On subsonic and subsonic-sonic flows in the infinity long nozzle with general conservatives force

Acta Mathematica Scientia 2017,37B(3):752–767 http://actams.wipm.ac.cn

ON SUBSONIC AND SUBSONIC-SONIC FLOWS IN THE INFINITY LONG NOZZLE WITH GENERAL CONSERVATIVES FORCE∗

RÞ)

Xumin GU (

Department of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China E-mail : [email protected]

U¶)

Tian-Yi WANG (

†

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China; Gran Sasso Science Institute, viale Francesco Crispi, 7, 67100 L’Aquila, Italy E-mail : [email protected]; [email protected]; [email protected] Abstract In this article, we study irrotational subsonic and subsonic-sonic flows with general conservative forces in the infinity long nozzle. For the subsonic case, the varified Bernoulli law leads a modified cut-off system. Because of the local average estimate, conservative forces do not need any decay condition. Afterwards, the subsonic-sonic limit solutions are constructed by taking the extract subsonic solutions as the approximate sequences. Key words

Steady flow; homentropic; irrotation; subsonic flow; subsonic-sonic limit

2010 MR Subject Classification

1

35Q31; 35L65; 76N10; 76G25; 35D30

Introduction

Here, we consider the steady homentropic Euler equations with extract forces, which are written as:    div(ρu) = 0, (1.1) div(ρu ⊗ u) + ∇p = ρF,    curlu = 0, where x = (x1 , · · · , xn ) ∈ Rn , n ≥ 3. u = (u1 , · · · , un ) ∈ Rn is the fluid velocity, while ρ, p, and F represent density, pressure, and extra forces, respectively. For the hometropic flow, the pressure p is a function of the density ρ, which is written as: p = p(ρ). As usual, we require p′ (ρ) > 0, ∗ Received

2p′ (ρ) + ρp′′ (ρ) > 0

for ρ > 0,

(1.2)

March 23, 2016; revised October 24, 2016. The research of first author was supported in part by NSFC (11601305). The research of second author was supported in part by NSFC (11601401), and the Fundamental Research Funds for the Central Universities (WUT: 2017IVA072 and 2017IVB066). † Corresponding author

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which include the γ-laws flow with p = κργ , for γ > 1 and κ > 0, and the isothermal flows, with p = κρ; see [11]. The Mach number is a non-dimensional ratio of the fluid velocity to the local sound speed, |u| M= , c where n 1/2 X |u| := u2i i=1

is the flow speed and c=

p p′ (ρ)

is the local sound speed. The flow is subsonic when M < 1, while M = 1 means that the flow is locally sonic. Otherwise, M > 1 implies that the flow is supersonic. Through this article, we consider the case where the extra force F is conservative, which means there exists a potential function ψ such that F = ∇ψ. There are many nature and important examples on this type of forces in reality, for instance, the gravity field. Another usual example is the electric field. One of classical problems on the steady compressible flows is the infinity long nozzle problem. Let Ω ⊂ Rn be an infinitely long nozzle, which is a homomorphism on the unit cylinder C = B(0, 1) × R in Rn . The compressible fluid fills in the region Ω. At the ∂Ω boundary, the flow satisfies the slip condition: u·ν =0 on ∂Ω, (1.3) where ν is the unit outward normal to the region Ω. One can derive the fixed mass flux property from (1.1)1 and (1.3): Z S0

l · ρu ds =: m,

where S0 is an arbitrary cross section of the nozzle, and l is the unit outer normal of the domain S0 . The theory of global subsonic flow in a variable nozzle was formulated by Bers in [4] in 1958. Then, the first rigorous proof of nozzle problem was achieved by Xie and Xin [27] by introducing the flux potential. Later, they extended it to the 3D axis-asymmetric nozzles case in [28]. The theorem for general infinitely long nozzle in Rn (n ≥ 2) was completed in Du-Xin-Yan [14]. The largely open nozzle case was proved by Liu and Yuan [23]. Besides the theory of nozzle problems, we also want to mention the study on another classical case: the airfoil problem. Frankl-Keldysh [24], Shiffman [25, 26], Bers [1–3], and Finn-Gilbarg [18] considered the irrotational, two-dimensional subsonic flow. Finn and Gilbarg [19] obtained the first result for higher dimensional subsonic flow past an obstacle under some restrictions on the Mach number. Then, for the three and higher dimensional case, Dong [12] and Dong-Ou [13] concluded these results to the situations where maximum Mach number is below 1. The respective case with extra conservative force is considered in [21]. For the rotational subsonic flows case, please refer to [5–7, 15–17, 29] . On the basis of the the existence of exact subsonic solutions, one could consider the subsonicsonic limit solutions by the compactness method. The compactness framework on sonic-subsonic irrotational flows in two dimension was introduced by [8, 27] independently. In [8], the general

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compactness framework was introduced by Chen, Dafermos, Slemrod, and Wang. While in [27], Xie and Xin investigated the subsonic-sonic limit of the two-dimensional irrotational flows for the infinitely long nozzle problem, and they extended the result to the three-dimensional axisymmetric flow through an axisymmetric nozzle in [28] later. Moreover, for the general multidimensional irrotational case, the compactness framework was established in [22]. The case of non-homentropic and rotation flows is concluded by Chen, Huang, and Wang [10]. In this article, we consider both subsonic flows and subsonic-sonic flows. For the subsonic case, in which the problem could be reduced to a second order elliptic equation, the major difference from the previous research is the effect of the extra force, especially the asymmetric conditions, which is quite crucial for the elliptic equation. By the proper modified cut off and the variation formulation, we construct the approximation solutions in the bounded domain. Then, the local average estimate shows the convergence to the infinity long nozzle without the decay condition to the infinity. It is worth mentioning that this is very different from our previous work [21] for the air foil problem, where the extra force will decay to zero at infinity with certain polynomial rates. Also, this result extends the pressure-density relation of the flow. The existence for the subsonic-sonic case is due to the convergence theorem in [10]. The rest of this article is organized as follows. In Section 2, we establish the formulation of the problem and state the main theorem. Next, we clarify the mathematical setting and introduce the cut-off by modifying the density function in Section 3. For the modified problem, the variation formulation is used to construct the solution in Section 4, and the local average estimate is also proven. In Section 5, the uniqueness of modified flows is proven. Finally, in Section 6, we complete the proof by the varied Bers skill and the subsonic-sonic compactness argument.

2

Formulation of the Problem and the Main Result

In this section, we introduce the mathematical formulation of problem and the main result. Firstly, the basic assumption on the multi-dimensional nozzle domain Ω is stated as follows. ¯ →C ¯ : x → y satisfying There exists an invertible C 2,α map T : Ω  ¯   T (∂Ω) = ∂ C, (2.1) For any k ∈ R, T (Ω ∩ {xn = k}) = B(0, 1) × {yn = k},    kT kC 2,α , kT −1kC 2,α ≤ K < ∞,

¯ = B(0, 1) × (−∞, +∞) is a unit cylinder in Rn , B(0, 1) is a where K is a uniform constant, C unit ball in Rn−1 centred at the origin, and xn is the axial coordinate and x′ = (x1 , · · · , xn−1 ) ∈ Rn−1 . Secondly, from the direct calculation [21], one can obtain the Bernoulli’s law: 1 2 |u| + h(ρ) = ψ, 2 after modified the constant, while h(ρ) =

Z

1

is the enthalpy.

ρ

p′ (τ ) dτ τ

(2.2)

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Now, we can settle the main problem considered in this article: Problem 1 (m): Find functions u = (u1 , · · · , un ) satisfying  div (ρu) = 0, curlu = 0,

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(2.3)

with the Bernoulli law (2.2) in Ω, the slip boundary condition (ρu) · ν = 0 on ∂Ω,

(2.4)

where ν denotes the unit inward normal of domain Ω; and the mass flux condition Z (ρu) · l ds = m > 0,

(2.5)

S0

where S0 is an arbitrary cross section of the nozzle, and l is the unit outer normal of the domain S0 . Remark 2.1 If the flow is without vacuum, that is, inf ρ(x) > 0, (2.4) can be written x∈∂Ω

as u · ν = 0 on ∂Ω.

(2.6)

Our main result is the following theorem: Theorem 2.2 Suppose that the nozzle Ω satisfies the basic assumptions and ψ ∈ L∞ and ∇ψ ∈ Lqloc for q > n.

(2.7)

Then (1) There exists a positive number m ¯ depending on Ω and ψ such that if m < m, ¯ then there exists a uniformly subsonic flow through the nozzle, that is, Problem 1 (m) has a smooth solution u(x) ∈ C 1,α (Ω) with Mach number M < 1. (2) Let mε → m ¯ as ε → 0, with mε < m, ¯ and uε = (uε1 , · · · , uεn ) be the corresponding ε solutions to Problem 1 (m ), then, as ε → 0, the solution sequence uε (x) possesses a subsequence (still denoted by) converge a.e. in Ω to u ¯(x) = (¯ u1 , · · · , u ¯n )(x), which is a weak solution of Problem 1 (m). ¯ Furthermore, u ¯ and ρ¯, defined through (2.2), also satisfy (1.1)2 in the sense of distributions and the boundary condition (2.4) as the normal trace of the divergence-measure field on the boundary (see [9]). Remark 2.3 It is noticeable that the condition on ∇φ is local without decay behaviour to infinity. This is different from the air foil problem studied in [21]. In fact, for the infinitely long nozzle, the far field behaviour of flow can be treated as a quasi-one-dimensional problem, where the key point is the local average estimate. Remark 2.4 It is easy to check the gravity: ψ = gxi for i = 1, · · · , n − 1, satisfying the conditions (3.1) and (2.7) on ψ. It can also be applied to the electric field. Remark 2.5 In part (1) of Theorem 2.2, the regularity of u is limited by ψ. One can improve the regularity of u and ρ by imposing the further smooth condition on ψ.

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Mathematical Setting and Modification of the Density Function

In this section, we transfer Problem 1 (m) to a second order partial differential problem, and introduce a respective subsonic cut-off. For the irrotation equation (2.3)2 , we could introduce the flow potential φ, which satisfies u = ∇φ. Then, the slip condition (2.6) on the boundary Γ comes to ∂φ = 0. ∂ν Next, we present the density ρ as the function of ∇φ and ψ. Without loss of generality, we assume that lim h(ρ) > ψ > lim+ H(ρ), (3.1) ρ→+∞

ρ→0

where H(ρ) =



 p′ (ρ) + h(ρ) . 2

From (1.2) and (3.1), it is easy to see that h(ρ) has the respective inverse function h−1 , which leads to the presentation of density:   |u|2 2 ρ(|∇φ| − 2ψ) := h−1 ψ − , (3.2) 2 which is equivalent to (2.2). Then, Mach number can be regarded as the function of ∇φ and ψ, which is written as M (∇φ; ψ). Then, we come to the second order equation from (2.3)1 :   2 div ρ(|∇φ| − 2ψ)∇φ = 0,

and (2.5) comes to

Z

S0

ρ |∇φ|2 − 2ψ


∂φ ds = m. ∂l

Then, Problem 1 (m) comes to Problem 2 (m): Find φ(x) such that    2  div ρ(|∇φ| − 2ψ)∇φ = 0,       ∂φ = 0, ∂ν   Z  
∂φ    ds = m. ρ |∇φ|2 − 2ψ ∂l S0

in Ω, on ∂Ω,

(3.3)

From the direct calculating, (3.3)1 comes to n X i=1

n n X X

∂i ρ |∇φ|2 − 2ψ ∂i φ = aij ∂ij φ + bi ∂i φ = 0,

where aij = ρδi,j

i,j=1

i=1

  ∂i φ∂j φ , − ρ (|∇φ| − 2ψ)∂i φ∂j φ = ρ δi,j − c2 ′

2

and bi =

ρ∂i ψ . c2

(3.4)

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Then, for ξ ∈ Rn ,

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ρ(1 − M 2 )|ξ|2 ≤ aij ξi ξj ≤ ρ|ξ|2 .

Then, we could see that (3.3)1 is elliptic if and only if the flow is subsonic, and it will degenerate in the subsonic-sonic case. To guaranteed the uniform ellipticity of (3.3)1 before the estimate on |∇φ|, we need to introduce a proper subsonic cut-off. From (1.2), H(ρ) is an increase function with respect to ρ. For fixed ψ,

1 qcr (ψ) := 2ψ − 2h H −1 (ψ) 2

is the critical speed. By the direct calculation, one can show that the flow is total subsonic if and only if |∇φ|(x) < qcr (ψ)(x) for x ∈ Ω. Now, we introduce a modified problem of Problem 2 (q∞ ), which is uniformly elliptic by presenting a way to modify the density ρ. For any small θ > 0, we define ρ˜ as 
2  ρ |∇φ| − 2ψ , if |∇φ| ≤ (1 − 2θ)qcr (ψ),    otherwise, ρ˜(|∇φ|2 , ψ) := monotone smooth connection,  
 2 2   sup ρ (1 − θ) q (ψ(x)) − 2ψ(x) , if |∇φ| ≥ (1 − θ)qcr (ψ). cr

x∈Ω

And, we denote ρ˜v (v, w) :=

∂ ˜(v, w), ∂v ρ

and ρ˜w (v, w) :=

∂ ˜(v, w). ∂w ρ

Then, Problem 3 (m) is defined as follows: Find φ(x) such that 

 div ρ˜ |∇φ|2 , ψ ∇φ = 0, in Ω,       ∂φ = 0, on ∂Ω, ∂ν   Z  
∂φ    ρ˜ |∇φ|2 , ψ ds = m. ∂l S0

After the similar calculation with (3.4), φ satisfies n X

i,j=1

a ˜ij ∂ij φ +

n X

˜bi ∂i φ = 0,

i=1

where a ˜ij = ρ˜δi,j − 2ρ˜v ∂i φ∂j φ, and ˜bi = ρ˜w ∂i ψ. And, for ξ ∈ Rn ,

0 < λ|ξ|2 ≤ a ˜ij ξi ξj ≤ λ−1 |ξ|2 ,

and |˜bi ∂i φ| ≤ C|∂i ψ|, where λ and C are the positive numbers dependent on θ and ψ.

(3.5)

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The Variation Formulation in the Bounded Domain

Because of the fact that Ω is an unbounded domain, we also need to construct a series of truncated problems to approximate the Problem 3 (m) with subsonic truncation. Let L > 0,  ΩL = x ∈ Ω |xn | < L , and S ± = Ω ∩ {xn = ±L}. L

Consider the following truncated problem: Problem 4 (m, L): Find a φ such that   div(˜ ρ(|∇φ|2 , ψ)∇φ) = 0,       ∂φ   = 0,  ∂n  ∂φ m    ρ˜(|∇φ|2 , ψ) = + ,   ∂x |S n  L|    φ = 0,

x ∈ ΩL , ∂Ω ∩ ∂ΩL , on SL+ on SL− .

The additional boundary condition on SL+ implies that the mass flux of the flow remains m. It is easy to see that the truncated Problem 4 (m, L) is a strong quasilinear elliptic problem in a bounded domain. From now on, instead of the original Problem 3 (m), we consider a series of the truncated Problem 4 (m, L) for any fixed sufficiently large L. With some uniform estimates of the approximate solutions, we can conclude that the solution of the truncated Problem 4 (m, L) converges to the original Problem 3 (m). We solve the truncated Problem 4 (m, L) by a variational method. First, we need to introduce the following Hilbert space: define n o HL = ϕ ∈ H 1 (ΩL ) : ϕ S − = 0 , L

then, HL is a Hilbert space under H -norm. The additional boundary condition on SL− is treated in the sense of traces. For the given ψ, let Z 1 Λ G(Λ, ψ) = ρ˜(v, ψ)dv, (4.1) 2 0 1

and a functional J(ϕ) on HL is defined as Z Z m0 J(ϕ) = G(|∇ϕ|2 , ψ)dx − + ϕdx′ , + |S | ΩL SL L where x′ = (x1 , x2 , . . . , xn−1 ). The existence of solutions to Problem 4 (m, L) is equivalent to the following variational problem: Problem 5 (m, L): Find a minimizer φ ∈ HL such that J(φ) = min J(ϕ). ϕ∈HL

It is direct to check that equation (3.5) is the Euler-Lagrangian equation of our variation problem. For any t ∈ R+ and any φ ∈ HL , and η ∈ Cc∞ (ΩL ), which implies φ + tη ∈ HL , Z Z
mt 2 2 G(|∇φ + t∇η| , ψ) − G(|∇φ| , ψ) dx − + J(φ + tη) − J(φ) = ηdx′ . (4.2) |SL | SL+ ΩL

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Mean value theorem yields that Z
G(|∇φ + t∇η|2 , ψ) − G(|∇φ|2 , ψ) dx ΩL

= =

Z Z

1

ΩL

Z

1

ΩL

Z

0

0

Gv (θ|∇φ + t∇η|2 + |∇φ|2 (1 − θ), ψ)dθ(|∇φ + t∇η|2 − |∇φ|2 )dx Gv (|∇φ|2 + θ(t2 |∇η|2 + 2t∇η · ∇φ), ψ)dθ(t2 |∇η|2 + 2t∇η · ∇φ)dx.

As |Gv (·, ψ)| ≤ C, ∇φ, ∇η ∈ L2 (ΩL ), substituting (4.3) into (4.2) shows that 1 lim inf (J(φ + tη) − J(φ)) + t t→0 Z Z 1 Z 2 2 2 = lim inf Gv (|∇φ| + θ(t |∇η| + 2t∇η · ∇φ), ψ)dθ(2∇η · ∇φ)dx − + t→0

ΩL

+ SL

0

1

m = Gv (|∇φ| , ψ)dθ(2∇η · ∇φ)dx − + |S ΩL 0 L| Z Z m = ρ˜(|∇φ|2 , ψ)∇φ · ∇ηdx − + ηdx′ . |SL | SL+ ΩL Z

Z

2

Therefore, for any η ∈ HL , Z

ΩL

Z

ρ˜(|∇φ|2 , ψ)∇φ · ∇ηdx −

+ SL

(4.3)

m ηdx′ |SL+ |

ηdx′

m |SL+ |

Z

+ SL

ηdx′ = 0.

(4.4)

For our variational problem, we have the following theorem. Theorem 4.1 Problem 5 (m, L) has an unique minimizer φ ∈ HL . Moreover, Z 1 |∇φ|2 dx ≤ Cm2 , |ΩL | ΩL

(4.5)

where the constant C does not depend on L. Proof

Step 1.

J(ϕ) is coercive on HL . For any ϕ ∈ HL , ϕ|S − = 0 indicates L Z 1 ϕdx′ ≤ C|ΩL | 2 k∇ψkL2 . SL+

(4.6)

Therefore, the Cauchy inequality yields Z Z m 2 J(ϕ) = G(|∇ϕ| , ψ)dx − + ϕdx′ |SL | SL+ ΩL Z ≥λ |∇ϕ|2 dx − C(m, |SL+ |, |ΩL |)k∇ϕkL2 ΩL

1 λ ≥ k∇ϕk2L2 − C(m, |SL+ |, |ΩL |), 2 λ which implies that J(ϕ) is coercive. Step 2. The existence of the minimizer φ ∈ HL . As J(ϕ) is coercive in HL , there is a minimizer sequence {φn } ⊂ HL such that J(φn ) → α = inf J(ψ) > −∞. ψ∈HL

Then, there exists a subsequence, denoted by {φn }, that converges weakly to some φ ∈ HL and 1 k∇φk2L2 ≤ 2 C(m, |SL+ |, |ΩL |). λ

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By the lower semi-continuity, it is easy to check that Z Z G(|∇φ|2 , ψ)dx ≤ lim inf n→∞

ΩL

ΩL

G(|∇φn |2 , ψ)dx.

On the other hand, similar to (4.6), we have Z  12  Z Z 2 ′ 2 (φn − φ) dx ≤ C(ΩL ) |φn − φ| dx + SL

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ΩL

(4.7)

 12 |∇φn − ∇φ| dx . 2

ΩL

Then, by the L2 (ΩL ) strong convergence of {φn }, we have Z |φn − φ|dx′ → 0 as n → ∞.

(4.8)

+ SL

Therefore, it follows from (4.7) and (4.8) that J(φ) ≤ lim inf J(φn ) = α, n→∞

which concludes J(φ) = min J(ψ) = α. ψ∈HL

The uniqueness of the minimizer can be proved by pick a minimizing sequence composed of two minimizers alternatively. A minimizing sequence is always convergent, so any two minimizers are the same. Step 3. By direct computations, we obtain Z Z Z m m G(|∇φ|2 , ψ)dx = J(φ) + + φdx′ ≤ J(0) + + φdx′ + + |S | |S | ΩL SL SL L L 1 m ≤ C + |ΩL | 2 k∇φkL2 . |SL | It follows from (4.1) that k∇φk2L2 ≤ That is

1 λ

1 |ΩL |

Z

Z

ΩL

ΩL

G(|∇φ|2 , ψ)dx ≤ C

|∇φ|2 dx ≤ C

where Smin is defined as the minimal of |SL+ |.

1 1 m 2 + |ΩL | k∇φkL2 . λ |SL |

m2 λ2 |SL+ |2

≤C

m2 2 λ2 Smin

, 

For the further regularity of solution of Problem 5 (m, L), taking the k-th partial derivative of equation (3.5)1 formally comes to n X

i,j=1

∂i (˜ aij ∂j ϕ′ ) +

n X

∂i (˜ ρw ∂k ψ∂i φ) = 0,

i=1

where ϕ′ = ∂k φ for k = 1, · · · , n. By the definition of the cut-off density ρ, ˜ a ˜ij has uniformly positive eigenvalues, hence the equation is uniform elliptic. Also, for i, k = 1, · · · , n, |˜ ρw ∂k ψ∂i φ| ≤ C|∂k ψ|. By ∇ψ ∈ Lq , we can show that ρ˜w ∂k ψ∂i φ are bounded in Lq . By the standard elliptic estimate, which could be found in [20], and approximation of derivatives by finite differences, one can get the following regularity lemma.

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Lemma 4.2 For ∇ψ ∈ Lq , q > n, and (2.1) on Ω, then there exist constants 0 < α < 1 and C depending on λ and Ω such that   sup |∇φ| ≤ C ||∇φ||2L2 (ΩL ) + ||∇ψ||Lq (ΩL ) , x∈ΩL/2

and sup x1 ,x2 ∈ΩL/2

  |∇φ(x1 ) − ∇φ(x2 )| 2 q (Ω ) . ≤ C ||∇φ|| + ||∇ψ|| 2 L L (ΩL ) L |x1 − x2 |α

Next, to remove the cut-off on domain, one may need to get the uniform estimate with respects to the location. Before the local average estimate, we need to introduce the local set, for x0 = (x′0 , x0,n ) ∈ Ω, Ω(a,b) := {x = (x′ , xn ) ∈ Ω | a < xn < b}. and define the average quantities as Z P(a,b) = −

φdx.

Ω(a,b)

Then, from the properties of the nozzle, we have the following claims: Lemma 4.3 (Uniform Poincar´e Inequality) For any a ∈ R, 1 ≤ p < ∞, one has

φ(x) − P(a,a+1) p ≤ Ck∇φ(x)kLp (Ω(a,a+1) ) , L (Ω ) (a,a+1)

where C is a positive constant depending only on p, Ω, independent of a. Proposition 4.4 For a < b, one can obtain |P[a−1,a] − P[b,b+1] | ≤ C

Z

Ω(a−1,b+1)

|∇φ|dx,

where C just depends on Ω and does not dependent on a and b. We refer the readers to [14] for the proofs. Now, we introduce the local average lemma. Lemma 4.5 There exists an uniform l depended on Ω and λ, such that for any b − a > l and Ω(a−1,b+1) ⊂ Ω L , one has 2 Z − |∇φ|2 dx ≤ Cm2 , (4.9) Ω(a,b)

where C does not depend on L. Proof For any − L2 < a − 1 < a < b < b + 1 < L2 , define η ∈ C ∞ (ΩL ), 0 ≤ η ≤ 1, |∇η| ≤ 2, η|Ω(a,b) = 1, and η|Ω−Ω(a−1,b+1) = 0. Then, we choose the test function η 2 φˆ ∈ H 1 (ΩL ), while   φ(x) − P[a−1,a] , xn ≤ a,     x −a ˆ φ(x) = φ(x) − P[a−1,a] − (P[b,b+1] − P[a−1,a] ) n , a ≤ xn ≤ b,  b−a     φ(x) − P[b,b+1] , xn ≥ b. Because of η 2 φˆ ∈ HL , one can have Z ˆ ρ˜(|∇φ|2 , ψ)∇φ · ∇(η 2 φ)dx = 0. ΩL

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−P

Note that ∇φˆ = ∇φ − [b,b+1]b−a[a−1,a] χ(a,b) (x)~en , where ~en = (0, · · · , 0, 1), and χ(a,b) (x) is the characteristic function of Ω(a,b) . Then,   Z Z ∂φ P[b,b+1] − P[a−1,a] η 2 ρ˜(|∇φ|2 , ψ)|∇φ|2 dx − η 2 ρ˜(|∇φ|2 , ψ) dx ∂xn b−a Ω(a−1,b+1) Ω(a,b) Z ˆ = −2 η ρ˜(|∇φ|2 , ψ)∇φ · ∇η φdx. Ω(a−1,b+1)

As

R

S xn

∂φ dx′ = m, ρ˜(|∇φ|2 , ψ) ∂x n Z η 2 ρ˜(|∇φ|2 , ψ)|∇φ|2 dx Ω(a−1,b+1)

= −2

Z

Ω(a−1,b+1)

ˆ + (P[b,b+1] − P[a−1,a] )m. η ρ˜(|∇φ|2 , ψ)∇φ · ∇η φdx

Consequently, Z Z λ |∇φ|2 dx ≤ Ω(a,b)

η 2 ρ˜(|∇φ|2 , ψ)|∇φ|2 dx

Ω(a−1,b+1)

Z ˆ + P[b,b+1] − P[a−1,a] m ≤ 2 η ρ˜(|∇φ|2 , ψ)∇φ · ∇η φdx Ω(a−1,a) ∩Ω(b,b+1) Z −1 ˆ + P[b,b+1] − P[a−1,a] m ≤ Cλ |∇φ||φ|dx Ω(a−1,a) ∩Ω(b,b+1) 1 ! ! 12 Z Z 2

≤ Cλ−1

Ω(a−1,a)

Z

+Cλ−1

|∇φ|2 dx

Ω(b,b+1)

Ω(a−1,a)

! 12

|∇φ|2 dx

Z

|φ − P[a−1,a] |2 dx

Ω(b,b+1)

! 12

|φ − P[b,b+1] |2 dx

+ P[b,b+1] − P[a−1,a] m.

(4.10)

It follows the uniform Poincar´e inequality that Z Z 2 |φ − P[a−1,a] | dx ≤ C Ω(a−1,a)

Z

Ω(a−1,a)

2

Ω(b,b+1)

|φ − P[b,b+1] | dx ≤ C

Z

Ω(b,b+1)

|∇φ|2 dx, |∇φ|2 dx,

where C does not depend on a and b. Substituting into (4.10) with Proposition 4.4 yields Z Z Z |∇φ|2 dx ≤ C1 λ−2 |∇φ|2 dx + C2 λ−1 m |∇φ|dx, (4.11) Ω(a,b)

Ω(a−1,b+1) −Ω(a,b)

Ω(a−1,b+1)

where C1 and C2 are the uniform positive constants. Alternatively, Z Z 1 m |∇φ|dx ≤ ε |∇φ|2 dx + |Ω(a−1,b+1) |m2 . 4ε Ω(a−1,b+1) Ω(a−1,b+1) Here, |Ω(a−1,b+1) | is the measure of Ω(a−1,b+1) . Combining (4.11) and (4.12) leads to  Z Z C1 C2 2 −1 |∇φ| dx ≤ |Ω(a−1,b+1) |m2 . + C2 λ ε |∇φ|2 dx + 2 C + λ 4λε 1 Ω(a,b) Ω(a−1,b+1)

(4.12)

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By introducing A(a,b) =

1 b−a

Z

Ω(a,b)

763

|∇φ|2 dx,

it comes to A(a,b) ≤



C1 + C2 λ−1 ε C1 + λ2



C2 |Ω(a−1,b+1) | 2 b−a+2 A(a−1,b+1) + m . b−a 4λε b−a

(4.13)

By choosing the proper ε, one can find the uniform l such that, when b − a > l,   C1 b−a+2 −1 ϑ := + C2 λ ε < 1. C1 + λ2 b−a Hence, (4.13) comes to the iteration inequality:

A(a,b) ≤ ϑA(a−1,b+1) + C ′ m2 , which lead to conclusion (4.9) by the standard argument.

5



The Existence and Uniqueness of the Modified Flows in the Infinity Long Nozzle In this section, we come back to Problem 3 (m).

Theorem 5.1 For every m ≥ 0, there exists a unique classical solution of Problem 3 (m). Furthermore, the velocity field ∇φ depends on m continuously and in particular maxΩ |∇φ| is a continuous function of m. Proof First, we will show the existence part. For any fixed suitably large L, according to previous subsections, one can get a H 1 function φL (x) such that (φL (x) − φL (0)) ∈ HL is a weak solution to Problem 4 (m, L). Set φˆL (x) = φL (x) − φL (0). Moreover, φˆL ∈ C 1,α (ΩL/2 ) and k∇φˆL kC 0,α (ΩL/2 ) ≤ Cm. For any fixed K, if L > 2K, kφˆL kC 1,α (ΩK ) ≤ C, where C does not depend on L, and φˆL satisfies Z   ρ˜ |∇φˆL |2 , ψ ∇φˆL · ∇ηdx = 0, ΩK

∀η ∈ C0∞ (ΩK ).

As φˆL ∈ HL ∩ C 1,α (ΩK ) satisfies equation (4.4), one can check easily that Z   ∂ φˆ L dx′ = m, for any x0 ∈ ΩK . ρ˜ |∇φˆL |2 , ψ ∂xn S x0 By a standard diagonal argument, there exists a φ ∈ C 1,α (Ω) and a subsequence φˆLn such that ′ for any K, for some α′ < α, φˆLn → φ in C 1,α (ΩK ) as n → ∞. Therefore, one has Z ρ˜(|∇φ|2 , ψ)∇φ · ∇ηdx = 0, ∀η ∈ C0∞ (Ω), Ω

and

Z

S x0

ρ˜(|∇φ|2 , ψ)

∂φ ′ dx = m, ∂xn

for any x0 ∈ Ω.

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It is clear that 1 φ ∈ C 1,α (Ω) ∩ Hloc (Ω)

and k∇φkC 0,α (Ω) ≤ Cm.

(5.1)

Next, we will prove the uniqueness of the solution. Suppose that there are two different solution of Problems 3 (m): φ1 and φ2 . Set φ¯ = φ1 − φ2 . Then, φ¯ satisfies  n X   ¯ = 0,  ∂i (A¯ij ∂j φ) in Ω,  i,j=1 (5.2)  ¯  ∂ φ   = 0, on ∂Ω, ∂ν where Z 2
¯ Aij = ρ˜(|∇φs |2 , ψ)δij + 2ρ˜v (|∇φs |2 , ψ)∂i φs ∂j φs ds, 1

φs = (2 − s)φ1 + (s − 1)φ2 . Moreover, there exist a positive constant λ, such that for any vector ξ ∈ Rn , λ|ξ|2 < A¯ij ξi ξj < λ−1 |ξ|2 . Let η(x) be a C0∞ function satisfying η|Ω(−L,L) = 1, And

where

η|Ω−Ω(−L−1,L+1) = 0,

  ¯ φ(x) − φ−  L,     − φ+ ˆ − L − φL φ(x) = φ(x) ¯ − φ − (xn + L), L  2L      φ(x) ¯ − φ+ L, Z φ− = − L

¯ φ(x)dx,

Ω(−L−1,−L)

and |∇η| ≤ 2. x ∈ Ω(−L−1,−L), x ∈ Ω(−L,L), x ∈ Ω(L,L+1),

Z φ+ = − L

¯ φ(x)dx.

Ω(L,L+1)

ˆ and integrating it over Ω, one Multiplying both sides of the first equation in (5.2) by η 2 φ, obtains Z n X ¯ j φdx ¯ η 2 A¯ij ∂i φ∂ Ω(−L−1,L+1) i,j=1

φ+ − φ− L + L 2L Z = −2

Z

Ω(−L,L)

Ω(−L−1,−L)


η 2 ρ˜(|∇φ1 |2 , ψ)∇φ1 − ρ˜(|∇φ2 |2 , ψ)∇φ2 · ~en dx

η(φ¯ − φ− L)

n X

i,j=1

¯ −2 A¯ij ∂i η∂j φdx

Z

Ω(L,L+1)

Similar to the calculation in Lemma 4.5, we have Z Z C 2 ¯ |∇φ| dx ≤ λ Ω(−L−1,−L) ∪ Ω(−L,L)

η(φ¯ − φ+ L)

Ω(L,L+1)

n X

i,j=1

¯ 2 dx, |∇φ|

¯ A¯ij ∂i η∂j φdx.

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765

where C independents on L. Then, we can have the iteration inequality: Z Z ¯ 2 dx ≤ ϑ¯ ¯ 2 dx, |∇φ| |∇φ| Ω(−L,L)

Ω(−L−1,L+1)

where ϑ¯ < 1 is an uniform constant. By repeating the previous argument, for n ≥ 1, one can have Z Z ¯ 2 dx ≤ ϑ¯n−1 C ¯ 2 dx. |∇φ| |∇φ| (5.3) λ Ω(−L,L) Ω(−L−n,−L−n+1) ∪ Ω(L+n−1,L+n) On the other hand, by the local average estimate (4.5), for k = 1, 2, n ≥ 1, one has Z |∇φk |2 dx ≤ Cm2 , Ω(−L−n,−L−n+1) ∪ Ω(L+n−1,L+n)

which implies that Z

Ω(−L−n,−L−n+1) ∪ Ω(L+n−1,L+n)

where C is independent of L and n. Taking L → ∞ in (5.3) yields

∇φ¯ = 0

¯ 2 dx ≤ C, |∇φ|

in Ω.

Finally, we show the continuity with respect to m. We take a sequence mk → m and φk as the unique solution of Problem 3 (mk ). Then, from the uniform estimate (5.1), Arzela–Ascoli theorem leads to, for some α′ < α, ∇φk (x) → ∇φ0 (x)

′

in C 0,α (Ω),

where φ0 is the unique solution of Problem 3 (m). By this convergence, one can conclude that max |∇φk | → max |∇φ0 |. Hence, max |∇φ| is a continuous function of m.  x∈Ω

6

x∈Ω

x∈Ω

Subsonic Flow and Subsonic-Sonic Flow in Space

In this section, we complete the proof of Theorem 2.2. The first step is to release the cut-off constrain by the Bers skill and complete the proof of the subsonic part of main theorem. Then, we take the subsonic-sonic limit by the compactness theorem in [10]. Proof of Theorem 2.2 First, we prove the part (1): the subsonic case. Up to now, we have shown that, for fixed cut off parameter θ, there exists an unique solution of Problem 3 (m), which is denoted as φ(x; m, θ). In order to remove the cut off, which is introduced in Section 3, we define the quantity by   |∇φ(x; m, θ)| M(m, θ) = max , x∈Ω qcr (ψ)(x) which is the equivalence of maximum Mach number of the field. It is noticeable that for certain θ, if M(m, θ) < 1 − 2θ, φ(x; m, θ) is the unique solution of Problem 2 (m). By the similar argument in Theorem 5.1, one can show that M(m, θ) also depends on m continuously. Let {θi }∞ n=1 be a strictly decreasing sequence of positive numbers, such that δi → 0 as i → ∞. For fixed i, there exists a maximum interval [0, mi ) such that, for m ∈ [0, mi ), M(m, θ) < 1 − 2θi .

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Then, for m ∈ [0, mi ), ∇φ(x; m, θi ) is the solution of Problem 1 (m). From the uniqueness of Problem 3 (m), we can see mi ≤ mj for i < j. So, {mi }∞ i=1 is an increasing sequence with the upper bounded mmax := |Smax | max(ρcr (ψ)qcr (ψ)(x)), which implies the convergence of the x∈Ω

sequence. As a consequence, we can have

m ˆ := lim mi . i→∞

i

If m < m ˆ for any i, then for any m ∈ [0, m), ˆ there exist an index i such that m ≤ mi . The solution of Problem 1 (m) is ∇φ(x; m, θi ), which could be written as ∇φ(x; m). Then, we have M (∇φ(x; m); ψ) → 1 as m → m. ˆ It means that subsonic flows will become subsonic-sonic flows. The uniqueness of Problem 1 (m) is already contained in Theorem 5.1. Next, we prove the part (2): the subsonic-sonic case. The strong solutions uε satisfy (2.3) and the Bernoulli’s law (2.2), and are uniform subsonic solutions of Problem 1 (mε ). Hence, Theorem 2.2 in [10] immediately implies the strong convergence of uε in Ω. As a consequence, the density function ρε (x), defined by (3.2), is convergence to ρ¯(x). The boundary conditions are satisfied for ρ¯ ¯u in the sense of Chen-Frid [9]. On the other hand, Because (1.1)2 holds for the sequence of subsonic solutions ρε (x) and uε (x), it is straightforward to see that ρ¯ and u¯ also satisfies (1.1)2 in the sense of distributions. This completes the proof of Theorem 2.2.  Acknowledgements This work was done partially when authors were visiting the Institute of Mathematical Sciences, the Chinese University of Hong Kong, whose hospitality is gratefully acknowledged. References [1] Bers L. An existence theorem in two-dimensional gas dynamics. Proc Symposia Appl Math, 1949, 1: 41–46 [2] Bers L. Boundary value problems for minimal surfaces with singularities at infinity. Transactions of the American Mathematical Society, 1951, 70(3): 465–491 [3] Bers L. Existence and uniqueness of a subsonic flow past a given profile. Communications on Pure and Applied Mathematics, 1954, 7(3): 441–504 [4] Bers L. Mathematical aspects of subsonic and transonic gas dynamics. New York: John Wiley & Sons, Inc, 1958 [5] Chen C, Xie C. Existence of steady subsonic Euler flows through infinitely long periodic nozzles. Journal of Differential Equations, 2012, 252(7): 4315–4331 [6] Chen C. Subsonic non-isentropic ideal gas with large vorticity in nozzles. Mathematical Methods in the Applied Sciences, 2015 [7] Chen C, Du L, Xie C, et al. Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body. Archive for Rational Mechanics and Analysis, 2016, 221(2): 559–602 [8] Chen G Q, Dafermos C M, Slemrod M, et al. On two-dimensional sonic-subsonic flow. Communications in Mathematical Physics, 2007, 271(3): 635–647 [9] Chen G Q, Frid H. Divergence-Measure Fields and Hyperbolic Conservation Laws. Archive for Rational Mechanics and Analysis, 1999, 147(2): 89–118 [10] Chen G Q, Huang F M, Wang T Y. Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations. Archive for Rational Mechanics and Analysis, 2016, 219(2): 719–740 [11] Courant R, Friedrichs K O. Supersonic flow and shock waves. Springer Science & Business Media, 1999 [12] Dong G C. Nonlinear partial differential equations of second order. American Mathematical Soc, 1991 [13] Dong G, Ou B. Subsonic flows around a body in space. Communications in Partial Differential Equations, 1993, 18(1/2): 355–379

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[14] Du L, Xin Z, Yan W. Subsonic flows in a multi-dimensional nozzle. Archive for Rational Mechanics and Analysis, 2011, 201(3): 965–1012 [15] Du L, Xie C, Xin Z. Steady subsonic ideal flows through an infinitely long nozzle with large vorticity. Communications in Mathematical Physics, 2014, 328(1): 327–354 [16] Duan B, Luo Z. Three-dimensional full Euler flows in axisymmetric nozzles. Journal of Differential Equations, 2013, 254(7): 2705–2731 [17] Duan B, Luo Z. Subsonic non-isentropic Euler flows with large vorticity in axisymmetric nozzles. Journal of Mathematical Analysis and Applications, 2015, 430(2): 1037–1057 [18] Finn R, Gilbarg D. Asymptotic behavior and uniqueness of plane subsonic flows. Communications on Pure and Applied Mathematics, 1957, 10(1): 23–63 [19] Finn R, Gilbarg D. Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Mathematica, 1957, 98(1): 265–296 [20] Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. Springer, 2015 [21] Gu X, Wang T.-Y. On subsonic and subsonic-sonic flows with general conservatives force in exterior domains. Preprint, 2015 [22] Huang F M, Wang T, Wang Y. On multi-dimensional sonic-subsonic flow. Acta Mathematica Scientia, 2011, 31(6): 2131–2140 [23] Liu L, Yuan H. Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles. Calculus of Variations and Partial Differential Equations, 2014, 49(1/2): 1–36 [24] Frankl F, Keldysh M. Die ussere neumannshe aufgabe fr nichtlineare elliptische differentialgleichungen mit anwendung auf die theorie der flugel im kompressiblen gas. Bull Acad Sci, 1934, 12: 561–687 [25] Shiffman M. On the existence of subsonic flows of a compressible fluid. Proceedings of the National Academy of Sciences, 1952, 38(5): 434–438 [26] Shiffman M. On the existence of subsonic flows of a compressible fluid. J Rational Mech Anal, 1952, 1: 605–652 [27] Xie C, Xin Z. Global subsonic and subsonic-sonic flows through infinitely long nozzles. Indiana University Mathematics Journal, 2007, 56(6): 2991–3024 [28] Xie C, Xin Z. Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles. Journal of Differential Equations, 2010, 248(11): 2657–2683 [29] Xie C, Xin Z. Existence of global steady subsonic Euler flows through infinitely long nozzles. SIAM Journal on Mathematical Analysis, 2010, 42(2): 751–784