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Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle

This paper focuses on two-dimensional continuous subsonic-sonic potential flows in a semi-infinitely long nozzle with a straight lower wall and an upper wall which is convergent at the outlet while straight at the far fields. It is proved that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique continuous subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the sonic curve. Furthermore, the sonic curve is free, where the flow is singular in the sense that the flow speed is only Holder continuous and the flow acceleration blows up. Additionally, the asymptotic behaviors of the flow speed at the far fields is shown.


Introduction. The Euler system
(ρu) + ∂ ∂y (ρv) = 0, ∂ ∂x (p + ρu 2 ) + ∂ ∂y (ρuv) = 0, ∂ ∂x (ρuv) + ∂ ∂y (p + ρv 2 ) = 0 (1) is usually used to describe the two-dimensional steady isentropic inviscid compressible flow, where (u, v), p and ρ represent the velocity, pressure and density of the flow, respectively, and p(ρ) = ρ γ /γ for a polytropic gas with the adiabatic exponent γ > 1 after the nondimensionalization. Suppose that the flow is irrotational, i.e., Then the density ρ can be formulated as a function of the flow speed q = √ u 2 + v 2 according to the Bernoulli law ( [2]): The sound speed c is defined as c 2 = p (ρ). At the sonic state, the flow speed is c * = 2/(γ + 1), which is critical in the sense that the flow is subsonic when q < c * , sonic when q = c * , and supersonic when q > c * . The system (1), (2) can be transformed into the full potential equation div(ρ(|∇ϕ| 2 )∇ϕ) = 0, where ϕ is the velocity potential with ∇ϕ = (u, v), ρ is the function given by (3). It is noted that (4) is elliptic in the subsonic region, degenerate at the sonic state, while hyperbolic in the supersonic region. Subsonic-sonic flow is one of the most interesting aspects in the mathematical theory of compressible flows. The related problems are usually raised in physical experiments and engineering designs, and there are a lot of numerical simulations and rigorous theory involved in this field (see, e.g., [2,8,15]). Two kinds of subsonicsonic flows have been intensively studied for decades: the flow past a profile and the flow in a nozzle. The outstanding work [1] by L. Bers proved that there exists a unique two-dimensional subsonic potential flow past a profile provided that the freestream Mach number is less than a critical value and the maximum flow speed tends to the sound speed as the freestream Mach number tends to the critical value. Later, the similar results for multi-dimensional cases were established in [13,9] by G. Dong, R. Finn and D. Gilbarg. These three works did not cover the flow with the critical freestream Mach number. It was shown in [3] based on a compensated compactness framework that the two-dimensional flow with sonic points past a profile may be realized as the weak limit of a sequence of strictly subsonic flows. However, all the subsonic-sonic flows above are obtained in the weak sense and their smoothness and uniqueness are unknown yet, so are the subsonic-sonic flows in an infinitely long nozzle. For a two-dimensional infinitely long nozzle, C. Xie et al. ( [22]) proved that there exists a critical value such that a strictly subsonic flow exists uniquely as long as the incoming mass flux is less than the critical value, and a subsonic-sonic flow exists as the weak limit of a sequence of strictly subsonic flows. The multi-dimensional cases were investigated in [24,12,14]. A typical subsonicsonic flow with precise regularity is a radially symmetric subsonic-sonic flow in a convergent straight nozzle. The structural stability was initially proved in [20] for the case of two-dimensional finitely long nozzle, and some new results can be found in [16,17,18,21,19]. In the recent decade, there are also some studies on rotational subsonic and subsonic-sonic flows, see [4,6,11,7,5,23] and the references therein.
The upper and lower wall of the nozzle are described as Γ up : y = f k (x) (x ∈ (−∞, 0]), and Γ low : y = −l 1 (x ∈ R), respectively, where k ∈ (0, 1] and The sonic curve of the flow is a free boundary intersecting the upper wall at the origin, which is chosen as the outlet of the nozzle and is denoted by Γ out : x = S(y), y ∈ [−l 1 , 0], S(0) = 0.
It is assumed further that the subsonic-sonic flow satisfies the slip conditions at Γ up and Γ low , and its velocity is along the normal direction at Γ out . See the following figure for an intuition.
As in [18,21], the subsonic-sonic flow problem can be formulated in the physical plane as div(ρ(|∇ϕ| 2 )∇ϕ) = 0, (x, y) ∈ Ω k , |∇ϕ(S(y), y)| = c * , ϕ(S(y), y) = 0, y ∈ (−l 1 , 0), where (ϕ, S) is a solution and Ω k is the semi-infinitely long nozzle bounded by Γ up , Γ low and Γ out . The problem (7)-(10) is a free boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain, whose degeneracy occurs at the free boundary and is characteristic. As mentioned in Remark 1.6 of [22], one can not require in advance that the flow tends to be uniformly subsonic at the far fields, otherwise, the elliptic problem may be overdetermined. In the paper, we prove that the subsonic-sonic flow in the nozzle is uniformly subsonic at the far fields, and the uniqueness of the flow results from this property. Similar to [20,18,16,17,21], we still solve the problem in the potential plane for the reason that the shape of the sonic curve is unknown in the physical plane while known in the potential plane, and the estimates of the flow speed can be made conveniently.
In the potential plane, the subsonic-sonic flow problem (7)-(10) can be transformed into a quasilinear degenerate elliptic problem with free parameters and nonlocal boundary conditions in unbounded domain. The unboundedness of the domain makes the problem more difficulty than the ones in [20,18,16,17,21]. The Schauder fixed point theorem is employed to prove the existence of subsonic-sonic flows. For a given incoming mass flux and flow speed at the upper wall, we solve a fixed boundary problem of a quasilinear degenerate elliptic equation. If the solved incoming mass flux and flow speed at the upper wall are just the given ones, we get the solution.
Note that the problem we concerns is in unbounded domain, we get the solution to the fixed boundary problem by taking limits of the sequences of the solutions to the truncated problems. Like that in [21], it seems very hard to construct appropriate super and sub solutions to prove the existence of solutions to truncated problems without sufficiently small (−x) −1/2 f L ∞ ((−l0,0)) . The method in [21] is used here: we first solve every regularized truncated problem when the flow speed at the outlet is suitable small and get priori estimates for the average and the derivatives of the solution, then we show the existence of the solution to the regularized truncated problem by use of the preliminaries obtained above, and finally we prove that their limit as the flow speed tends to be sonic at the outlet is a desired solution to the truncated problem. The difficulty here is that in order to get the solution to the fixed boundary problem by taking the limit of the solutions to the truncated problems, we must seek a suitable variation rate k 0 such that the solutions to all the truncated problems exit provided that k ∈ (0, k 0 ]. We overcome this difficulty by constructing complicated super and sub solutions to all the truncated problems. The Harnack's inequality is used to achieve the regularities and the asymptotic behaviors of the solution to the fixed boundary problem. As to the uniqueness of the subsonic-sonic flow, we first fix the free boundaries into fixed ones and transform the nonlocal boundary conditions into common ones by a proper coordinates transformation, and then we estiblish the uniqueness theorem by the energy estimates. Summing up, it is proved in this paper that if f satisfies (5) and (6), then there exists a unique subsonic-sonic flow to the problem (7)-(10) for suitably small k, and the flow speed is only C 1/2 Hölder continuous and the flow acceleration blows up at the sonic curve. Furthermore, the flow is uniformly subsonic at the far fields. The paper is arranged as follows. In Section 2, we formulate the subsonic-sonic flow problem (7)-(10) in the potential plane. Then in Section 3, we solve the fixed boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain. Finally in Section 4, we establish the well-posedness of the subsonic-sonic flow, and prove that the flow is uniformly subsonic at the far fields.
2. Formulation of the subsonic-sonic flow problem in the potential plane. Define a velocity potential ϕ and a stream function ψ, respectively, by where θ is the flow angle. The system (1), (2) can be reduced to the Chaplygin equations ( [2]): in the potential-stream coordinates (ϕ, ψ). The coordinates transformation (11) between the two coordinate systems are valid at least in the absence of stagnation points. Eliminating θ from (12) yields the following quasilinear equation of second It is obvious that B(·) is strictly increasing in (0, 2/(γ − 1)), while A(·) is strictly increasing in (0, c * ] and strictly decreasing in [c * , 2/(γ − 1)). It follows from [21] that there exist two constants 0 < N 1 < N 2 depending only on γ such that for c * /6 ≤ q ≤ c * , where E = A • B −1 and B −1 is the inverse function of B. We use A −1 (·) to denote the inverse function of A(·) (0,c * ] in this paper. Additionally, the flow angle at the upper and the lower wall are respectively. As in [18,21], in order to describe the problem in the potential plane, we denote the flow speed at the upper wall by then the potential function at the upper wall is expressed by with The inverse function of Φ up is denoted by X up . The subsonic-sonic flow problem (7)-(10) can be formulated in the potential plane as follows: where (q, m) is a solution with m > 0 being the incoming mass flux. Solutions to the problem (16)- (19) are defined as follows.
Definition 2.1. For m > 0, a function q ∈ L ∞ ((−∞, 0)×(0, m)) is called a solution to the fixed boundary problem (16)- (19), if The existence of solutions to the problem (16)- (20) will be proved by a fixed point argument. Give m and Q up in advance as follows: with For such Q up , it is clear that Φ up and X up are well determined. Direct calculations yield that where χ [ζ0,0] (ϕ) is the characteristic function of the interval [ζ 0 , 0], and (5) and (6), it follows from [21] that there exists a constantl 0 ∈ (0, l 0 ) depending only on f (0) and , and hence there exist two constants 3. Fixed boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain. In this section, we deal with the well-posedness of the fixed boundary problem. For the given m and Q up ∈ C 1/4 ((−∞, 0]) satisfy (21) and (22), respectively, we solve the degenerate elliptic problem (16)- (19). Since the problem is in an unbounded domain, we first deal with the truncated problem in with any sufficient large positive integer n, and make some useful compact estimates. Then we solve the problem (16)-(19) by a limit process. The key of the proof is seeking the variation rate k, which ensures the solutions to the truncated problems exist, is independent of n.

3.1.
Well-posedness of the truncated problem. The truncated problem is written as Note that (26) is degenerate at q n = c * , we replace (30) with the following boundary condition where c ∈ [c * /3, c * ) is a constant, and consider the regularized truncated problem (26)-(29), (31). Then we solve the problem (26)-(30) by a limit process. The proof can be divided into four steps.
Step 2. A priori estimates of the average of solutions to the problem (26)-(29), .

Extending the problem (68)-(70) into the domain [3k
if i is even.
Using (73) and the same operation on q leads to that Then the arbitrariness of n ≥ 2 leads to (65), and hence (66) holds.

Remark 1.
Through the similar process of the proof of Proposition 2, one can show that for any positive integer λ, it holds that l0,0)) .