Two-dimensional jet flows for compressible full Euler system with general vorticity
Yan Li

two-dimensional jet flows for full Euler system]Two-dimensional jet flows for compressible full Euler system with general vorticity

School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China
liyanly@sjtu.edu.cn



	In this paper, we consider the well-posedness theory of two-dimensional compressible subsonic jet flows for steady full Euler system with general vorticity. Inspired by the analysis in <cit.>, we show that the stream function formulation for such system admits a variational structure. Then the existence and uniqueness of a  smooth subsonic jet flow can be established by the variational method developed by Alt, Caffarelli and Friedman. Furthermore, the far fields behavior of the flow and the existence of a critical upstream pressure are also obtained. 

		

[2010]
	35Q31, 35R35, 35J20, 35J70, 35M32, 76N10

Updated on April 29, 2024
		
[
    Yan Li
    April 29, 2024
==================


		


§ INTRODUCTION AND MAIN RESULTS


Steady compressible subsonic flows in infinitely long nozzles have been studied extensively. In <cit.> Bers first asserted that there exists a unique subsonic irrotational flow in an infinitely long nozzle if the mass flux is less than a critical value. The rigorous mathematical proof of this assertion was given in <cit.> for two-dimensional flows, in <cit.> for three-dimensional axisymmetric flows, and in <cit.> for arbitrary multi-dimensional flows. As for rotational flows (with non-zero vorticity), the well-posedness and critical mass flux of two-dimensional subsonic flows were obtained in <cit.> by using stream function formulations. This method also works for non-isentropic flows (<cit.>), three-dimensional axisymmetric flows (<cit.>), flows with large vorticity (<cit.>), flows in periodic nozzles (<cit.>) and non-smooth nozzles (<cit.>) and so on.

Compared with flows in fixed nozzles, jet flows are usually more difficult to analyze, since free boundaries appear when the flows  issue from the nozzles. Early researches on such free boundary problems relied on hodograph tranformation and comformal mappings, which were restricted to two-dimensional flows and special nozzles, see <cit.>. Alt, Caffarelli and Friedman developed a new method to solve two-dimensional and three-dimensional axisymmetric jet problems for irrotational

flows (<cit.>) by investigating the associated Lagrangian functionals (<cit.>). This variational approach turns out to be very powerful to deal with different kinds of free boundary problems, cf. <cit.> and references therein. More results about subsonic-sonic jet flows and jet flows in bounded domains can be found in <cit.> and references therein.

Based on the variational method mentioned above, the well-posedness theory of two-dimensional isentropic subsonic jet flows with general incoming flows  was  established in <cit.>. It was shown that the jet problem for isentropic flows enjoys a variational structure even when the vorticity is non-zero, so that the framework developed by Alt, Caffarelli and Friedman can be applied to obtain the existence and uniqueness of smooth solutions to the subsonic jet problem. Furthermore, a critical mass flux of the flows was obtained in the sense that as long as the incoming mass flux is larger than the critical value, the well-posedness theory holds true. It is natural to ask whether this variational method can be adapted to study the non-isentropic jet flows. In this paper, we give a positive answer and establish the well-posedness of two-dimensional non-isentropic subsonic jet flows with general vorticity.


Two-dimensional steady compressible jet flows satisfy the following full Euler system

    {[      ·(u)=0,; ·(u⊗𝐮)+ P=0,;  ·(̆̊E+P)̆=0 ].

in the flow region,
where =̆(u_1,u_2) denotes the velocity, $̊ is the density,Pis the pressure, andEis the total energy of the flow. For the polytropic gas, one hasE=||̆^2/2+P/(-1),where the constantγ>1is called the adiabatic exponent. The local sound speed and the Mach number of the flow are defined as

    c=√( P/)   and   M=||̆/c,

respectively. The flow is called subsonic ifM<1, sonic ifM=1, and supersonic ifM>1, respectively.


Now consider a nozzle in^2bounded by two solid boundaries, which is symmetric about thex_1-axis. Denote

    _0:={(x_1,0): x_1∈} and_1:={(x_1,x_2): x_1=Θ(x_2)}

as the symmetry axis and the solid upper boundary of the nozzle, respectively. HereΘ∈C^1,([1,H̅])for some givenH̅>1and∈(0,1), moreover, it satisfies

    Θ(1)=0  andlim_x_2→H̅-Θ(x_2)=-∞,

i.e., the outlet of the nozzle is atA:=(0,1)and the nozzle is asymptotically horizontal with heightH̅at upstreamx_1→-∞. In order to study the uniqueness of solution, we further assume that there exists anh_*∈[1,H̅)such that the nozzle is monotone decreasing forx_2∈(h_*,H̅), i.e.,

    Θ'(x_2)≤ 0   for  x_2∈ (h_*,H̅).



The main goal of this paper is to study the following jet problem.



Given a density ρ̅=ρ̅(x_2),
a horizontal velocity u̅=u̅(x_2) and a constant pressure P̅>0 of the flow at the upstream, find (,̊,̆ P), the free boundary , and the  outer pressure P which we assume to be a positive constant, such that the following statements hold:

   (i) The free boundary  joins the outlet of the nozzle as a continuous differentiable curve and tends asymptotically horizontal at downstream as x_1→∞.

  
   (ii) The solution (,̊,̆P) solves the Euler system (<ref>) in the flow region 𝒪 bounded by _0, _1 and . It
  takes the incoming data at the upstream, i.e.,
  
    ρ(x_1,x_2)→ρ̅(x_2), u_1(x_1,x_2)→u̅(x_2),
      P(x_1,x_2)→P̅,
      as  x_1→-∞.

  
  
  
  Moreover, it satisfies the boundary conditions
  
    P=P on anḑ̆=0  on _1∪,

  where  is the unit normal along _1∪.



The main results in this paper can be stated as follows.


Let the nozzle boundary _1 defined in (<ref>) satisfy (<ref>)-(<ref>) and P̅>0 be a constant. Given a density ρ̅∈ C^1,1([0,H̅]) and a horizontal velocity u̅∈ C^1,1([0,H̅]) of the flow at the upstream. Suppose

    ρ̅_*:=inf_x_2∈[0,H̅]ρ̅(x_2)>0,
    ρ̅'(0)=0, ρ̅'(H̅)=0,
       u̅_*:=inf_x_2∈[0,H̅]u̅(x_2)>0,
    u̅'(0)=0,u̅'(H̅)≥0.

There exist constants ̨̅, P̅_c>0 depending on ρ̅_*, u̅, γ and the nozzle, such that the following statements hold:

   (i) (Existence and properties of solutions) For any pressure P̅>P̅_c at the upstream, if
   
    P̅ρ̅'_C^0,1([0,H̅])<̨̅,

  then there exist functions  ρ,𝐮,P∈ C^1,α(𝒪)∩ C^0(𝒪) (for any α∈ (0,1)) where 𝒪 is the flow region, the free boundary Γ, and the outer pressure P such that (,̊,̆P,, P) solves Problem <ref>. Furthermore, the following properties hold:
  
  *   (Smooth fit) The free boundary  joins the outlet of the nozzle as a C^1 curve.
  	
  	
  *   The free boundary Γ is given by a graph x_1=Υ(x_2), x_2∈ ( H, 1], where Υ is a C^2,α function, H∈(0,1), and lim_x_2→ H+Υ(x_2)=∞. For x_1 sufficiently large, the free boundary  can also be written as x_2=f(x_1), where f is a  C^2,α function and satisfies
  	lim_x_1→∞f(x_1)= H
  	  and  lim_x_1→∞f'(x_1)=0.
  *  The flow is globally uniformly subsonic and has negative vertical velocity in the flow region , i.e.,
  	
    sup_||̆^2/c^2<1 and u_2<0  in .

  *  (Upstream and downstream asymptotics) The flow satisfies the asymptotic behavior
  	
    (,̊u_1,u_2,P)(x_1,)̧-(()̧,u̅()̧,0,P̅)_C^1,α([0,H̅))→0 as  x_1→-∞

  	and
  	
    (,̊u_1,u_2,P)(x_1,)̧-(()̧,u()̧,0,P)_C^1,α([0,H))→0 as  x_1→∞,

  	where (,u)∈ (C^1,α([0,H]))^2 are positive functions. Moreover, the downstream asymptotics , u, P and H are uniquely determined by ρ̅, u̅, P̅, γ, and H̅.
  
   (ii) (Uniqueness) The Euler flow which satisfies all properties in part (i) is unique.

  
   (iii) (Critical pressure) P̅_c is the critical pressure at the upstream for the existence of subsonic jet flows in the following sense: either
  
    sup_||̆^2/c^2→1 as P̅→P̅_c+,

  or there is no σ>0 such that for all P̅∈(P̅_c-σ, P̅_c), there are Euler flows satisfying all properties in part (i) and
  
    sup_P̅∈(P̅_c-σ,P̅_c)sup_||̆^2/c^2<1.

The condition (<ref>) of the nozzle boundary _1 at far field is only used for the uniqueness of the solution and the existence of the critical pressure.

 The conditions of u̅ in Theorem <ref> are the same as that in <cit.>, where the density ρ̅ is a positive constant.

		Suppose the flow satisfies the asymptotic behavior with high order compatibility at the downstream, i.e.,
	
    (ρ(x_1,x_2),u_1(x_1,x_2),u_2(x_1,x_2),P(x_1,x_2))→(ρ(x_2), u_1(x_2), u_2(x_2), P(x_2))

	and
	
    (ρ(x_1,x_2),u_1(x_1,x_2),u_2(x_1,x_2),P(x_1,x_2))→(ρ(x_2), u_1(x_2), u_2(x_2), P(x_2))

	for some smooth functions (ρ, u_1, u_2, P) as x_1→∞. It follows from the Euler system (<ref>) that
	
    (ρ u_2)'=0, (u_1 u_2)'=0, (u_2^2)'+ P'=0,(u_2( u_1^2+ u_2^2/2+ P/(-1)))'=0.

	The first equation in (<ref>) shows that ρ u_2≡ C for some constant C. Since u_2=0 on the x_1-axis (since the flow is symmetric about the x_1-axis) implies C=0 and since the solution is supposed to satisfy ρ>0, one has u_2=0. Then it follows from the third equation in (<ref>) that P'=0.
	This means that the downstream pressure P is a constant. Consequently, in view of (<ref>), the downstream density ρ of a non-isentropic flow is a function. Same arguments hold for the upstream flow states.





To prove Theorem <ref>, we first establish the existence of solutions to the jet problem with sufficiently large pressure at the upstream. We derive an equivalent formulation of the full Euler system in terms of the stream function, which is a second order quasilinear equation with an inhomogeneous term coming from the vorticity and the entropy. Moreover, the equation is elliptic in the subsonic region and degenerate when the flow approaches the sonic state. To deal with this possible degeneracy, we use a subsonic truncation to get a uniformly elliptic equation. A domain truncation is also used to avoid the difficulties caused by the unboundedness of the domain.
Then we reformulate the non-isentropic jet problem for compressible Euler flows with non-zero vorticity into a variational problem, which has exactly the same form as in the isentropic case (cf. (<ref>)-(<ref>) below and (3.3)-(3.5) in <cit.>). Thus the existence and regularity of solutions to the variational problem can be obtained by using the same arguments as in <cit.>. Some fine properties of the solutions are also obtained. Then we remove the domain and subsonic truncations to get the desired solution to the original jet problem. Furthermore, the uniqueness of the solution to the jet problem and the existence of a critical upstream pressure are established.

This paper is organized as follows. In Section <ref>, we reformulate the jet problem for the full Euler flows in terms of the stream function formulations, and introduce the subsonic truncation. In Section <ref>, we use domain truncations and give the variational formulation for the truncated problems. The existence, regularity and fine properties of solutions to the truncated problems are investigated in Section <ref>, where the continuous fit and smooth fit of the free boundary are also established. In Section <ref>, we remove the domain and subsonic truncations, then we obtain the existence and uniqueness of the subsonic solution to the jet problem. The existence of the critical pressure is given in Section <ref>.

In the rest of the paper, for convention the repeated indices mean the summation.



§ STREAM FUNCTION FORMULATION AND SUBSONIC TRUNCATION


In this section, inspired by <cit.> we introduce the stream function to show that the full Euler system is equivalent to a second order quasilinear equation, which is elliptic in the subsonic region and becomes singular at the sonic state. Then a subsonic truncation is used to modify this quasilinear equation into a uniformly elliptic equation.



 §.§ The equation for the stream function


It follows from the continuity equation (<ref>)_1that there is a stream functionψsatisfying

    ψ=(-ů_2,ů_1).

Moreover, by (<ref>), if the flow is away from vacuum, then

    ̧̆ B=0 anḑ̆ S=0,

whereBis the Bernoulli function defined by

    B:=||̆^2/2+ P/(-1),

andSis the entropy function defined by

    S:= P/(-1)^̊.

This implies the Bernoulli functionBand the entropy functionSare constants along each streamline. Now we show that actuallyBandScan be expressed as functions of the stream functionψunder suitable assumptions. For this we denote the mass flux of the flow as

    Q:=∫_0^H̅(ρ̅u̅)(x_2)dx_2.

	Given a constant pressure P̅>0, a density ρ̅∈ C^1,1([0,H̅]) and a horizontal velocity u̅∈ C^1,1([0,H̅]) of the flow at the upstream, which satisfy (<ref>). Suppose that (ρ, 𝐮,P) is a solution to the Euler system (<ref>), and each streamline is globally well-defined in the flow region. Then there exist C^1,1 functions ℬ:[0,Q]→ and  𝒮:[0,Q]→ such that
	
    (B,S)(x_1,x_2)=(,)(ψ(x_1,x_2))

	and
	
    (ψ)=|∇ψ|^2/2ρ^2+^̊-1(ψ).

	Moreover, denote
	
    κ_0:=u̅'_C^0,1([0, H̅])andκ_1:=ρ̅'_C^0,1([0,H̅]).

	Then
	
    ℬ'_L^∞([0, Q])≤C/ρ̅_*(_̨0+P̅_̨1/(ρ̅_*)^2),𝒮'_L^∞([0, Q])≤CP̅_̨1/(ρ̅_*)^+2,
    		
       ℬ”_L^∞([0,Q])≤C/(ρ̅_*)^2(_̨0+_̨0_̨1/ρ̅_*+P̅_̨1/(ρ̅_*)^2(1+_̨0+_̨1/ρ̅_*)),
       𝒮”_L^∞([0,Q])≤CP̅_̨1/(ρ̅_*)^+3(1+_̨0+_̨1/ρ̅_*),

	where C=C(γ,u̅_*)>0, ρ̅_* and u̅_* are defined in (<ref>).

	Since the Bernoulli function B and the entropy function S are conserved along each streamline, which is globally well-defined in the flow region, then B and S are uniquely determined by their values at the upstream. Let (z):[0,Q]→[0,H̅] be the position of the streamline at the  upstream where the stream function takes the value ψ=z, i.e.

    z=∫_0^(z)(u̅)(x_2)dx_2.

Note that (z) is well-defined as ρ̅, u̅>0. Since the Bernoulli function and the entropy function at the upstream are

    B̅(x_2):=lim_x_1→ -∞ B(x_1,x_2)=u̅(x_2)^2/2+P̅/(-1)(x_2)

and

    S̅(x_2):=lim_x_1→ -∞ S(x_1,x_2)=P̅/(-1)(x_2)^

for x_2∈[0,H̅], if we define two functions ℬ and  as

    (z):=B̅((z))=u̅((z))^2/2+P̅/(-1)((z))

and

    (z):=S̅((z))=P̅/(-1)((z))^

for z∈[0,Q], then one has (<ref>). Since by (<ref>) the flow velocity :̆=(u_1,u_2) satisfies ||̆^2=|ψ|^2/^̊2, the equality (<ref>) follows from the definitions of the Bernoulli function in (<ref>) and the entropy function in (<ref>), as well as (<ref>).

To estimate the first and second derivatives of the functions  and , we note that by




differentiating (<ref>) with respect to z one gets
'(z)=1/(̊̅u̅)((z)).
Thus differentiating (<ref>) and (<ref>) directly yield

    '(z)=u̅'((z))/ρ̅((z))-P̅/-1(ρ̅'/ρ̅^3u̅)((z)),
    		'(z)=-^2P̅/-1(ρ̅'/ρ̅^+2u̅)((z)),
       ”(z)=1/ρ̅^2u̅[u̅”-ρ̅'u̅'/ρ̅-P̅/-1(ρ̅”/ρ̅^2u̅-3(ρ̅')^2/ρ̅^3u̅-ρ̅'u̅'/ρ̅^2u̅^2)]((z)),
       ”(z)=-^2P̅/-1[1/^+3u̅^2(ρ̅”-ρ̅'u̅'/u̅-(+2)(ρ̅')^2/ρ̅)]((z)).

Then (<ref>) follows from the definitions of _̨0 and _̨1 in (<ref>).




Now we deduce the connection between(,)and the vorticity functionø:=_x_1u_2-_x_2u_1. By (<ref>) and (<ref>) one has

    B=u_1^2+u_2^2/2+( P/-1)^-1/S^1/.

Differentiating (<ref>) and noting that=̊( P/(-1)S)^1/,one gets

    _x_1B=u_1_x_1u_1+u_2_x_1u_2+^̊-1/_x_1S+_x_1P/

and

    _x_2B= u_1_x_2u_1+u_2_x_2u_2+^̊-1/_x_2S+_x_2P/.

These together with

    u_1_x_1u_1+u_2_x_2u_1+_x_1P/=0
      and    u_1_x_1u_2+u_2_x_2u_2+_x_2P/=0,

which are obtained from the first three equations in (<ref>), give

    _x_1B-^̊-1/_x_1S =u_2ω and_x_2B-^̊-1/_x_2S=-u_1ω.

Since it follows from (<ref>) and (<ref>) that

    (_x_1B,_x_2B)='(ψ)(-u_2,u_1) and 	(_x_1S,_x_2S)='(ψ)(-u_2,u_1),

substituting (<ref>) into (<ref>) yields

    ø=-'(ψ)+^̊/'(ψ)

provided|𝐮|≠0. Thus smooth solutions of the full Euler system  (<ref>) satisfy a system consisting of the continuity equation (<ref>)_1, (<ref>) and (<ref>). The following proposition shows that the two systems are actually equivalent under appropriate conditions.


	Let Ø̃⊂^2 be the domain bounded by two streamlines _0 and
	={(x_1,x_2): x_1=Θ̃(x_2),  h<x_2<H̅},
	where 0<h<H̅<∞ and Θ̃:( h,H̅)→ℝ is a C^1 function with
	lim_x_2→H̅-Θ̃(x_2)=-∞ andlim_x_2→h+Θ̃(x_2)=∞.
	Let ρ:Ω̃→(0,∞), 𝐮=(u_1,u_2):Ω̃→ℝ^2 and P:Ω̃→(0,∞)
	be C^1,1 in Ω̃ and continuous up to ∂Ω̃ except finitely many points.
	Suppose $̆ satisfies the slip boundary conditioņ̆=0onØ̃,(ρ,,̆P)satisfies the upstream asymptotic behavior  (<ref>) for some positiveC^1,1functions(ρ̅,u̅)and a positive constantP̅. Moreover, suppose that
    
    u_2<0 in Ø̃.

	Then(ρ,,̆P)solves the full Euler system (<ref>) inΩ̃if and only if(ρ,,̆P)solves the system consisting of the continuity equation (<ref>)_1, (<ref>) and (<ref>) inØ̃.

	In view of (<ref>), through each point in Ø̃, there is one and only one streamline which satisfies
	
    {   dx_1/ds=u_1(x_1(s),x_2(s)),
       dx_2/ds=u_2(x_1(s),x_2(s)),
    		.

	and can be defined globally in the domain. Furthermore, any streamline through some point in Ø̃ cannot touch Ø̃. To prove it, without loss of generality we assume that there exists a streamline through (-x̅_1,x̅_2) (with x̅_1>0 sufficiently large) in Ø̃ passing through (x̃_1, 0). By  the continuity equation (<ref>)_1 and the slip boundary condition, along each streamline one has
	0=∫_0^x̅_2(ů_1)(-x̅_1,s)ds,
	which contradicts the asymptotic behavior (<ref>).
	As each streamline is globally well-defined in Ø̃, from Proposition <ref> and previous analysis, it suffices to derive the full Euler system (<ref>) from the system consisting of (<ref>)_1, (<ref>) and (<ref>).
	
	It follows from (<ref>) and (<ref>) that
̆̊(ø/-^̊-1/'(ψ))=0.
With the help of (<ref>), (<ref>) and (<ref>), one can check that the above equality is equivalent to

    ̆̊(ø/)+(_x_2P/,-_x_1P/)=0.

Since the continuity equation (<ref>)_1 leads to
̆̊(ø/)=̧̆ø-ø̧̆/=̧̆ø+ø̧̆,
it follows from (<ref>) that
_x_1(u_1_x_1u_2+u_2_x_2u_2+_x_2P/)-_x_2(u_1_x_1u_1+u_2_x_2u_1+_x_1P/)=0.
Therefore there exists a function Φ such that
_x_1Φ=u_1_x_1u_1+u_2_x_2u_1+_x_1P/ and_x_2Φ=u_1_x_1u_2+u_2_x_2u_2+_x_2P/.
Moreover, combining (<ref>), (<ref>) and (<ref>)  together yields
̧̆(u_1_x_1u_1+u_2_x_2u_1+_x_1P/, u_1_x_1u_2+u_2_x_2u_2+_x_2P/)=0,
that is
̧̆Φ=0.
This means Φ is a constant along each streamline, in particular along _0 and . Notice that (<ref>) implies _x_2Φ→0 as x_1→-∞, hence one has

    Φ→ Cas  x_1→-∞.

Therefore, one concludes that Φ≡ C in the whole domain Ø̃. This implies that _x_1Φ=_x_2Φ≡0 in Ø̃, i.e., (<ref>)
holds globally in Ø̃.

Combining the continuity equation (<ref>)_1,  (<ref>) and the Bernoulli law (the first equality in (<ref>))  together gives the full Euler system (<ref>).



For later purpose, we extend the Bernoulli functionand the entropy functionfrom[0,Q]toas follows: firstly in view of (<ref>),ρ̅andu̅can be extended toC^1,1functions in, which are still denoted byρ̅andu̅respectively, such thatρ̅, u̅>0   on ,   ρ̅'=u̅'= 0   on  (-∞,0],  ρ̅'=0,  u̅'≥ 0

  on  [H̅,∞).Furthermore, the extension can be made such thatρ̅_C^1,1()=ρ̅_C^1,1([0,H̅])andu̅_C^1,1()≤ Cu̅_C^1,1([0,H̅])and

    ρ̅_∗=inf_ρ̅≤sup_ρ̅=ρ̅^∗andu̅_∗=inf_u̅≤sup_u̅:=ũ^∗<Cu̅^*,

where the constantsρ̅_*andu̅_*are defined in (<ref>),ρ̅^*andu̅^*are defined by

    ρ̅^*:=sup_x_2∈[0,H̅]ρ̅(x_2)andu̅^*:=sup_x_2∈[0,H̅]u̅(x_2),

andC=C(u̅)>0.
Consequently, using (<ref>) and (<ref>) one naturally gets extensions ofℬandtoC^1,1functions in, which are still denoted byℬandrespectively, such that

    ℬ'(z)=𝒮'(z)=0   on  (-∞, 0], '(z)≥0,  '(z)=0   on  [Q,∞),

and

    ℬ'_C^0,1()≤ C ℬ'_C^0,1([0, Q]), 𝒮'_C^0,1()= 𝒮'_C^0,1([0, Q])

for someC=C(u̅)>0. Moreover,andare bounded in, i.e.,

    0<B_*≤(z)≤ B^*<∞and
    	0<S_*≤(z)≤ S^*<∞,  z∈,

where

    B_*:=(u̅_*)^2/2+P̅/(-1)^*,
    	B^*:=(ũ^*)^2/2+P̅/(-1)_*,

and

    S_*:=P̅/(-1)(^*)^,
    	
    	S^*:=P̅/(-1)(_*)^,

and whereρ̅^*,ρ̅_*,u̅_*,ũ^*are the same constants as in (<ref>).
	
Before formulating the Euler system into a quasilinear equation for the stream functionψ, let us digress for the study on the flow state.
It follows from (<ref>)-(<ref>) and Proposition <ref> that the flow speedqsatisfy

    q=𝔮(ρ,z)=√(2((z)-^̊-1(z))).

Besides, using (<ref>) and the definition of the sound speed in (<ref>) one hasc=√((-1)ρ^-1𝒮(z)).Let

    ϱ_c(z):=(2(z)/(+1)(z))^1/-1andϱ_m(z):=((z)/(z))^1/-1

be the critical density and maximum density respectively. The flow speedq=𝔮(ρ,z)is well-defined if and only ifρ≤ϱ_m(z). The flow is subsonic, i.e.,q∈[0,c), if and only ifq∈[0,𝔮(ϱ_c(z),z)), or equivalently∈̊(ϱ_c(z),ϱ_m(z)]. Denote the square of the momentum and the square of the critical momentum by

    (ρ, z):=ρ^2𝔮^2(ρ, z)=2ρ^2((z)-ρ^-1(z))

and

    _c(z):=(ϱ_c(z),z)=(-1)(2(z)/+1)^+1/-1(z)^-2/-1,

respectively.

By the definition of the sonic speed in (<ref>), the flow is subsonic at the upstream if(ũ^*)^2<P̅/ρ̅^*, whereũ^*is defined in (<ref>). This inequality holds provided

    P̅>P̃_*,   whereP̃_*:=1/ρ̅^*(ũ^*)^2≥P̅_*:=1/sup_x_2∈[0,H̅](ρ̅u̅^2)(x_2).

Then by (<ref>) and (<ref>) one has

    B^*<(+1)P̅/2(-1)ρ̅_*and
    	
    	(ρ̅_*)^-1<(z)/(z)<+1/2(ρ̅^*)^-1,  z∈.

Thus in view of (<ref>) and (<ref>), forz∈,

    0<(2/+1)^1/-1ρ̅_*<ϱ_c(z)<ϱ_m(z)<(+1/2)^1/-1ρ̅^*,

and

    0<(2/+1)^+1/-1P̅(ρ̅_*)^2/ρ̅^*≤𝔱_c(z)≤P̅(ρ̅^*)^2/ρ̅_*.

Furthermore, since

    _c'(z)=(2/γ+1)^γ+1/γ-1((γ+1)((z)/(z))^2/γ-1'(z)-2((z)/(z))^γ+1/γ-1'(z)),

by theL^∞-estimates for the derivatives ofandin (<ref>), and the upper bound of/in (<ref>), one has

    |'_c(z)|≤ C(ρ̅^*)^2/ρ̅_*(_̨0+P̅_̨1/(ρ̅_*)^2(1+(ρ̅^*/ρ̅_*)^-1))

for some constantC=C(,u̅_*)>0.
Similarly, by the explicit expression of_c”(z)one has

    |”(z)|≤    C(ρ̅^*)^2/(ρ̅_*)^2(_̨0+_̨0_̨1/ρ̅_*+P̅_̨1/(ρ̅_*)^2(1+_̨0+_̨1/ρ̅_*)(1+(ρ̅^*/ρ̅_*)^-1))
       +C(ρ̅^*)^+2/P̅(ρ̅_*)^+1(_̨0+P̅_̨1/(ρ̅_*)^2(1+(ρ̅^*/ρ̅_*)^-1))(_̨0+P̅_̨1/(ρ̅_*)^2),

for some constantC=C(,u̅_*)>0.


	Suppose the density function $̊ and the stream functionψsatisfy (<ref>). Then the following statements hold:
	
  (i) The density function $̊ can be expressed as a function of|ψ|^2andψin the subsonic region, i.e.,
		
    =̊1/g(|ψ|^2,ψ), if ∈̊(ϱ_c(ψ),ϱ_m(ψ)],

		whereϱ_candϱ_mare functions defined in (<ref>), andg:{(t,z):0≤ t<𝔱_c(z),  z∈}→is a function smooth intandC^1,1inzwith𝔱_cdefined in (<ref>). Furthermore,
		
    (sup_z∈ϱ_m(z))^-1=:g_*≤ g(t,z)≤ g^*:=(inf_z∈ϱ_c(z))^-1.

  (ii) The functiongsatisfies the identity
		
    _z g(t,z)=-2_t g(t,z)'(z)-g(t,z)^1-'(z)/g(t,z)^2,  t∈[0,𝔱_c(z)),  z∈.
(i). From the expression (<ref>), straightforward computations give

    _(,̊ z)=4((z)-+1/2^̊-1(z)).

Now, with ϱ_c and ϱ_m defined in (<ref>) one has

   (a)↦̊(,̊z) achieves its maximum _c(z) at =̊ϱ_c(z);
  
   (b)_(,̊z)<0 when ϱ_c(z)<≤̊ϱ_m(z).

Thus by the inverse function theorem, if ρ∈(ϱ_c(z),ϱ_m(z)], one can express ρ as a function of t:=(ρ,z)∈[0,_c(z)) and z, that is, ρ=ρ(t,z). Let

    g(t,z):=1/(̊t,z),  t∈[0,𝔱_c(z)),  z∈.

The function g is smooth in t by the inverse function theorem and C^1,1 in z by the C^1,1 regularity of  and .
This together with (<ref>) gives (<ref>). Consequently (<ref>) holds.

(ii).
Differentiating (<ref>) with respect to t:=(ρ,z) and z, respectively, one gets

    1/2=_t/(t-(-1)^̊+1(z))

and

    _z/(t-(-1)^̊+1(z))=-^̊2('(z)-^̊-1'(z)).

Combining (<ref>) and (<ref>) together gives
_z=̊-2^̊2_t(̊'(z)-^̊-1'(z)).
Thus by (<ref>), the equality (<ref>) holds. This completes the proof of the lemma.


The following lemma gives the equation of the stream function in the subsonic region.


Let (,̊,̆P) be a solution to the system consisting of  (<ref>)_1, (<ref>) and (<ref>). Assume (,̊,̆P) satisfies the assumptions in Proposition <ref>. Then in the subsonic region |∇ψ|^2<_c(ψ) where _c is defined in (<ref>), the stream function ψ solves

    (̧g(|ψ|^2,ψ)ψ)='(ψ)/g(|ψ|^2,ψ)-'(ψ)/ g(|ψ|^2,ψ)^,

where g is defined in (<ref>),  is the Bernoulli function  defined in (<ref>), and  is the entropy function defined in (<ref>). Moreover, the equation (<ref>) is elliptic if and only if |ψ|^2<_c(ψ).

Expressing the vorticity ø in terms of the stream function ψ, and using (<ref>) one has

    -(ψ/)=ø=-'(ψ)+^̊/'(ψ).

In view of (<ref>) the above equation can be rewritten as  (<ref>).

The equation (<ref>) can be written in the nondivergence form as follows
a^ij(ψ,ψ)_ijψ+_zg(|ψ|^2,ψ)|ψ|^2='(ψ)/g(|ψ|^2,ψ)-'(ψ)/ g(|ψ|^2,ψ)^,
where the matrix
(a^ij)=g(|ψ|^2,ψ)I_2+2_t g(|ψ|^2,ψ)ψ⊗ψ
is symmetric with the eigenvalues
_̱0=g(|ψ|^2,ψ)  and_̱1=g(|ψ|^2,ψ)+2_tg(|ψ|^2,ψ)|ψ|^2.
Differentiating the identity t=(1/g(t,z),z) gives

    _tg(t,z)=-g(t,z)^2/_(1/g(t,z),z)for  t∈[0,_c(z)),  z∈.

This implies
_tg(t,z)≥0  andlim_t→_c(z)-_tg(t,z)=+∞.
Thus _̱0 has uniform upper and lower bounds

(cf. (<ref>) and (<ref>)),
and _̱1 has a uniform lower bound but blows up when |ψ|^2 approaches _c(ψ). Therefore, the equation (<ref>) is elliptic as long as |ψ|^2<_c(ψ), and is singular when |ψ|^2=_c(ψ). This completes the proof of the lemma.


 §.§ Reformulation for the jet flows in terms of the stream function


Let

    :=_̊0√(2(Q)-2_̊0^-1(Q)) with_̊0:=( P/(-1)(Q))^1/

be the constant momentum on the free boundary, whereis defined in (<ref>),is defined in (<ref>), and P>0is the pressure on the free boundary as in Problem <ref>.

Then Problem <ref> can be solved as long as the following problem in terms of the stream function is solved.


Given a density ρ̅∈ C^1,1([0,H̅]) and a horizontal velocity u̅∈ C^1,1([0,H̅]) satisfying (<ref>). Given a constant pressure P̅>0 of the flow at the upstream. Find a triple (ψ,_ψ,) satisfying ψ∈ C^2,α({ψ<Q})∩ C^1({ψ<Q}) (α∈(0,1)), _x_1ψ>0 in {0<ψ<Q} and

    (̧g(|ψ|^2,ψ)ψ)='(ψ)/g(|ψ|^2,ψ)-'(ψ)/ g(|ψ|^2,ψ)^       in {ψ<Q},
       ψ=0        on _0,
       ψ=Q        on _1∪_ψ,
       |ψ|=       on _ψ,

where the free boundary _ψ:={ψ<Q}\_1. Furthermore, the free boundary _ψ and the flow

    (,̊,̆ P)=(1/g(|ψ|^2,ψ),   g(|ψ|^2,ψ)∇^⊥ψ,  (γ-1)(ψ)/γ g(|ψ|^2,ψ)^γ)

are expected to have the following properties:

  *  The free boundary Γ_ψ is given by a graph x_1=Υ(x_2), x_2∈ (H, 1] for some C^2,α  function Υ and some H∈ (0,1).
	
	
  *  The free boundary Γ_ψ fits the nozzle at A=(0,1) continuous differentiably, i.e. Υ(1)=Θ(1) and Υ'(1)=Θ'(1).

	
	
  *  For x_1 sufficiently large, the free boundary is also an x_1-graph, i.e., it can be written as x_2=f(x_1) for some C^2,α function f. Furthermore, at downstream x_1→∞, one has
	lim_x_1→∞f(x_1)=Handlim_x_1→∞f'(x_1)=0.
  *  The flow is subsonic in the flow region, i.e. |∇ψ|^2<_c(ψ) in {ψ<Q}, where _c is defined in (<ref>).
	
	
  *  At the upstream the flow satisfies the asymptotic behavior (<ref>). At the downstream the flow satisfies 
	lim_x_1→∞(,̊,̆P)=(,u,0, P)
	for some positive functions =(x_2) and u=u(x_2), and some constant P>0.
	


As in <cit.>, we have the following observation, which asserts that(ρ,𝐮,P)defined in (<ref>) is a solution of the Euler system (<ref>) even ifρ,𝐮,Pare onlyC^1,α.
 

Given a density ρ̅∈ C^1,1([0,H̅]) and a horizontal velocity u̅∈ C^1,1([0,H̅]) satisfying (<ref>). Given a constant pressure P̅>0 of the flow at the upstream. Assume that a function ψ∈ C^2,α({ψ<Q})∩ C^1({ψ<Q}) is a subsonic solution of (<ref>), where α∈(0,1). Then the flow (ρ,𝐮,P) defined in (<ref>) solves the Euler system (<ref>) in the flow region {ψ<Q}.

	Due to the regularity of the function ψ,  the functions ρ, 𝐮:=(u_1,u_2) and P defined in (<ref>) belong to C^1,α({ψ<Q})∩ C({ψ<Q}). Since ρ and 𝐮  satisfy (<ref>), it follows from the definition of the function g in Lemma <ref> (i) that  the density ρ=1/g satisfies
	
    u_1^2+u_2^2/2+^̊-1(ψ)=	|∇ψ|^2/2ρ^2+^̊-1(ψ)=ℬ(ψ)

	in the flow region.
	Differentiating the above equality with respect to x_1 and x_2, respectively, one gets
	
    u_1∂_x_1u_1+u_2∂_x_1u_2 +^̊-1/'(ψ)_x_1ψ+_x_1P/='(ψ)∂_x_1ψ,

	and
	
    u_1∂_x_2u_1+u_2∂_x_2u_2+^̊-1/'(ψ)_x_2ψ+_x_2P/='(ψ)∂_x_2ψ.

	Note that by the quasilinear equation in (<ref>) one has
	'(ψ)-^̊-1/'(ψ)=-_x_1u_2-_x_2u_1/ρ.
	Thus the equalities (<ref>) and (<ref>) together with (<ref>) yield (<ref>). Note that ρ and 𝐮 satisfy the continuity equation (<ref>)_1. Then the Euler system (<ref>) follows from (<ref>)_1, (<ref>) and the equality ρ𝐮·∇(ψ)=0 where (ψ) satisfies (<ref>). This finishes the proof of the proposition.


 §.§ Subsonic truncation


One of the major difficulties to solve Problem <ref> is that the equation (<ref>) becomes degenerate as the flow approaches the sonic state, so we use a subsonic truncation to avoid this degeneracy. Letϖ:→[0,1]be a smooth nonincreasing function such thatϖ(s)=
1   if  s≤-1,

0   if  s≤-1/2,
 and
|ϖ'|+|ϖ”|≤8.For fixed∈̌(0,1/4), 
letϖ_(̌s):=ϖ((s-1)/)̌. Define

    g_(̌t,z)=g(t,z)ϖ_(̌t/_c(z))+(1-ϖ_(̌t/_c(z)))g^*,

where_c(z)is given by (<ref>) andg^*is the upper bound ofgin (<ref>). The properties ofg_$̌ are summarized in the following lemma.


Let g be the function defined in Lemma <ref> and let g_$̌ be the subsonic truncation ofgdefined in (<ref>). Then the functiong_(̌t,z)is smooth with respect totandC^1,1with respect toz. Furthermore, suppose the upstream  pressureP̅of the flow satisfies (<ref>). There exist positive constantsc_*=c_*()andc^*=c^*(,u̅_*)such that for allt≥ 0, one has
  
    c_*(ρ̅^*)^-1≤ g_(̌t,z)≤ (c_*ρ̅_*)^-1, 
    
         c_*(ρ̅^*)^-1≤ g_(̌t,z)+2_tg_(̌t,z)t≤ (c_*̅̌ρ_*)^-1, 
       |_zg_(̌t,z)|≤  c^*^̌-1(ρ̅^*)^3/P̅(ρ̅_*)^4(_̨0+P̅_̨1/(ρ̅_*)^2(1+(ρ̅^*/ρ̅_*)^-1)),
       |t_zg_(̌t,z)|≤ c^*^̌-1(ρ̅^*/ρ̅_*)^5(_̨0+P̅_̨1/(ρ̅_*)^2(1+(ρ̅^*/ρ̅_*)^-1)),

 whereρ̅_*is defined in (<ref>), andρ̅^*is defined in (<ref>).

	
	It follows from Lemma <ref> and the definition of ϖ_$̌ thatg_$̌ is smooth with  respect to t and C^1,1 with respect to z.
	
    (i).  Clearly (<ref>) follows directly from (<ref>) and (<ref>).

	To show (<ref>),  we first claim that if 0≤ t≤(1-ϵ/2) _c(z) then 
    0<_t g(t,z)≤g(t,z)/ϵ_c(z),
where _c is defined in (<ref>). In view of the expression of _t g(t,z) in (<ref>) it suffices to estimate _ρ(ρ,z) at ρ=ρ(t,z) for 0≤ t≤(1-ϵ/2) _c(z). In fact, it follows from (<ref>) and (<ref>) that for 0≤ t≤(1-ϵ/2) _c(z)  and ρ≥ϱ_c(z) it holds that
	
    _(ρ,z)
    		   =2/ρ((ρ,z)-(-1)ρ^+1(z))≤2/ρ((ρ,z)-(-1)ϱ_c(z)^+1(z))
       = 2/ρ((ρ,z)-_c(z))≤-_c(z)/ρ.

	Thus from (<ref>) we conclude that
	
    0<_tg(t,z)=-g^2(t,z)/_ρ(ρ, z)|_ρ=1/g(t,z)≤g(t,z)/ϵ_c(z),

	which is the claimed inequality (<ref>).
	With (<ref>) at hand, we thus have
	
    0≤_tg_ϵ(t,z)    = ∂_t g (t, z) ϖ_ϵ(t/_c(z))+(g(t, z)-g^*) ϖ'_ϵ(t/_c(z))1/_c(z)
       ≤g(t,z)/ϵ_c(z)+8g^∗/ϵ_c(z)≤9 g^∗/ϵ_c(z).

	Note that
	
    _tg_ϵ(t,z)=0  in {(t,z):t>(1-ϵ/2)𝔱_c(z),  z∈}.

	Thus
	
    0≤ t_tg_ϵ(t,z)≤9 g^∗_c(z)/ϵ_c(z)= 9 g^∗/ϵ.

	Then the estimate (<ref>) follows directly from  (<ref>).
	
	(ii).
	 Direct computations yield
	
    _z g_ϵ (t,z)= _z g(t,z)ϖ_ϵ(t/_c(z)) + (g^∗-g(t,z))ϖ_ϵ'(t/_c(z)) t_c'(z)/_c(z)^2
    (<ref>)=    -2_t g(t,z)'(z)-g(t,z)^1-'(z)/g(t,z)^2ϖ_ϵ(t/_c(z))+ (g^∗-g(t,z))ϖ_ϵ'(t/_c(z)) t_c'(z)/_c(z)^2.

	
	
	
	Note that both sums in the expression of _tg_ϵ, cf. (<ref>), are nonnegative. Thus we get from  the expression of _zg_ϵ in (<ref>) that
	
    |_zg_ϵ(t,z)|≤   max{2|'(z)-g(t,z)^1-'(z)|/g(t,z)^2,t𝔱_c'(z)/𝔱_c(z)}_tg_ϵ(t,z).

	By the L^∞-bounds of ' and ' in (<ref>), the bounds of g in (<ref>) and (<ref>), one has
	
    2|'(z)-g(t,z)^1-'(z)|/g(t,z)^2≤C(ρ̅^*)^2/ρ̅_*(_̨0+P̅_̨1/(ρ̅_*)^2(1+(ρ̅^*/ρ̅_*)^-1)),

	where C=C(,u̅_*).
	This together with the estimate for |_c'| in (<ref>), the estimate for _tg_$̌ in (<ref>), the lower bound of_cin (<ref>), and the support condition  of_tg_ϵin (<ref>) yields  (<ref>).
	To show (<ref>), first we note that from (<ref>) and (<ref>) one has
	
    _zg_ϵ(t,z)=0  in {(t,z):t>(1-ϵ/2)𝔱_c(z),  z∈}.

	Then (<ref>) follows from (<ref>) and the upper bound of_cin (<ref>). 
	
	
	
	
	This finishes the proof of the lemma.


§ VARIATIONAL FORMULATION FOR THE TRUNCATED  PROBLEM


In this section we show that the equation (<ref>), which has a complicated memory term, can be written as an Euler-Lagrange equation of an energy functional. Thus  Problem <ref> can be transformed into a domain variation problem as in the isentropic case (cf. <cit.>). This is the key to take advantage of the variational method developed by Alt, Caffarelli and Friedman to solve the jet problem.

In this section, we always assume that the upstream pressureP̅of the flow satisfies (<ref>).

LetØbe the domain bounded by_0and_1∪([0,∞)×{1}). Noticing thatØis unbounded, we make an approximation by considering the problem in a series of truncated domainsØ_μ,R:=Ø∩{-μ<x_1<R}, whereμandRare two large positive numbers. To define the energy functional, we set

    G_(̌t,z):=1/2∫_0^tg_(̌τ,z)dτ+-1/(g_(̌0,z)^-(z)-g_(̌0,Q)^-(Q)),

which is smooth intandC^1,1inzby the regularity ofg_$̌ (cf. Lemma <ref>), and set
Φ_(̌t,z):=-G_(̌t,z)+2_tG_(̌t,z)t.
Given a function ψ^♯_μ,R∈ C(Ø_μ,R)∩ H^1(Ω_μ,R) with 0≤ψ^♯_μ,R≤ Q, we consider the minimization problem:

    find ψ∈_ψ^♯_μ,R  s.t.  J_μ,R,^(̌ψ)= inf_φ∈_ψ^♯_μ,R J_μ,R,^(̌φ),

where the energy functional

    J^_̌μ,R,(φ):=∫_Ω_μ,RG_(̌|φ|^2,φ)+_^̌2χ_{φ<Q}dx, _:̌ =√(Φ_(̌^2,Q)),

with  defined in (<ref>), and the admissible set

    _ψ^♯_μ,R:={φ∈ H^1(Ø_μ,R):φ=ψ^♯_μ,R on Ø_μ,R}.

Note that Φ_(̌^2,Q)>0. Indeed, with the aid of the elliptic condition (<ref>), a straightforward computation shows that t↦Φ_(̌t,z) is monotone increasing:

    0<1/2c_*(ρ̅^*)^-1≤_tΦ_(̌t,z)=1/2g_(̌t,z)+_tg_(̌t,z)t≤1/2 (c_*̅̌ρ_*)^-1.

Since Φ_(̌0,Q)=-G_(̌0,Q)=0, one has Φ_(̌t,Q)>0 for all t>0.

The existence and Hölder regularity of minimizers for (<ref>) will be shown in Section <ref>. Let us assume the existence and continuity of a minimizer ψ firstly. We derive the Euler-Lagrange equation for the variational problem (<ref>) in the open set {ψ<Q}.



Let ψ be a minimizer of (<ref>). Assume that Ø_μ,R∩{ψ<Q} is open. Then ψ is a solution to

    (̧g_(̌|ψ|^2,ψ)ψ)-_zG_(̌|ψ|^2,ψ)=0 in Ø_μ,R∩{ψ<Q}.

Furthermore, if |ψ|^2≤(1-)̌_c(ψ), it holds that

    _zG_(̌|ψ|^2,ψ)='(ψ)/g(|ψ|^2,ψ)-'(ψ)/ g(|ψ|^2,ψ)^.

Let η∈ C^∞_0(Ø_μ,R∩{ψ<Q}). Direct computations give
d/dϑJ_μ,R,^(̌ψ+ϑη)|_ϑ=0=∫_Ø_μ,R2_tG_(̌|ψ|^2,ψ)ψη+_zG_(̌|ψ|^2,ψ)η.
By the definition of G_$̌ in (<ref>), minimizers of (<ref>) satisfy the equation (<ref>).

In addition, note that

    g_=̌g  in ℛ_:̌={(t,z): 0≤ t≤(1-)̌_c(z),  z∈}.

By the definitions ofG_$̌ in (<ref>) and g_$̌ in (<ref>), for(t,z)∈ℛ_ϵ, one has

    _zG_(̌t,z)   =1/2∫_0^t_z g(τ,z)dτ-(-1)g(0,z)^--1_zg(0,z)(z)+-1/g(0,z)^-'(z)
       =: I+II+III.

Using (<ref>) yields

    I   =-∫_0^t_t g(τ,z)'(z)-g(τ,z)^1-'(z)/g(τ,z)^2dτ
       ='(z)∫_0^t_t(̊τ,z)dτ-1/'(z)∫_0^t_t((̊τ,z)^)dτ
       ='(z)((̊t,z)-(̊0,z))-1/'(z)((̊t,z)^-(̊0,z)^).

Since takingt=0in (<ref>) gives

    _z(̊0,z)='(z)-(̊0,z)^-1'(z)/(-1)(̊0,z)^-2(z),

one has

    II=(-1)(̊0,z)^-1_z(̊0,z)(z)
    =(̊0,z)('(z)-(̊0,z)^-1'(z)).

Moreover,III=-1/(̊0,z)^'(z).Combining the above equalities together yields

    _zG_(̌t,z)='(z)/g(t,z)-'(z)/ g(t,z)^,     (t,z)∈ℛ_.

This finishes the proof for the lemma.


The following lemma implies that minimizers for (<ref>) satisfy the free boundary condition in a weak sense.
The proof is the same as that in <cit.>, so we omit it here.


Let ψ be a minimizer for (<ref>). Assume that Ø_μ,R∩{ψ<Q} is open. Then
Φ_(|ψ|^2,ψ)=_^̌2   on _ψ:=Ø_μ,R∩{ψ<Q}
in the sense that
lim_s→0+∫_{ψ<Q-s}(Φ_(|ψ|^2,ψ)-_^̌2)(ην̧)dℋ^1=0for any η∈ C_0^∞(Ø_μ,R;^2),
where ℋ^1 is the one-dimensional Hausdorff measure.

In view of Lemma <ref>, if a minimizer ψ is smooth near the free boundary Γ_ψ and Γ_ψ is also smooth, then by the monotonicity of t↦Φ_(̌t,z) for each z (cf. (<ref>)) and the definition of _$̌ in (<ref>) one has|ψ|=on_ψ.
Moreover, sinceλ_ϵ^2=Φ_ϵ(Λ^2, Q) =Φ_ϵ(0, Q)+∫_0^Λ^2_tΦ_ϵ(s,Q)ds,it follows from (<ref>) andΦ_ϵ(0,Q)=0that
	
    1/2c_*(ρ̅^*)^-1Λ^2≤λ_ϵ^2≤1/2 (c_*̅̌ρ_*)^-1Λ^2,

wherec_*=c_*()>0is defined in Lemma <ref>,ρ̅_*andρ̅^*are the lower and upper bounds ofρ̅defined in (<ref>) and (<ref>), respectively.


At the end of the section, we summarize the properties of the functionG_$̌. For this we denote

    (∓,z):=G_ϵ(|∓|^2,z).

	Let G_ϵ be defined in (<ref>) and  be defined in (<ref>). Then (∓,z) is smooth in ∓ and C^1,1 in z. Furthermore, suppose that the density ρ̅ and pressure P̅ of the flow at the upstream satisfy (<ref>) and (<ref>), where the constant ̨̅=̨̅(ρ̅_*, u̅,,H̅) in (<ref>) is sufficiently small.
	
	Then the following properties hold:
	
  (i)There exist  positive constants  𝔟_∗=c_*(ρ̅^*)^-1 and 𝔟^∗=(c_*ρ̅_*)^-1, where  c_∗>0 is a constant depending only on γ, such that
    𝔟_*|∓|^2    ≤ p_i ∂_p_i (∓,z) ≤𝔟^*|∓|^2,
    𝔟_* |ξ|^2    ≤ξ_i∂_p_ip_j(∓,z)ξ_j ≤𝔟^* ϵ^-1 |ξ|^2   for all ξ∈^2.

  (ii) One has
		
    _z(,z)= 0   in {(, z): ∈^2, z∈ (-∞,0]},
       _z(,z)≥ 0   in {(, z): ∈^2, z∈ [Q,∞)}.

  (iii) There exist constants
		
    δ':=
    			C^̌-1_̨0 andδ:=(1+_̨0)δ'/ρ̅_*,

		where κ_0 and _̨1 are defined in (<ref>) and C=C(,u̅_*)>0, such that
		
    ϵ^-1|_z (∓,z)|+|∓·_∓ z(∓,z)|≤δ',
       |_∓ z(∓,z)|+|_zz(∓,z)|≤δ, 
       (0,Q)=0,  (∓,z)≥𝔟_*/2|∓|^2-
    			̣̌'min{Q, (Q-z)_+}.
(i). In view of the definition of G_$̌ in (<ref>), straightforward computations yield
	
    p_i_p_i(∓,z)=g_(̌|∓|^2,z)|∓|^2,
       _p_ip_j(∓,z)=g_(̌|∓|^2,z)_̣ij+2_tg_(̌|∓|^2,z)p_ip_j.

	Thus from (<ref>) and (<ref>) one immediately gets  (<ref>) and (<ref>).
	
	(ii). From the definition ofand the equality (<ref>) one has
	
    _z(∓,z)   =1/2∫_0^|∓|^2_zg_(̌τ,z)dτ-(-1)g(0,z)^--1_zg(0,z)(z)+-1/g(0,z)^-'(z)
       (<ref>)=1/2∫_0^|∓|^2_zg_(̌τ,z)dτ+'(z)/g(0,z)-'(z)/ g(0,z)^.

			
			
			
	This, together with the explicit expression for_zg_ϵin (<ref>) and an integration by parts yields that
	
    _z (,z)=   -∫_0^||^2(
    			ℬ'(z)/g(τ,z)-𝒮'(z)/ g(τ,z)^-1/2
    			(g^∗ - g(τ,z)) τ_c'(z)/_c(z))ϖ'_ϵ(τ/_c(z))/_c(z) dτ
       +(ℬ'(z)/g(||^2, z)-𝒮'(z)/ g(||^2, z)^)ϖ_ϵ(||^2/_c(z)).

	Note thatℬ'(z)=𝒮'(z)=0on(-∞, 0]and'(z)≥0,'(z)=0on[Q,∞), cf. (<ref>). If we can show that
		
    g(τ,z)(g^∗ - g(τ,z)) (2(z)/(+1)(z))^2/-1≤ 1

	in the setℛ:={(τ,z):τ∈ [0,_c(z)),  z∈(-∞,0]∪[Q,∞)}, then it follows from the expression of_c'in (<ref>),ϖ'_≤̌0,ϖ_≥̌0andg≥ 0that_z (, z)is a positive multiple ofℬ'(z). In view of (<ref>), we consequently conclude that (<ref>) holds true.
	
	
	
	
    Now it remains to prove (<ref>). Using (<ref>) and (<ref>) yields
	
    0≤ 4g(τ,z)(g^∗-g(τ,z))≤ (g^∗)^2 = (2/+1)^-2/-1(ρ̅_*)^-2.

	By (<ref>) and the upper bound of/in (<ref>), one can check that the inequality (<ref>) is equivalent to
	
    ρ̅^*/ρ̅_*≤(2^/+1)^1/-1.

	SinceP̅>P̅_*>ρ̅_*(u̅_*)^2/(cf. (<ref>)), the inequality (<ref>) is guaranteed by the condition (<ref>) provided̨̅=̨̅(ρ̅_*, u̅,,H̅)is sufficiently small.
	This finishes the proof of (ii).
	
	(iii). In order to prove (<ref>)-(<ref>), we let the constant̨̅in (<ref>) be sufficiently small such that̨̅≤_̨0(ρ̅_*)^2. Consequently,
	
    P̅_̨1/(ρ̅_*)^2≤_̨0.
Proof for (<ref>): It follows from the estimate for_c'in (<ref>), the bounds ofgin (<ref>), the estimate (<ref>), and theL^∞-bounds of'and'in (<ref>) that
	
    |g^∗_c'(z)|+|ℬ'(z)/g(τ, z)|+|𝒮'(z)/ g(τ, z)^|≤ C C^*,

	whereC=C(,u̅_*)>0and
	
    C^*:=ρ̅^*/ρ̅_*(1+ρ̅^*/ρ̅_*)(_̨0+P̅_̨1/(ρ̅_*)^2(1+(ρ̅^*/ρ̅_*)^-1))(<ref>)-(<ref>)≤ C(γ)_̨0.

	Thus by the expression of_zin (<ref>) and the support condition ofϖ_ϵ',
	
	
	
	
	
	one has
	
    |_z(∓,z)|≤ CC^*

	for someC=C(,u̅_*)>0.
	Besides, a direct differentiation of_zyields
	
    _p_iz(∓,z) = p_i _zg_ϵ (|∓|^2,z).

	Thus by (<ref>) and (<ref>)-(<ref>) one has
	
    |∓·_∓ z(∓,z)|=|∓|^2|_zg_ϵ(|∓|^2,z)|≤
    		C^̌-1_̨0

	withC=C(,u̅_*).
	This finishes the proof of (<ref>).
	
	Proof for (<ref>): Using (<ref>), the estimates of_zg_$̌ in (<ref>) and (<ref>), the upper bound of _c in (<ref>), and (<ref>)-(<ref>), one gets
	
    |_∓ z(∓,z)|=|∓||_zg_ϵ(|∓|^2,z)|
    		
    		   ≤ C^̌-1(P̅ρ̅_*)^-1/2_̨0

	with C=C(,u̅_*).
	
	For the estimate of _zz, note that by the expression of _z in (<ref>) and the expression of _zρ(0,z) in (<ref>) it holds
	
    _zz(∓,z)=   1/2∫_0^|∓|^2_zzg_(̌τ,z)dτ+”(z)/g(0,z)-”(z)/ g(0,z)^
       +('(z)-'(z)/g(0,z)^-1)^2g(0,z)^-2/(-1)(z).

	In view of the estimates about derivatives of  and  in (<ref>), the bounds of g in (<ref>), the lower bound of  in (<ref>), and (<ref>)-(<ref>), the last three terms on the right-hand side of (<ref>) are bounded by
	
	
	
	
    C_̨0/ρ̅_*(1+_̨0+_̨1/ρ̅_*)+C_̨0^2/P̅,

	where C=C(γ,u̅_*). To estimate the first term on the right-hand side of (<ref>), noting that from (<ref>) and that  _z g (t,z)=  2('(z)-g(t,z)^1-'(z))_t(1/g(t,z)), cf. (<ref>), one has
	
    _zzg_(̌τ,z) =   2(ℬ”(z)_t(1/g(τ,z)) + ℬ'(z) _tz(1/g(τ,z)) ) ϖ_(τ/_c(z)) 
       -2/γ(𝒮”(z)_t(1/g(τ,z)^γ) + 𝒮'(z) _tz(1/g(τ,z)^γ) ) ϖ_(τ/_c(z)) 
       +  2_z g(τ,z)d/dzϖ_(τ/_c(z)) +(g(τ,z)-g^∗)d^2/dz^2ϖ_(τ/_c(z)).

	We plug it into the expression of _zz. For the term involving _tz(1/g(τ,z)), an integration by parts yields
	
    ∫_0^|∓|^2_tz(1/g(τ,z))ϖ_ϵ(τ/_c(z))  dτ
     =    -_zg/g^2(|∓|^2, z)ϖ_ϵ(|∓|^2/_c(z))+_zg/g^2(0,z)
    		+∫_0^|∓|^2_zg/g^2(τ,z) ϖ'_ϵ(τ/_c(z))/_c(z) dτ.

	The term involving _tz(1/g^γ(τ,z)) can be  estimated similarly as above. Note that
	|ϖ”_ϵ(τ/_c(z))|≤ 8ϵ^-2  and   | supp(ϖ”_ϵ(τ/_c(z)))|≤.̌
	Thus we infer from the estimates (<ref>), the bounds for g in (<ref>), the upper bound for _tg in (<ref>), the relation between _zg and _tg in (<ref>), the estimates of _c' and _c” in (<ref>)-(<ref>) and (<ref>)-(<ref>) that
	
	
	
	
	
    |_zz|≤   C^̌-1_̨0/ρ̅_*(1+_̨0+_̨1/ρ̅_*)+C^̌-1_̨0^2/P̅,

	where C=C(γ,u̅_*). Note that P̅ satisfies (<ref>) and _̨1<̨̅/P̅
	by (<ref>) for ̨̅≤_̨0(ρ̅_*)^2.
	The inequality (<ref>) together with (<ref>) gives the estimate (<ref>).
	
	Proof for (<ref>): In view of the definition of  in (<ref>) and the definition of G_ϵ in (<ref>), one gets (0,Q)=0. The second inequality in (<ref>) follows directly from the strong convexity in ∓ and the estimates of derivatives in z in (<ref>). More precisely, the strong convexity property (<ref>) gives that
	
    (∓,z)≥(0,z)+ 𝔟_∗/2|∓|^2,

	where we have also used that ∇_𝐩(0,z)=0 (since  ∇_𝐩(∓,z)=2𝐩_tg_ϵ(|∓|^2,z)). To estimate (0,z) we use a first order Taylor expansion at z=Q.
	Then for z∈ [0,Q], one has
		
    |(0,z)|=|(0,Q)+ _z(0,ξ_z)(z-Q)|≤̣̌' (Q-z),

		where |_z(∓, z)|≤̣̌' from (<ref>) is used in the last inequality. Moreover, we infer from 
		(<ref>) that
		𝒢(0,z)=𝒢(0,0)≥ - ̣̌'Q
		for z<0, and 𝒢(0,z)≥𝒢(0,Q)=0 for z>Q.
	Combining the above estimates together we obtain the second inequality in (<ref>).
	This completes the proof of the proposition.


§ EXISTENCE, REGULARITY AND FINE PROPERTIES OF SOLUTIONS TO THE TRUNCATED PROBLEM


In the previous sections, we reformulate the truncated free boundary problem into a variational problem (<ref>), which has exactly the same formulation as in <cit.>. With Proposition <ref> at hand, one can argue along the same lines as in <cit.> to obtain the existence, regularity and fine properties of a minimizer and its free boundary for the variational problem (<ref>). For completeness we briefly states the conclusions here.




 §.§ Existence and regularity of solutions to the truncated problem

The existence of minimizers for the variational problem (<ref>) follows from standard theory for calculus of variations.
(<cit.>)
	Assume  defined in (<ref>) satisfies (<ref>) and (<ref>). Then the problem (<ref>) has a minimizer.


Throughout the rest of this section, we always assume that the conditions (<ref>) and (<ref>) hold, where the constant ̨̅=̨̅(ρ̅_*, u̅,,H̅) in (<ref>) is sufficiently small. Then the function  defined in (<ref>) satisfies all properties in Proposition <ref>.
In view of <cit.>, a minimizer ψ has the following properties.


	Let ψ be a minimizer for (<ref>). Then the following statements hold:
	
  (i)ψ is a supersolution of the elliptic equation
		
    ∂_i(_p_i(∇ψ, ψ))- _z (∇ψ,ψ) = 0,

		in the sense of
		
    ∫_Ø_μ,R(_p_i(∇ψ, ψ)∂_i ζ + _z (∇ψ,ψ) ζ)≥ 0,  for all ζ≥ 0, ζ∈ C_0^∞(Ø_μ,R).

  (ii)ψ satisfies 0 ≤ψ≤ Q.	
		
  (iii)ψ∈ C^0,α_loc(Ø_μ,R) for any α∈ (0,1).
	


Note that by the definition of Q in (<ref>) and the upper bound of ρ̅^*/ρ̅_* in (<ref>) one has

    ρ̅_*u̅_L^1([0,H̅])≤ Q≤ρ̅^*u̅_L^1([0,H̅])≤ Cρ̅_*u̅_L^1([0,H̅]),

where the constant C>0 only depends on . Then with the help of the comparison principle for the elliptic equation (<ref>) (cf. <cit.>), one can prove the following (optimal) Lipschitz regularity and the nondegeneracy property of the minimizers as in <cit.>.
	Let ψ be a minimizer for (<ref>), then ψ∈ C^0,1_loc(Ω_μ,R). Moreover, for any connected domain K⋐Ω_μ,R containing a free boundary point, the Lipschitz constant of ψ in K is estimated by CΛ, where C depends on γ, ϵ, u̅, Λ/Q,  K, and dist(K,Ω_μ,R).

	Let ψ be a minimizer for (<ref>). Then for any p>1 and any 0<r<1, there exist positive constants c_r=c_r(γ,ϵ,u̅, r,p) small
		and R_0=R_0(γ,ϵ,u̅, r)∈ (0,1), 
		such that for any B_R(x̅)⊂Ω_μ,R with R≤min{R_0, c_rΛ/Q},
	if
	
    1/R( 
    		_B_R(x̅)|Q-ψ|^p)^1/p≤ c_rΛ,

	then ψ=Q in B_rR(x̅).


The measure theoretic properties of the free boundary follows from the Lipschitz regularity and nondegeneracy of the minimizers.
Then with the help of the estimates for |ψ| near the free boundary and the improvement of flatness arguments, one obtains the following regularity of the free boundary.

	Let ψ be a minimizer for (<ref>). Let  be the function defined in (<ref>) which  satisfies all properties in Proposition <ref>. Then the free boundary Γ_ψ is locally C^k+1,α if (,z) is C^k, α in its components (k≥1, 0<α<1), and it is locally real analytic if (,z) is real analytic.

Under our assumptions that ρ̅, u̅∈ C^1,1([0,H̅]), the function (∓,z) is smooth in ∓ and C^1,1 in z. Thus in our case the free boundary is locally C^2,α for any α∈ (0,1).


 §.§ Fine properties of the free boundary.

In order to show the fine properties of the free boundary, we take a specific boundary value on Ø_μ,R for the variational problem (<ref>).




Let s∈ (1/2,1) be a fixed constant and b_μ∈(1,H̅) such that Θ(b_μ)=-μ, where Θ is defined in (<ref>). Choose a point (-μ, b'_μ) with 1<b'_μ< b_μ and k_μ:=b_μ-b'_μ small depending on u̅, γ, and $̌. Defineψ^†(x_2)as the minimizer of∫_0^H̃G_ϵ(|v'|^2,v)in the admissible setK^†:={v∈ C^0,1([0,H̃];ℝ): v(0)=0, v(H̃)=Q}, whereH̃∈(0,1)is small enough such that(ψ^†)'(H̃)>Λ.
Set

    ψ^♯_μ,R(x_1,x_2):={   0       if  x_1=-μ,    0<x_2<b'_μ,
       Q(x_2-b'_μ/k_μ)^1+s      if  x_1=-μ,    b'_μ≤ x_2≤ b_μ,
       min{ψ^†,Q}      if  x_1=R,    0<x_2<1,
       0       if  x_2=0,
       Q       if  (x_1, x_2)∈_1∪([0,R]×{1}).
    	.

Note thatψ^♯_μ,Ris continuous and it satisfies0≤ψ^♯_μ,R≤ Q.
Ifk_μis sufficiently small,ψ^♯_μ,R(-μ,)̧is a subsolution andψ^♯_μ,R(R,)̧is a supersolution to (<ref>) inØ_μ,R.

Letψ_μ,R,be a minimizer for (<ref>) with the boundary valueψ_μ,R^♯.
By the strong maximum principle, one hasψ^♯_μ,R(-μ,x_2)<ψ_μ,R,(x_1,x_2)≤ψ^♯_μ,R(R,x_2) for all  (x_1,x_2)∈Ø_μ,R.Then one can show thatψ_μ,R,Λis the unique minimizer for  (<ref>), andψ_μ,R,Λis monotone increasing in thex_1-direction, i.e.,_x_1ψ_μ,R,≥0inØ_μ,R. Consequently, the free boundaryΓ_ψ_μ,R,Λ(which is not empty with the boundary valueψ^♯_μ,R) is anx_2-graph. More precisely, there exists a function_,∈(-μ,R]and H_μ,R,∈[H̃,1)such that_ψ_μ,R,Λ={(x_1,x_2): x_1=_μ,R,(x_2),  x_2∈( H_μ,R,,1)}.By the non-oscillation property (<cit.>) for the free boundary and the unique continuation property (cf. <cit.> and <cit.>) for the elliptic equation, one concludes that_μ,R,is continuous andlim_x_2→1-_μ,R,(x_2)exists.



 §.§ Continuous fit and smooth fit.

The following proposition shows the existence of a suitable>0such that the free boundary_ψ_μ,R,fits the outletA=(0,1)of the nozzle continuously.


	Let ψ:=ψ_μ,R,Λ be the minimizer of (<ref>) with the boundary value ψ^♯_μ,R as in (<ref>). There exists Λ=Λ(μ, R)>0 such that Υ_μ,R,Λ(1)=0. Moreover, Λ satisfies C^-1 Q ≤Λ(μ, R)≤ C Q for positive constant C depending on γ, ϵ, and u̅, 
	but independent of μ and R.


The continuous fit implies the following smooth fit.


		Let ψ:=ψ_μ,R,Λ be the minimizer of (<ref>) with the boundary value ψ^♯_μ,R as in (<ref>) and  Λ as in Proposition <ref>.
	Then _1∪Γ_ψ is C^1 in a neighbourhood of the point A=(0,1), and ∇ψ is continuous in a {ψ<Q}-neighborhood of A.


§ EXISTENCE AND UNIQUENESS OF THE SOLUTION TO THE JET PROBLEM

In this section, we first remove the domain and subsonic truncations, and then study the far fields behavior of the jet flows. Consequently the existence and uniqueness of the solution to Problem <ref> are obtained.
The proof is essentially the same as that in <cit.>. Here we only give the details for the removal of subsonic truncation.


 §.§ Remove the the domain truncations.


Let the nozzle boundary_1defined in (<ref>) satisfy (<ref>).
Given a densityρ̅∈ C^1,1([0,H̅]), a horizontal velocityu̅∈ C^1,1([0,H̅])and a pressureP̅>0of the flow at the upstream, which satisfy  (<ref>)-(<ref>) and (<ref>) with the constant̨̅=̨̅(ρ̅_*, u̅,,H̅)in (<ref>) sufficiently small.
Letψ_μ,R,Λbe the minimizer of the problem (<ref>)
withψ^♯_μ,Rdefined in (<ref>). Then for anyμ_j, R_j→∞, there is a subsequence (still labelled byμ_jandR_j) such thatΛ_j:=Λ(μ_j, R_j)→Λ_∞for someΛ_∞∈ (0,∞)andψ_μ_j, R_j, Λ_j→ψ_∞inC^0,α_loc(Ω)for anyα∈ (0,1).
Furthermore, the following statements hold:

  (i) The function ψ:=ψ_∞ is a local minimizer for the energy functional, i.e., for any D⋐Ω,  one has  J^ϵ(ψ_∞)≤ J^ϵ(φ) for all  φ=ψ_∞ on ∂ D, where
	
    J^ϵ(φ):=∫_D G_ϵ(|∇φ|^2,φ)+λ_ϵ,∞^2 χ_{φ<Q} dx
    		  and  λ_ϵ,∞:=√(Φ_ϵ(Λ_∞^2,Q)).

	Besides, ψ solves
	
    {   ∇·(g_ϵ(|∇ψ|^2,ψ)∇ψ)-_z G_ϵ (|∇ψ|^2,ψ)=0        in 𝒪,
       ψ =0        on _0,
       ψ =Q        on _1 ∪Γ_ψ,
       |∇ψ| =Λ_∞       on Γ_ψ,
    		.

	where 𝒪:=Ω∩{ψ<Q} is the flow region, Γ_ψ:={ψ<Q}\_1 is the free boundary, and  _zG_ϵ(|∇ψ|^2,ψ) satisfies (<ref>) in the subsonic region |∇ψ|^2≤ (1-ϵ)_c(ψ).
	
  (ii) The function ψ is in C^2,α(𝒪)∩ C^1(𝒪), and it satisfies _x_1ψ≥ 0 in Ω.
	
  (iii) The free boundary Γ_ψ is given by the graph x_1=Υ(x_2), where Υ is a C^2,α function as long as it is finite.
	
  (iv) At the orifice A=(0,1) one has lim_x_2→ 1-Υ(x_2)=0. Furthermore, _1∪Γ_ψ is C^1 around A, in particular, Θ'(1)=lim_x_2→ 1-Υ'(x_2).
	
  (v) There is a constant H∈ (0,1) such that Υ(x_2) is finite if and only if x_2∈ ( H, 1], and lim_x_2→ H+Υ(x_2)=∞. Furthermore, there exists a constant a>0 sufficiently large, such that Γ_ψ∩{x_1>a}= {(x_1, f(x_1)): a<x_1<∞} for some C^2,α function f and lim_x_1→∞f'(x_1)=0.


 §.§ Remove the subsonic truncation

In this subsection, we remove the subsonic truncation introduced in Section <ref> provided the upstream pressure of the flow is sufficiently large. 


Let ψ:=ψ_∞ be a solution obtained in Section <ref>.
Suppose that P̅>P̅^* for some P̅^*=P̅^*(ρ̅_*,u̅, , ,̌_1)≥P̃_*, where P̃_* is defined in (<ref>).
Then
	
    |ψ|^2≤(1-)̌_c(ψ)

	where _c is defined in (<ref>).

	In the flow region 𝒪:=Ω∩{ψ<Q}, ψ satisfies the following equation of non-divergence form
	
    a_ϵ^ij(∇ψ,ψ) _ijψ=F_ϵ(|∇ψ|^2, ψ),

	where the matrix
	(a_ϵ^ij)=g_ϵ(|∇ψ|^2, ψ)I_2+2_t g_ϵ(|∇ψ|^2, ψ)∇ψ⊗∇ψ
	is symmetric with the eigenvalues
	
    β_0,ϵ  =g_ϵ(|∇ψ|^2, ψ)  and  β_1,ϵ =g_ϵ(|∇ψ|^2, ψ)+2_t g_ϵ(|∇ψ|^2, ψ)|∇ψ|^2,

	and 
	F_ϵ(|∇ψ|^2, ψ)=-_z g_ϵ(|∇ψ|^2, ψ) |∇ψ|^2+_zG_ϵ(|∇ψ|^2, ψ).
	
	It follows from Lemma <ref>, (<ref>) and (<ref>)-(<ref>) that there exist constants C_1=C_1(γ)>1 and C_2=C_2(,u̅_*)>0 such that
	
    1≤β_1,ϵ/β_0,ϵ≤ C_1 ϵ^-1  and  |F_ϵ/β_0,ϵ|≤ C_2^̌-1_̨0ρ̅^*.

	Note that Q∼ρ̅^* by (<ref>).
	With (<ref>) at hand, one can use similar arguments as in <cit.> to get
	
    ψ_C^1(𝒪)≤ C(1+Q+F_ϵ/β_0,ϵ_L^∞(ℝ×[0,Q]))≤ C(1+(1+_̨0)ρ̅^*)

	where C=C(u̅, γ,ϵ,_1). This together with the lower bound of _c in (<ref>) and (<ref>) yields (<ref>), provided P̅=P̅(ρ̅_*, u̅, ,,̌_1) is sufficiently large.
	
Note that the upstream density ρ̅ and the upstream pressure P̅ of the flow need satisfy (<ref>) with ̨̅=̨̅(ρ̅_*, u̅, ,H̅)>0, cf. Section <ref>. For fixed ∈̌(0,1/4), if 

    ρ̅'_C^0,1([0,H̅])<̨̅/P̅^*,

where P̅^* is defined in Proposition <ref>, then (<ref>) holds as long as the upstream pressure P̅ satisfies 

    P̅^*<P̅< P̂:=̨̅/ρ̅'_C^0,1([0,H̅])

(without loss of generality we assume ρ̅'_C^0,1([0,H̅])>0). Therefore, by (<ref>) the limiting solution obtained in Section <ref> is exactly a solution of  (<ref>). 


 §.§ Asymptotic behavior of the jet flow.

The solution obtained in Section <ref> has the following asymptotic behavior asx_1→±∞.


Let the nozzle boundary _1 defined in (<ref>) satisfy (<ref>). Given a density ρ̅∈ C^1,1([0,H̅]), a horizontal velocity u̅∈ C^1,1([0,H̅]), and
a pressure P̅>P̅_* for P̅_* defined in (<ref>), such that (<ref>)-(<ref>) hold with the constant ̨̅=̨̅(ρ̅_*,u̅,,H̅) sufficiently small. Let ψ be a solution of (<ref>) with Λ:=Λ_∞>0. Assume that ψ and the free boundary Γ_ψ satisfy the properties (ii)-(v) in Section <ref> and (<ref>). Then for any ∈(0,1), as x_1→ -∞,
	
    ψ(x_1,x_2)→ψ̅(x_2):=∫_0^x_2(ρ̅u̅)(s)ds,   in  C_loc^2,α([0,H̅));

	as x_1→∞,
	
    ψ(x_1,x_2)→ψ(x_2):=∫_0^x_2(ρ u)(s)ds,   in C_loc^2,α([0, H)),

	where ρ∈ C^1,α([0, H]) is the positive downstream density, u∈ C^1,α([0, H]) is the positive downstream horizontal velocity, and H>0 is the downstream asymptotic height. Furthermore, the downstream states (ρ,u, H) are uniquely determined by P̅, ρ̅, u̅, γ, ϵ, and Λ.

	Under the assumptions of Proposition <ref>, we refer from Proposition <ref> and Remark <ref> that the downstream pressure P, which is defined by
	
    P=(γ-1)(ψ)/γ g(|ψ|^2,ψ)^γ,

	must be a constant.
	Thus using (<ref>) one has
	 P=(-1)ρ^( H)(Q)/.


From Propositions <ref>, we get the asymptotics for the solution obtained in Section <ref>. As shown in <cit.>, one has_x_1ψ_∞>0in the flow region. Thus in view of Proposition <ref>, we have proved the existence of solutions to Problem<ref> when the upstream pressureP̅is sufficiently large.
Then the existence of solutions to Problem <ref> follows from Proposition <ref>.



 §.§ Uniqueness of the jet flow.

The following proposition gives the uniqueness of the solution to the jet problem. For the proof we refer to <cit.>.


		Let the nozzle boundary _1 defined in (<ref>) satisfy (<ref>) and (<ref>).
		Given a density ρ̅, a horizontal velocity u̅, and a pressure P̅ of the flow at the upstream as in Proposition <ref>.
		If (ψ_i, Γ_i, Λ_i) (i=1, 2) are two uniformly subsonic solutions to Problem <ref>, then they must be the same.


§ EXISTENCE OF THE CRITICAL PRESSURE


For given densityρ̅, horizontal velocityu̅, and pressureP̅of the flow at the upstream as in Section <ref>, we have shown that there exist subsonic jet  flows as long asP̅>P̅^*for someP̅^*sufficiently large. In this section, we decreaseP̅^*as small as possible until some critical pressureP̅_c, such that forP̅≤P̅_ceither the flow fails to be global subsonic or the stream functionψfails to be a solution in the sense of Problem <ref>.


Let the nozzle boundary _1 defined in (<ref>) satisfy (<ref>) and (<ref>).
Given a density ρ̅∈ C^1,1([0,H̅]) and a horizontal velocity u̅∈ C^1,1([0,H̅]) satisfying (<ref>). Assume that ρ̅ satisfies (<ref>).  
There exists a critical pressure P̅_c∈[P̅_*,P̅^*] such that if P̅∈(P̅_c,P̂), where P̅_* is defined in (<ref>), P̅^* is defined in Proposition <ref> and P̂ is defined in (<ref>), then there exists a unique solution ψ to Problem <ref>, which satisfies

    0< ψ< Q   in 𝒪  and  𝔐(P̅):=sup_𝒪|∇ψ|^2/_c(ψ)<1,

where the domain 𝒪 is bounded by _0, _1, and Γ. Furthermore, either 𝔐(P̅)→ 1 as P̅→P̅_c+, or there does not exist a σ>0 such that Problem <ref> has solutions for all P̅∈ (P̅_c-σ,P̅_c) and
	
    sup_P̅∈(P̅_c-σ,P̅_c)𝔐(P̅)<1.

	The proof is basically the same as in <cit.>. For completeness we give the details.
	
    
	
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 Section <ref>.
Letψ_n(;̧P̅)be the function (if it exists) such that

  (i)ψ_n(x; P̅)∈  C^2,α({ψ_n(x;P̅)<Q})∩ C^1({ψ_n(x;P̅)<Q}) satisfies

    (̧g__̌n(|ψ|^2,ψ)ψ)-_zG__̌n(|ψ|^2,ψ)=0    in {ψ<Q},
    ψ=0    on _0,
    ψ=Q    on _1∩_ψ,
    
    		|ψ|=   on _ψ,

	where the mass flux Q is defined in (<ref>), the functions g_ϵ_n and G_ϵ_n are defined in (<ref>) and (<ref>), respectively;
	
  (ii)ψ_n(x; P̅) satisfies ∂_x_1ψ_n(x;P̅)>0 in {ψ_n(x;P̅)<Q};
	
  (iii)  the free boundary Γ_ψ_n satisfies the properties (1)-(3) in Problem <ref>; furthermore, there exist constants C_1=C_1(ϵ_n, P̅, τ) and C_2=C_2(ϵ_n, P̅, a) (with τ, a>0) depending on P̅ continuously such that
	
    Υ_C^2, α([H+τ, 1-τ ])≤ C_1(ϵ_n, P̅, τ)   and  f_C^2,α((a, ∞))≤ C_2(ϵ_n, P̅, a),

	where Υ and f are the functions appeared in the properties (1) and (3) in Problem <ref>, respectively.

Moreover, if|ψ_n(x;P̅)|^2/_c(ψ_n(x;P̅))≤1-_̌n,thenψ_n(x;P̅)solves (<ref>), and it satisfies the far fields behavior claimed in Proposition <ref> and

    0≤ψ_n(x;P̅)≤ Q.

Set𝔖_n(P̅):={ψ_n(x;P̅):ψ_n(x;P̅)  satisfies the properties (i)–(iii)}.Define_n(P̅):=inf_ψ_n∈𝔖_n(P̅)sup_|ψ_n(x;P̅)|^2/_c(ψ_n(x;P̅))and𝔗_n:={s:s≥P̅_*, _n(P̅)≤1-_̌n  if P̅∈(s,P̂)}.It follows from Proposition <ref> and Remark <ref> that[P̅^*,P̂)⊂𝔗_nfor sufficiently largen. Hence𝔗_nis not an empty set. DefineP̅_n:=inf𝔗_n.

The sequence{P̅_n}has some nice properties.

First,_n(P̅)is right continuous forP̅∈[P̅_n,P̂). Let{P̅_n,k}⊂(P̅_n,P̂)andP̅_n,k↓P̅ask→∞. Since_n(P̅_n,k)≤1-_̌n, one hasψ_n(x;P̅_n,k)_C^0,1(_n,k)≤ C  andψ_n(x;P̅_n,k)_C^2,(_n,k\ B_r(A))≤ C(r)  for any r>0,where_n,kis the associated domain bounded by_0,_1and the free boundary, andB_r(A)is the disk centered atA=(0,1)with radiusr. Therefore, there exists a subsequenceψ_n(x;P̅_n,k_m)such thatψ_n(x;P̅_n,k_m)→ψand_n,k_m→asm→∞. Furthermore, together with the estimates (<ref>),ψsatisfies the properties (i)–(iii) mentioned above. Thus_n(P̅)≤lim_m→∞_n(P̅_n,k_m)≤1-_̌n. It follows from Proposition <ref> thatψis unique. Hence_n(P̅)=lim_m→∞_n(P̅_n,k_m).

Second,P̅_n>P̅_*. Suppose not, by the definition ofP̅_none hasP̅_*∈𝔗_n. It follows from the right continuity of_nthat_n(P̅_*)≤1-_̌n. Thusψ_n(x;P̅_*)satisfies the far field behavior claimed in Proposition <ref>.
However, it follows from the definition ofP̅_*thatsup_x∈|ψ_n(x;P̅_*)|^2/_c(ψ_n(x;P̅_*))≥sup_x_2∈[0,H̅]|(̊̅u̅)(x_2;P̅_*)|^2/_c(ψ̅(x_2;P̅_*))=1.Thus_n(P̅_*)≥1. This leads to a contradiction, which impliesP̅_n>P̅_*.

Finally, it follows from the definition of{P̅_n}that{P̅_n}is an decreasing sequence.

DefineP̅_c:=lim_n→∞P̅_n. Based on previous properties of{P̅_n},P̅_cis well-defined andP̅_c∈ [P̅_*,P̅^*]⊂ [P̅_*,P̂). Note that for anyP̅∈(P̅_c,P̂), there exists aP̅_n∈(P̅_c,P̅)and thus_n(P̅)≤1-_̌n. Thereforeψ=ψ_n(x;P̅)solves Problem <ref>.

If(P̅)→1does not hold asP̅→P̅_c+, thensup_P̅∈(P̅_c,P̂)(P̅)<1, and hence there existsnsuch thatsup_P̅∈(P̅_c,P̂)(P̅)<1-_̌n. As the same as the proof for the right continuity of_n(P̅)on[P̅_n,P̂),(P̅_c)≤1-_̌n. If in addition there existsσ>0such that Problem <ref> always has a solutionψforP̅∈(P̅_c-σ,P̅_c), andsup_P̅∈(P̅_c-σ,P̅_c)(P̅)=sup_P̅∈(P̅_c-σ,P̅_c)sup_|ψ|^2/_c(ψ)<1.Then there existsk>0such thatsup_P̅∈(P̅_c-σ,P̅_c)(P̅)=sup_Q∈(P̅_c-σ,P̅_c)sup_|ψ|^2/_c(ψ)≤ 1-_̌n+k.This yields thatP̅_n+k≤P̅_c-σand leads to a contradiction. Therefore, either(P̅)→1asP̅→P̅_c+, or there does not existσ>0such that Problem <ref> has solutions  for allP̅∈(P̅_c-σ,P̅_c)and the associated solutions satisfy (<ref>).

This completes the proof of the proposition.



Combining all the results in previous sections, Theorem <ref> is proved.


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