequence ψ (−n) converges toψ which solves For given Q and Λ ∞ > 0, there is a unique one dimensional solution ψ 1d to the above problem. Furthermore, sinceψ(x 1 , 0) = 0, the height l is uniquely determined by Q and Λ ∞ . Indeed, for k > 0, consider the variational problem where K k = {ϕ ∈ C 0,1 ([0, k]; R) : ϕ(0) = 0, ϕ(k) = Q}. There exists a unique minimizerψ k by the direct method in the calculus of variations. Moreover, by the boundary Hopf lemma Λ(k) := lim x→k−ψ k (x) is strictly monotone decreasing in k. Thus there exists a unique l such that Λ(l) = Λ ∞ . By the uniqueness of the problem for the downstream blowup limit, one can thus conclude that ψ ∞ → ψ as x 1 → +∞, and the asymptotic height is uniquely determined.
(ii). Upstream asymptotic behavior. Since the nozzle is asymptotically horizontal with the heightH, the limits of ψ ∞ (x 1 − n, x 2 ) solve the Dirichlet problem in the infinite strip Furthermore, it follows from the energy estimate (cf. [64,Lemma 1]) that the problem (100) has a unique solution. Clearlyψ defined in (99) satisfies (100). Therefore, ψ (n) →ψ. This proves the asymptotic behavior of the flows in the upstream and thus finishes the proof of the proposition.
In view of Proposition 2.1 and Lemma 2.4, with the regularity and asymptotic behavior of solutions at hand, the equivalence between the stream function formulation and the original Euler system is true as long as ∂ x 1 ψ > 0 inside the flow region. This fact is guaranteed by the following proposition. Proof. It follows from Proposition 6.5 (ii) that ∂ x 1 ψ ≥ 0 in O. The strict inequality can be proved by the strong maximum principle. Indeed, assume that ∂ x 1 ψ(x (0) ) = 0 for some x (0) ∈ Ω ∩ {ψ < Q}. Let V := ∂ x 1 ψ. Then V ≥ 0 in O and it satisfies The strong maximum principle (cf. [44,Theorem 2.2]) gives that V = 0 in O, and hence all streamlines are horizontal. This is however a contradiction. Hence V > 0 in O and thus the proof of the proposition is complete.
Remark 6.8. To show that this is the unique state at downstream, we make use of the flatness of the free boundary at x 1 → +∞. More precisely, consider the limits ψ(x 1 + k n , x 2 ) → ψ ∞ along sequences k n → ∞. Then ψ ∞ is a solution to the Cauchy problem in the infinite strip One knows thatψ(x 2 ) =ρ´x 2 0ū (s)ds is a solution to the Cauchy problem with g(|∇ψ| 2 ,ψ) = 1 ρ . Since the flow is subsonic so that the governing equation is elliptic, by the unique continuation we infer thatψ is the unique solution to the above Cauchy problem (101).

Remove the subsonic truncation when the flux is small
In this section, the subsonic truncation is removed. We show that if (29) holds for sufficiently small κ * and Q is suitably small, then one always has |∇ψ| 2 < t c (B(ψ)) − , where t c (B(z)) is defined in (19). The proof is based on the classical Bernstein's technique.  Remark 7.2. Note that in view of (19), one can fix a truncation parameter > 0 small depending only on B * and γ, such that t c (B(ψ)) is uniformly bounded from below by a positive constant depending only on B * and γ.
Proof of Proposition 7.1. By the boundary estimate for the elliptic equation, for any where C depends on B * , γ and the nozzle. Since 0 ≤ ψ ≤ Q in O and it follows from Propositions 2.2 and 4.1 and Corollary 6.3 that one has (102) provided that (29) holds with κ 0 sufficiently small and Q ∈ (Q * ,Q) withQ sufficiently small. Both κ 0 andQ depend on B * , γ and the nozzle.
In fact, when the flows are subsonic, the downstream states of the flows are uniquely determined by the upstreaming incoming data B, Q as well asH and Λ. Proposition 7.3. Let (ψ, Γ ψ , Λ) be a smooth subsonic triple to the Problem 2 with the given incoming Bernoulli function B and the mass flux Q ∈ (Q * , Q * ), i.e., both ψ and Γ ψ are smooth, and that ∂ x 1 ψ > 0 in the fluid region {ψ < Q}. Then the states at downstream, (ρ,ū), and the asymptotic heightH are uniquely determined by B, Q,H and Λ.
It follows from (104) that one has Using the first equation in (105) yields , Substituting (106) into (107) yields that θ satisfies the Cauchy problem for a standard first order ODE. Therefore, θ, in particularH = θ(1), can be uniquely determined. The downstream velocityū is thus also determined by (106). Proposition 7.1 together with Proposition 6.7 yields the existence of solutions to the Problem 1. The downstream asymptotic behavior of the solutions follows from Proposition 7.3 and Remark 6.8.

Uniqueness
In this section we prove the uniqueness of the solution to the Problem 1.
Proposition 8.1. Suppose that (ψ i , Γ i , Λ i ) (i = 1, 2) are two uniformly subsonic solutions to Problem 2, then they must be the same.
Proof. Let (ρ i ,ū i ,H i ), i = 1, 2, be the downstream asymptotics. The proof is divided into two steps.
Let O i be the domain bounded by S 0 , S 1 , and Γ i . SinceH 1 >H 2 and that S 1 ∪ Γ i is an x 2 -graph, one can find a k ≥ 0 such that the domain O ≤ ψ 2 in O 2 . Noting that at the touching point one has ψ (k * ) 1 (x * 1 , x * 2 ) = ψ 2 (x * 1 , x * 2 ), then by the boundary Hopf lemma we have This leads to a contradiction. Hence one has Λ 1 = Λ 2 .

Existence of Critical Mass Flux
For the given Bernoulli function at the upstream satisfying (8) with κ * sufficiently small depending on B * , γ and the nozzle, Proposition 7.1 shows that there exist subsonic jet flows as long as Q ∈ (Q * ,Q) for someQ > Q * sufficiently small. In this section, we increaseQ as large as possible until some critical mass flux Q c , such that for Q > Q c either the flow fails to be subsonic or the stream function ψ fail to be a solution in the sense of Problem 2. Proof. The proof for the proposition is inspired by [62,Proposition 6], where the existence of critical flux for subsonic solutions with nonzero vorticity in fixed nozzles was established.
For the given Bernoulli function B in the upstream satisfying (8) and any Q ∈ (Q * , Q * ), one can defineρ andū(x 2 ). Since B(ψ) andρ depend on Q by definition, in this section, we denote them byρ(Q) and B(ψ; Q), respectively.
Let { n } ∞ n=1 be a strictly decreasing sequence of positive numbers such that n ↓ 0 and 1 ≤ which is used for the subsonic truncations in previous sections. Let κ * be the constant defined in (8). Fix κ * > 0 sufficiently small depending on B * , γ and the nozzle (thus κ 0 given in (29) is small), let ψ n (·; Q) be the solution (if it exsits) to (111) such that the free boundary condition Γ joins S 1 as a C 1 smooth curve, where g n and G n are defined in (40) and (50), respectively, and moreover, if It follows from Proposition 7.1 that [Q * ,Q] ⊂ T n for sufficiently large n. Hence T n is not an empty set. Define Q n = sup T n .
The sequence {Q n } has some nice properties.
First, M n (Q) is left continuous for Q ∈ (Q * , Q n ]. Let {Q n,k } ⊂ (Q * , Q n ) and Q n,k ↑ Q n as k → ∞. Since M n (Q n,k ) ≤ − n , one has ψ n (x; Q n,k ) C 0,1 (O n,k ) ≤ C and ψ n (x; Q n,k ) C 2,α (O n,k \B δ (A)) ≤ C(δ) for any δ > 0, where O n,k is the associated domain bounded by S 0 , S 1 , and Γ, and B δ (A) is the disk centered at A = (0, 1) with radius δ. Therefore, there exists a subsequence ψ n (x; Q n,k l ) such that ψ n (x; Q n,k l ) → ψ and O n,k l → O as l → ∞. Furthermore, ψ solves (111). Thus M n (Q) ≤ lim l→∞ M n (Q n,k l ) ≤ − n . It follows from Proposition 8.1 that ψ is unique. Hence M n (Q) = lim l→∞ M n (Q n,k l ).
Second, Q n < Q * . Suppose not, by the definition of Q n , Q * ∈ T n . It follows from the left continuity of M n , M n (Q * ) ≤ − n . Thus ψ n (x;Q) satisfies far field behavior. However, it follows from the definition of Q * that sup x∈O |∇ψ n (x; Q * )| 2 − t c (B(ψ n (x; Q * ))) ≥ sup Thus M n (Q * ) ≥ 0. This leads to a contradiction, which implies Q n < Q * .
Finally, it follows from the definition of {Q n } that {Q n } is an increasing sequence.
Define Q c := lim n→∞ Q n . Based on previous properties of {Q n }, Q c is well-defined and Q c ≤ Q * . Note that for any Q ∈ (Q * , Q c ), there exists a Q n > Q and thus M n (Q) ≤ − n . Therefore ψ = ψ n (x; Q) solves (111) and This yields that Q n+k ≥ Q c +σ and leads to a contradiction. Therefore, either M(Q) → 0, or there does not exist σ > 0 such that Problem 2 has solution for all Q ∈ (Q c , Q c +σ) and the associated solutions satisfy (110).
This completes the proof of the proposition.
Combining all the results in previous sections, Theorem 1 is proved.