Low Mach number limit of steady flows through infinite multidimensional largely-open nozzles

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Low Mach number limit of steady flows through infinite multidimensional largely-open nozzles ✩ Tian-Yi Wang, Jiaojiao Zhang ∗ Department of Mathematics, Wuhan University of Technology, Wuhan, Hubei 430070, P. R. China Received 31 August 2019; revised 20 November 2019; accepted 14 January 2020

Abstract In this paper, we justify of the low Mach number limit of the steady irrotational Euler flow through infinite multidimensional largely-open nozzles. Firstly, the existence and uniqueness of the incompressible flow are proved. Then, the uniform estimates on the compressibility parameter ε, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is in the Hölder space and is unique. Moreover, the convergence rate is of order ε2 including the pressure. And, for the compressible case, the extra force influences the asymptotic behaver to the open end, while the effect vanishes in the limiting process from compressible flows to the incompressible ones. Also, for the compressible flows with general conservative forces, the well-posedness of subsonic solution are given, which leads to the existence of subsonic-sonic limit solutions. © 2020 Elsevier Inc. All rights reserved. MSC: 35Q31; 35L65; 76N15; 35B40 Keywords: Multidimensional; Low Mach number limit; Steady flow; Homentropic Euler equations; Convergence rate

1. Introduction Euler equation describes the ideal fluid motion without viscous effect, which could be divided into incompressible case and compressible case up to compressibility. The incompressible ✩

This work is supported in part by NSFC grant 11601401 and 11971024.

* Corresponding author.

E-mail addresses: [email protected] (T.-Y. Wang), [email protected] (J. Zhang). https://doi.org/10.1016/j.jde.2020.01.023 0022-0396/© 2020 Elsevier Inc. All rights reserved.

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limit is devoted to building a bridge to fill the gap between those two different types of fluids by determining in which sense that the compressible flows tend to the incompressible ones as the compressibility parameter tends to zero. A typical domain to study the compressibility is called the De Laval nozzles, which have the convergent part and the divergence part connected by the throat part. Here the convergent part indicates the compression of the flows, while the divergence part indicates the rarefaction. To show the sufficient compression and divergence, the largely-open nozzles, which means the cross section of the nozzles tends to infinity as the length goes to infinity, are considered here. This type of problem also links to the jet flows. For rigorous formulation of domain, please see Section 2.1. The steady compressible Euler equation reads 

div (ρu) = 0, div (ρu ⊗ u) + ∇p = ρF,

(1)

where x = (x1 , · · · , xn ) ∈ Rn , n ≥ 3, u = (u1 , · · · , un ) is the velocity, ρ, p and F represent the density, pressure and extra force, respectively. Moreover, u ⊗ u = (ui uj )n×n is an n × n matrix. As the homentropic flow, the pressure is a function of density p(ρ) satisfying: p  (ρ) > 0,

2p  (ρ) + ρp  (ρ) > 0

for ρ > 0.

(2)

We remark that condition (2) holds for the flows which satisfy the γ -law that p = κ1 ρ γ + κ2 with γ > 1, and theisothermal flows that p˜ = κ1 ρ + κ2 , for κ1 > 0 (see [16]). The sound speed of the flow is c := p  (ρ), and the Mach number is defined as M :=

|u| , c

where

n  ui 2 )1/2 . |u| := (

(3)

i=1

The flow is subsonic when M < 1, sonic when M = 1, and supersonic when M > 1. The problem of subsonic flows through nozzles in two dimension was proposed by Bers [4] in 1950s. Due the strong convergent, the shock may appear. And, the flow behind the shock-front could be subsonic from divergence of nozzle, and it turns out that this subsonic flow determines many important features of transonic shocks. The studies on transonic shocks in nozzles are started by Chen and Feldman [6]. It is noticeable that Wang and Xin [44,45] recently made a great progress on smooth transonic flow in the finite length De Laval nozzle. Xie and Xin [46] firstly gave a rigorous proof of Bers’ problem for potential flows in two-dimensional bounded nozzles. Later, they extended it to the 3D axis-asymmetric case in [47]. The theorem for general infinite long nozzle in Rn , n ≥ 2 was completed in Du-Xin-Yan [19], while the result was extended in [25] to the extra force case. However, all these works only considered nozzles with uniform bounded cross section even at far field. So it is quite different from the largely-open nozzles we consider here. Yuan-Liu [34] is the first result on the subsonic flows through infinite multidimensional largely-open nozzles. Besides the infinite long nozzle problem, there is another classical problem: the airfoil problem. Shiffman [41], Bers [2,3], and Finn-Gilbarg [22] considered the two-dimensional irrotational subsonic flow. Finn and Gilbarg [23] got the first result for three-dimensional subsonic flow under some restrictions on the Mach number. Then Dong-Ou [18] extended these results to the case when Mach number M < 1 for arbitrarily dimensional case, while the case with conservative force effect was considered in [26]. The respective

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subsonic-sonic flow was considered in [27]. For the jet flow problem, please see [12,15]. For the rotational subsonic flows, one can refer [7–9,17,20,48]. The low Mach limits formulate the pressure as: p(ρ) =

p(ρ) ˜ − p(1) ˜ , ε2

(4)

whileε is the compressible parameter [40], and take ε going to zero. Formally, if |u| is bounded and p˜  (ρ) does not vanish, the Mach number will go to 0 as ε → 0. For this reason, the limit ε → 0 is called the low Mach number limit. It is easy to see, for the low Mach number limit, one should start from the case with sufficiently small Mach number, in which the flow is subsonic. The first theory of the low Mach number limit is due to Janzen and Rayleigh (see [39, Sect. 47], [43]), which is concerned with the steady irrotational flow. Their method of the expansion of solutions in power with respect to the Mach number was used both as a computational tool and as a method for the proof of the existence of solutions of the compressible flow. Klainerman and Majda [30] proved the convergence of compressible to incompressible flow by directly obtaining estimates for the scaled form of the partial differential equations. In particular, they established the incompressible limit of local smooth solutions of the unsteady Euler equations for compressible fluids with well-prepared initial data, i.e., some smallness assumption on the divergence of initial velocity. By using the fast decay of acoustic waves, Ukai [42] verified the low Mach number limit for the general data. The major breakthrough on the ill-prepared data is due to Métivier and Schochet [37], in which they proved the low Mach number limit of full Euler equations in the whole space by a significant convergence lemma on acoustic waves. Later, Alazard [1] extended [37] to the exterior domain. For the one dimensional Euler equation, the low Mach number limit have been proved under the BV space in [5]. Recently, [32] considered the low Mach number limits of steady Euler flow in the bounded infinite long nozzle, while [31] studied airfoil problem. For other fluid models, see [21,28,29,33,36,37] and the references therein. From [31,32], the expected corresponding homogeneous incompressible Euler equations as ε → 0 are written as: 

div u¯ = 0, div (u¯ ⊗ u) ¯ + ∇ p¯ = 0,

(5)

where u¯ = (u¯ 1 , · · · , u¯ n ) and p¯ represents the velocity and pressure, respectively, while density ρ¯ ≡ 1. As far as we know, up to now there is no mathematical result on the multidimensional incompressible flow through infinite largely-open nozzles. The airfoil problem of incompressible steady Euler equations is considered by Ou in [35,38]. For recent progress on the jet problem and related topic, please see [10,11,13,14]. In this paper, we first prove the existence and uniqueness of the incompressible flow in largelyopen nozzle by employ the variation method in Yuan-Li [34]. Then, we introduce the variational functional with respect to the compressible-incompressible difference function to obtain the uniform L2 estimates. Comparing with the previous result of [34], our compressible case includes the effect of extra force, which dominates the asymptotic rate to the infinity. Since the incompressible flows are free from this effect, the vanishing phenomenon is rigorously justified by the proper expansion based on a suitable formulation on the Bernoulli’s law and a refined cut-off function for the low Mach number limits. Another difficulty is to find a proper way to show

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Fig. 1. Largely-open nozzle.

the convergence of the pressure. Based on the fine analysis in the phase plane, we establish the strong convergence of pressure rather the gradient in [31,32]. It is noticeable that for any fixed compressibility parameter, we extend the previous result [34] to the case with conservative force effect. The rest of this paper is organized as follows. In Section 2, we introduce mathematics formulation of the low Mach number limit of the steady irrotational Euler flows in the infinite multidimensional largely-open nozzles problem, and then present the main theorems of this paper. In Sections 3, the existence and uniqueness of the incompressible flow is established. In Section 4, we show existence, uniform estimates, and further properties of compressibleincompressible difference function. In Section 5, the convergence rate of the low Mach number limit of the steady irrotational Euler flows is proven. In Section 6, for the compressible flow with general conservative forces, the well-posedness of subsonic solution and the construction of subsonic-sonic limit solutions are presented. 2. Infinite multidimensional largely-open nozzles problem and the low Mach number limit In this section, we will first introduce both the incompressible and compressible largely-open nozzles problem, then the low Mach number limit, and finally the main theorem of this paper. 2.1. Infinite multidimensional largely-open nozzle problem The De Laval nozzle we consider in this work is a C 2,α smooth domain D in space Rn (n ≥ 3) with convergent part C− and divergent part C+ , connected by another part T called throat. Here C− (resp. C+ ) is the infinite part of a cone with a cross-section − (resp. + ) and center at a point A− ∈ Rn (resp. A+ ∈ Rn ) so that r− := |x − A− | (resp. r+ := |x − A+ |) is larger than some fixed positive number b− (resp. b+ ), C− ∩ C+ = ∅, and T is a bounded cylinder-like domain so R = C ∩ (B R )c that D = C− ∪ C+ ∪ T is a C 2,α smooth simply connected domain in Rn . And C+ + + R R c R R (resp. C− = C− ∩ (B− ) ), with B+ (resp. B− ) the ball centered at A+ (resp. A− ) with radius R R )c (resp. (B R )c ) its complement. in Rn (n ≥ 3) and (B+ − Without loss of generality, we may assume A± = (±a, 0, . . . , 0) for a number a > 0, and − (resp. + ) be a C ∞ smooth simply connected domain in the unit (n-1)-sphere Sn−1 ⊂ Rn . Note that T could be a curved pipe in space with quite complex position. The boundary of D is assumed to be a solid wall, and denoted by W. It is a connected C ∞ smooth surface in Rn , and its outward unit normal vector is written as n. Let S be a connected component of S  ∩ D so that D\S consists of two and only two connected components. Then we call S a cross-section of D. Here S  is a smooth (n − 1)-dimensional hypersurface in Rn that intersects with D (see Fig. 1).

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The fluid fills in the region D. At the boundary W, the flow satisfies the slip boundary condition: u·n=0

on W,

(6)

where n is the unit outward normal of domain D. Due to (1)1 and (6), one can obtain the fixed mass flux property:  ρu · lds = m,

(7)

S0

where S0 is cross-section of D and l its unit outer vector. m is called the mass flux. Now the incompressible infinite multidimensional largely-open nozzle problem can be formulated as the following problem. Problem 1. Find functions (ρ¯ ≡ 1, u, ¯ p), ¯ which satisfies (5), with the slip boundary condition (6) and mass flux condition (7). For the compressible Euler, the infinite multidimensional largely-open nozzle problem can be formulated as the following problem. Problem 2. Find functions (ρ, u, p), which satisfies (1), with the slip boundary condition (6) and mass flux condition (7). 2.2. A Hilbert space To solve the infinite multidimensional largely-open nozzle problem, we first recall the following lemma and theorem [34], which is adapted from [18,35,38] on the airfoil problem. Lemma 1. There holds  R C±

|v|2 dx ≤ |r± |2



2 n−2

2  |∇v|2 dx, while r± = |x − A± |,

(8)

R C±

where R > b± , and v ∈ C0∞ (Rn ). Lemma 2. There is a constant C depending on D so that for any v ∈ C0∞ (Rn ), there holds  D

|v|2 dx ≤ C(D) 1 + |x|2

 |∇v|2 dx.

(9)

D

∞ n Theorem 3. Let V0 = {v : v is  the restriction of a function u ∈ C0 (R ) on D} be equipped with an inner product (v1 , v2 ) = D ∇v1 · ∇v2 dx. Then it is a pre-Hilbert space. Its completion is written as V, with the norm  · . Inequalities (8) and (9) also hold for any v ∈ V.

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Then we will study Problem 1 and Problem 2 for the incompressible and compressible cases separately. 2.3. Formulation of incompressible flow Let us consider the incompressible case first. The steady irrotational incompressible Euler flow is governed by the following equation: ⎧ ⎨ div u¯ = 0, div (u¯ ⊗ u) ¯ + ∇ p¯ = F, ⎩ curl u¯ = 0.

(10)

Here, the conservative force F could be written as F = ∇ϕ. By (10)2 , we have the Bernoulli law for the incompressible irrotational case that:  ∇

 |u| ¯2 + p¯ − ϕ = 0, 2

(11)

which equals to: p¯ = ϕ −

|u| ¯2 , 2

(12)

up to a constant. Theorem 4. For any fixed m > 0, there exists a unique solution (ρ¯ ≡ 1, u, ¯ p) ¯ of Problem 1 corresponding to equations (10), with u¯ ∈ L2 (D) and u ¯ C α (D) < C|m|, satisfies: |u| ¯ ≤

C n

(1 + |x|) 2

.

Moreover, suppose ϕ ∈ L∞ (D)

q

and ∇ϕ ∈ Lloc (D)

for q > n,

then p¯ ∈ C α (D). 2.4. Formulation of the compressible flow and the low Mach number limit Now let us consider Problem 2 for the compressible case and the low Mach number limit. The irrotational compressible Euler flow with low Mach number can be written as ⎧ ⎨ div (ρ ε uε ) = 0, div (ρ ε uε ⊗ uε ) + ∇p ε = ρ ε F, ⎩ curl uε = 0,

(13)

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where p ε = p (ε) (ρ ε ) = The Mach number is defined as M ε =

˜ p(ρ ˜ ε ) − p(1) . 2 ε

ε  |u | (p (ε) ) (ρ ε )

=

(14)

ε ε|u | . p˜  (ρ ε )

By (13)3 , we have the following Bernoulli law that ⎛ ⎜ ∇⎝

|uε |2 2

ρ

ε

+

⎞ (p (ε) ) (s) s

⎟ ds − ϕ ⎠ = 0,

(15)

1

which is equivalent to |uε |2 + 2

ρ

ε

(p (ε) ) (s) ds = ϕ, s

(16)

1

up to a constant. ˜ which satisfies: Let us introduce the rescaled enthalpy function h, p˜  (ρ) . h˜  (ρ) = ρ

(17)

|uε |2 + h(ε) (ρ ε ) − h(ε) (1) = ϕ, 2

(18)

Then, (16) becomes

˜ ˜ where h(ε) (ρ) = ε −2 h(ρ). By (2), we can see that h(ρ) is a strictly increasing function with respect to ρ, so does h(ε) (ρ ε ). Let p˜  (ρ) ˜ H˜ (ρ) := + h(ρ). 2

(19)

ε which satisfies that Then for each fixed ε, we can introduce the critical density ρcr ε ˜ ) = h(1) + ε 2 ϕ. H˜ (ρcr

(20)

ε ˜ ρcr (ϕ) = H˜ −1 (h(1) + ε 2 ϕ),

(21)

So

and the critical speed is ε qcr (ϕ) =

 1 ˜ p˜  ◦ H˜ −1 (h(1) + ε 2 ϕ), ε

(22)

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which will go to the infinity as ε goes to zero. ε holds if and only if the flow is subsonic, i.e., M < 1. Similarly, It is easy to see that |uε | < qcr for each θ ∈ (0, 1), there exists qθε such that |uε | ≤ qθε holds if and only if M ε ≤ θ . Moreover, ε , for fixed θ ∈ (0, 1), qθε is monotonically increasing with respect to θ ∈ (0, 1). Similar to qcr ε and εq ε are uniformly bounded respect to ε. Then qθε → ∞ as ε → 0. Also, for ε > 0, both εqcr θ density ρ ε can be represented as the function of |uε |2 , i.e.,  −1  |uε |2 (ε) 2 ρ = ρ (|u | , ϕ) = h +ε ϕ h (1) − 2     ε 2 |uε |2 −1 2 ˜ = h˜ +ε ϕ . h(1) − 2 ε

ε

ε 2



(ε)

(23)

For fixed 0 < ε < 1, when flow is subsonic, it holds that   ε ˜ ≤ ρcr ≤ ρ ε (|uε |2 , ϕ). H˜ −1 ◦ h(1) When ε → 0, we could expect the compressible Euler flow will converge to the corresponding incompressible Euler flow. More precisely, there is the following result as the main theorem of the paper. Theorem 5. For a given L∞ function ϕ which satisfies: ϕ ∈ L2 (D),

(1 + |x|β )∇ϕ ∈ Lq (D),

q > n,

β >1−

n . q

(24)

For any fixed m > 0, there exists constant εc such that when 0 < ε < εc there exists a unique solution (ρ ε , uε , p ε ) of Problem 2 where p ε satisfies (14) and M ε < 1. For any 0 < ε < εc , ε compressible-incompressible difference velocity u˜ (ε) := u ε−2 u¯ ∈ C α (D) satisfies: |u˜ (ε) | ≤ where β  = min{ n2 , β + ρ (ε) = 1 + O(ε 2 ),

n q

C , (1 + |x|)β

− 1}. Furthermore, as ε → 0, we have that u(ε) = u¯ + O(ε 2 ),

p (ε) = p¯ + O(ε 2 ),

M ε = O(ε),

(25)

where (ρ¯ ≡ 1, u, ¯ p) ¯ is the classical solution of Problem 1. Theorem 6. For ε = 1 and ϕ which satisfies (24), there exists a positive number m ¯ depending on D and ϕ. Then (1) if m < m, ¯ then there exists a unique solution (ρ, u, p) of Problem 2 where p satisfies (2) and M < 1. And, |u| ≤

C , (1 + |x|)β

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where β  = min{ n2 , β + qn − 1}. (2) Let m → m ¯ as  → 0, with m < m. ¯ And (ρ  , u , p  ) be the corresponding solutions to  Problem 2 as m = m . Then, as  → 0, the solution sequence (ρ  , u , p  ) possess a subsequence (still denoted by) converge a.e. in D to (ρ, ˇ u, ˇ p) ˇ which is a weak subsonic-sonic limiting solution of Problem 2 as m = m. ¯ 3. Incompressible irrotational flow In this section, we will study the existence, regularity, and uniqueness of steady subsonic incompressible potential flows in a class of multidimensional infinite De Laval nozzles with largely-open convergent and divergent parts. Now let us consider the incompressible case. By (10)3 , we can introduce the velocity potential φ¯ for the incompressible case such that ¯ u¯ = ∇ φ.

(26)

Therefore, the incompressible irrotational Euler flows of Problem 1 can be reformulated into the following problem. Problem 3. Let n ≥ 3. Find function φ¯ such that ⎧ ¯ ⎪ ⎨ ¯φ = 0, ∂φ ∂n = 0, ⎪ ⎩  ∂ φ¯ ds = m, S0 ∂l

x ∈ D, x ∈ W,

(27)

where S0 is any arbitrary cross section of the nozzle, and n and l are the unit outer normals of the nozzle wall W and S0 respectively. 3.1. Existence of weak solution Let us consider the existence of the incompressible flow for infinite multidimensional largelyopen nozzle problem. Now we review some properties of the special symmetric solutions to (27)1 . Suppose a solution φ¯ depends only on r = |x|, then um =

m , ||r n−1

(28)

where  is a domain in Sn−1 with || its area. Direct computation leads to: u0 (r) ≡ 0, |um (r) − um (r)| =

(29) |m − m |

||r n−1 (n − 1)|m| |∂r um (r)| = , ||r n

|∂r um (r) − ∂r um (r)| =

,

(n − 1)|m − m | . ||r n

(30) (31) (32)

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We introduce now an auxiliary function φ m . Apply the above conclusion to the cones C± , that is, to r = r± , and  = ± . Let R˘ = 4b, we may solve symmetric solutions u−m (r− ) ≤ 0 (resp. ≥ 0) for r− > We then define,

R˘ 4

and um (r+ ) ≥ 0 (resp. ≤ 0) for r+ >

⎧ ⎪ ⎨ 0, φ m =  r± ± ⎪ ⎩ R˘ ζ (s± )u±m (s± )ds± ,

R˘ 4.

R˘

if x ∈ D 2 ,

R˘

if x = (r± , w± ) ∈ C±2 ,

(33)

2

where ζ (s) ∈ C ∞ (R) is a nonnegative monotonic function that equals 1 for s > R˘ and vanishes ˘ for s < R2 . It is clear that φ m belongs to C ∞ (D). We list some useful properties below. Lemma 7. For any m, there hold: (1) φ 0 = 0; (2) it satisfies the Neumann condition on W : ∇φ m · n = 0; ˘

R (3) ∂r± φ m = u± ±m (r± ) in C± ;

˘ (4) ∇φ m ∈ L2 (D) and ∇φ m − ∇φ m L2 (D) ≤ C(n, R)|m − m |; ˘ n)|m − m | and  φ − φ  L2 (D) ≤ C(R, ˘ n)|m − m |; (5) ∇φ − ∇φ   ∞ ¯ ≤ C(R, m L (D )

m

m

(6) φ m − φ m does not belong to V, useless m = m .

m

Proof. (1) For m = 0, we can get u0 ≡ 0 and φ 0 ≡ 0. (2) By definition, we can get ⎧ ⎪ x−A+ ⎪ ⎪ ζ (r+ )u+ +m (r+ ) r+ , ⎨ ∇φ m = 0, ⎪ ⎪ ⎪ ⎩ x−A− ζ (r− )u− −m (r− ) r− ,

R˘

if x = (r+ , w+ ) ∈ C+2 , R˘

if x ∈ D 2 ,

(34) R˘ 2

if x = (r− , w− ) ∈ C− .

(3) We easily get (3), by (34). (4) By direct calculation, we have  |∇φ m − ∇φ m |2 dx = D

 +,−

 2 ± ζ 2 (r± ) u± (r ) − u (r ) dx ±m ± ±m ±

R˘

C±2

≤

 +,−

=

±∞ 2 ± |± | (r ) − u (r ) r± n−1 dr± u±  ± ± ±m ±m R˘ 2

±∞  |m − m |2  +,−

|± |

R˘ 2

1 dr± r± n−1

˘ |± |)|m − m |2 . = C(n, R,

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(5) By calculation, we show ± ∇φ m − ∇φ m L∞ (D¯ ) ≤ sup |ζ (r± )||u± ±m (r± ) − u±m (r± )|

˘ ≤ C(n, R)|m − m |, and  | φ m − φ m |2 dx D

≤

 

+,−

R˘

± 2 |∂r± ζ (r± )|2 |u± ±m (r± ) − u±m (r± )| dx ˘

R C±2 \C±

+

 +,−

± 2 |ζ (r± )|2 |∂r± u± ±m (r± ) − ∂r± u±m (r± )| dx

R˘

C±2

≤

 |m − m |2 +,−

|± |2

 r± 2−2n dx + R˘

 |m − m |2 (n − 1)2  |± |2

+,−

r± −2n dx

R˘

˘

R C±2 \C±

C±2

˘ |± |)|m − m |2 . ≤ C(n, R, ˘ we have (6) For r+ > R,

˘ + φ m (r+ ) = φ m (R)

r+

u+ +m (s)ds

R˘

˘ + = φ m (R)

r+ R˘

|m| ds |+ |s n−1

˘ + (2 − n)|m| = φ m (R) |+ |



1 r+ n−2

−

1



R˘ n−2

For m > m , we have φ m (r+ ) − φ m (r+ ) ˘ − φ  (R) ˘ + = φ m (R) m

r+  R˘

 + u+ +m (s+ ) − u+m (s+ ) ds+

> 0.

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R˘ =

 ζ (s+ )



+ u+ +m (s+ ) − u+m (s+ )

R˘ 2

m − m = |+ |

R˘ ζ (s+ )s+

1−n

R˘ 2

m − m ds+ + |+ |

m − m ds+ + |+ |

r+ s+ 1−n ds+ R˘

r+ s+ 1−n ds+ R˘

˘ m) > 0. ≥ C(R, Recalling Theorem 3, it is easy to see φ m − φ m does not belong to V. Let φ¯ = ψ + φ m and   I [ψ, φ m ] = D

 1 |∇ψ|2 − ψ φ m dx. 2

(35)

We consider the following variational problem: min I [ψ, φ m ].

(36)

ψ∈V

For simplicity, in the following, we temporarily drop the parameter m and write φ m as φ, um as u, etc. Lemma 8. The critical point of the energy functional satisfies (27). Proof. Suppose that ψ ∈ V is a critical point of the functional (35). For any ϑ ∈ V, define I (τ ) = I [ψ + τ ϑ, φ] for τ > 0. Then  

 1 1 |∇(ψ + τ ϑ)|2 − (ψ + τ ϑ) φ − |∇ψ|2 + ψ φ dx 2 2

I (τ ) − I (0) = D

 

= D

τ2 = 2 =

Thus

τ2 2

 τ2 2 τ ∇ψ · ∇ϑ + |∇ϑ| − τ ϑ φ dx 2 

 |∇ϑ| dx − τ D

D





|∇ϑ|2 dx − τ D

 ϑ φdx + τ

2

D

∇ψ · ∇ϑdx D

¯ ϑ φdx.

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.13 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

I (τ ) − I (0) τ = τ 2





¯ ϑ φdx

|∇ϑ|2 dx − D

13

D

= J 1 + J2 . Here limτ →0 J1 = 0. Since I  (0) = 0, this gives J2 = 0. Therefore we proved that once ψ is a critical point, then φ¯ = ψ + φ is a weak solution to (27)1 . Next we will consider the solution of variational functional. Firstly, since D is unbounded, we need to check that the energy functional I [ψ, φ] is well-defined (finite) for any ψ ∈ V. Secondly, we need to prove the existence of minimizer of variational functional. Finally, the uniqueness of the solution will be proved. Lemma 9. The energy functional I [ψ, φ m ] is coercive on V. Proof. By Lemma 1, (30) and (31), we have          ±  ψ · φ  ≤ |ψ| |ζ  (r± )||u± ±m (r± )| + |ζ (r± )||∂r± u±m (r± )| dx     D

R˘

C±2



|m| |ψ| dx + |± |r± n−1

≤ R˘

 |ψ| ˘

(n − 1)|m| dx |± |r± n

R C±

˘

R C±2 \C±





≤ CR,n ˘ |m|

|ψ|dx + CR,n ˘ |m|

D R˘

⎛ ⎜ ≤ CR,n ˘ |m| ⎝

˘

|ψ| 1 dx r± r± n−1

R C±

⎞1



2

|ψ|2

⎟ dx ⎠ 1 + |x|2

D R˘

⎛ ⎜ ⎜ +CR,n ˘ |m| ⎝



⎞1 ⎛ 2 ∞ ⎟ ⎜ 2 |∇ψ| dx ⎟ ⎠ ⎝

˘

R C±

R˘

⎞1

2

⎟ dr 2n−2 ± ⎠ 1

r±

≤ CR,n ˘ |m|ψ. Thus 1 2 2 I [ψ, φ] ≥ ψ2 − CR,n ˘ |m|ψ ≥ ψ − CR,n ˘ m , 2 hence the energy functional is coercive on V.

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.14 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

By coerciveness of the functional I [·, φ] on V, we know l = infψ∈V I [ψ, φ] exists. Observing I [0, φ] ≡ 0, we have l ≤ 0. By definition of infimum, there is a minimizing sequence {ψk }∞ k ⊂V 1 2 so that limk→∞ I [ψk , φ] = l. So for k large, there holds 0 ≥ I [ψk , φ] ≥ 2 ψk  − CR˘ m2 and hence a uniform bound ψk  ≤ CR˘ |m|. Since V is a Hilbert space, there is a subsequence of {ψk } (still denoted by {ψk }), that converges weakly to some ψ¯ ∈ V. By weak lower semi-continuous of norms, we see ¯ ≤ C ˘ |m|. ψ R

(37)

Lemma 10. For the energy functional I [ψ, φ m ], ψ¯ is a minimizer. Proof. Firstly, we need to show that the functional I [·, φ] is sequentially lower semi-continuous with respect to the weak convergence of V, that is, ¯ φ] ≤ lim inf I [ψk , φ], if I [ψ,

ψk  ψ¯

k→∞

weakly in V.

(38)

˘ For an increasing sequence {Rj }∞ j =1 ⊂ R with R1 > R and Rj → ∞ as j → ∞, recall 1 (D), using compact embedding H 1 (D Rj ) →→ L2 (D Rj ), by taking a diagonal subψk ∈ Hloc sequence of {ψk } (still denote it to be {ψk }), we may assume ψk → ψ¯ in L2 (DRj ) for any j ≥ 1. By (31), we have 

 

φ ψk − ψ¯ dx

Rj

C±



≤

   ¯ |ζ (r± )| ∂r± u± ±m (r± ) |ψk − ψ|dx

Rj

C±



≤ CR˘ |m| Rj

¯ |ψk − ψ| dx n r±

C±

1− 2 ¯ ≤ CR,n ˘ |m|ψk − ψRj

n

n

≤ C  Rj 1− 2 . For any  > 0, we choose j large so that C  Rj 1− 2 < . While on DRj , using the fact ψk → ψ¯ in L2 (DRj ), and note φ ∈ L2 (D) by Lemma 7, we have n

 D

Rj

¯

φ(ψk − ψ)dx →0

as k → ∞.

(39)

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T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

15

Therefore we proved  lim

k→∞

¯

φ(ψk − ψ)dx = 0.

D

¯ 2 , we know the functional is lower semi-continuous. Thus ψ¯ ∈ V is Due to ψk 2 ≥ ψ actually a minimizer. Lemma 11. For any fixed φ, there is only one ψ ∈ V so that φ¯ = ψ + φ is a weak solution to (27)1 . In other words, the energy functional I [·, φ] has only one critical point, namely, the minimizer. Proof. Suppose that there are ψj ∈ V so that φ¯ j = ψj + φ (j = 1, 2) are weak solutions to (5) (hence ψj are critical points of I [·, φ]). Set φ˚ = φ¯ 1 − φ¯ 2 = ψ1 − ψ2 , which belongs to V. Note that φ˚ satisfies  ˚ ∇ φ∇vdx = 0,

v ∈ C0∞ (Rn ).

(40)

D

Since φ˚ ∈ V, by definition, there exists a sequence {φ˚ k } ⊂ C0∞ (Rn ) so that    ˚k  ∇ φ − ∇ φ˚ 

L2 (D )

→ 0, as k → ∞.

(41)

Using Cauchy-Schwarz inequality,              ˚ 22 ∇ φ ≤  ∇ φ˚ · ∇ φ˚ k dx  +  ∇ φ˚ · ∇ φ˚ k − φ˚ dx  L (D )     D D          ≤ ∇ φ˚  2 ∇ φ˚ k − φ˚  2 L (D )

 2   Taking k → ∞, ∇ φ˚  2

L (D )

L (D )

(42)

→ 0, which implies φ˚ = 0.

Summarizing the results obtained above, we have proved the following theorem. Lemma 12. For any fixed φ m , the functional (35) has uniquely one critical point ψm ∈ V, which is the minimizer. Also, φ¯ m = ψm + φ m is a weak solution to Problem 3. The minimizer (weak solution) also satisfies ˘ ψm  ≤ C(n, R)|m|.

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T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

16

Remark 1. For any fixed m > 0, the weak solution to Problem 3 satisfies ˘ ∇ φ¯ m L2 (D) ≤ C(n, R)|m|.

(43)

3.2. Regularity of weak solution In this section, we consider regularity of gradient of weak solution to Problem 3. Firstly, we need the following proposition. Proposition 1. Let aij for i, j = 1, . . . , n be L∞ functions on B1 , and λ be a positive constant. Assume that ∀ ξ ∈ Rn , λ|ξ |2 ≤

n 

aij ξi ξj ≤ λ−1 |ξ |2 , and fi ∈ Lq , q > n.

i,j =1

Let w(y) be a function in H 1 . Let fi be functions in Lq with q > n. Suppose n 

n   ∂i aij (y)∂j w(y) + ∂i fi = 0,



i,j =1

i=1

holds in the distribution sense. Then w(y) is Hölder continuous in B1/2 and there exist two constants 0 < α ≤ 1, k, depending on λ such that   sup |w(y)| ≤ k ||w||L2 (B1 ) + ||fi ||Lq (B1 ) ,

y∈B1/2

sup

y1 ,y2 ∈B1/2

  |w(y1 ) − w(y2 )| ≤ k ||w||L2 (B1 ) + ||fi ||Lq (B1 ) . α |y1 − y2 |

The proof of this proposition can be found in [24] (see Theorem 8.24). ¯ For the regularity, we propose the following theorem: Then we consider regularity of ∇ φ. Lemma 13. Suppose that φ¯ = ψ + φ is a weak solution to Problem 3. Then there is a constant ¯ there holds α ∈ (0, 1) depending on n so that for any open set U ⊂⊂ O ⊂⊂ D, ¯ C α (U ) ≤ C|m|, ∇ φ

(44)

and that ¯ |∇ φ(x)| ≤

C|m| n

(1 + |x|) 2

.

(45)

Proof. Firstly, to solve boundary regularity, we recall the straighten boundary [34]. For any point x0 ∈ W, set Tx0 be the tangent space of W at xo , and n(x0 ) the outward unit normal vector. Choose a coordinate system y  = (y1 , . . . , yn−1 ) on Tx0 with origin at x0 , by combining them with the coordinate z along −n(x0 ), we find a local coordinate system y = (y  , z) of Rn at x0 .

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.17 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

17

By definition of C k boundary, there is a neighbourhood Nx0 of x0 in Rn so that W ∩ Nx0 is the graph of a function z = f (y  ), and  n(x) = (∇f, −1)/ 1 + |∇f |2 ,

∇f = (∂y1 f, . . . , ∂yn−1 f ),

for x = (y  , f (y  )) ∈ Nx0 ∩ W. Now, for yn = d(x), we define a change of variables (coordinates)  : x → y = (y  , d) given by x = (y  , f (y  )) − n(y  )d.

(46)

Here n(y  ) = n(y  , f (y  )). Under the transform (46), W ∩ Nx0 becomes yn = 0, and the direction −n(x) on W ∩ Nx0 becomes (0, · · · , 0, 1). Without loss of generality, we may assume that Nx0 ∩ W becomes the half-ball Br+ (Br± = {y ∈ Rn : |y| < r, ±yn > 0}) with r < k, and Nx0 ∩ W becomes r = {y ∈ Rn : |y| < r, yn = 0}. ¯ In the y−coordinates, write φ¯  (y) = φ((y)), where  = −1 . We have

y φ¯  = 0, in Br+ , n n  

∂yj φ¯ 

j =1 i=1

(47)

∂yj ∂yn · n = 0, on r . ∂xi xi

(48)

We will use a reflection argument to reduce the boundary L∞ and Hölder regularity of gradient of corresponding interior regularity. Here we propose the following somewhat restrictive structure conditions at yn = 0: n  n 

∂yj φ¯ 

j =1 i=1

n n   j =1 i=1

∂yj φ¯ 

∂yj ∂yn = ∇n φ¯  ∂xi ∂xi

 n  n−1  ∂yj ∂yk ∂yj ∂yk   ¯ = ∂yj φ ∂xi ∂xi ∂xi ∂xi j =1

(k = 1, · · · , n − 1).

i=1

By reflection, we know that φ˜  (y  , yn ) =



φ¯  (y  , yn ), in Br+ , φ¯  (y  , −yn ), in Br− ,

belongs to H 1 (Br ) and is a weak solution to

y φ¯  = 0 in Br . Applying Proposition 1 to (49), we get ∇y φ¯  C α (B + ) ≤ C  ∇y φ¯  Br+ , r/2

(49)

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.18 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

18

and ¯ ∞ + ≤ C  r − 2 ∇y φ ¯ 2 +. |∇y φ| L (B ) L (Br ) n

r/2

+ Returning to the x-coordinates, we find for U = (Br/2 ) and O = (Br+ ), the following boundary Hölder gradient regularity:

¯ C α (U ) ≤ C∇ φ ¯ L2 (O) ∇ φ and ¯ L∞ (U ) ≤ C  r − 2 ∇ φ ¯ L2 (O) . |∇ φ| n

For interior regularity, we can get conclusion by applying Proposition 1. ˘ Next let us consider the estimates of ∇ φ¯ when |x| is sufficiently large. For a fixed R > 4R, R/2,2R R/2 c R/4,4R 2R = C± ∩ B± ∩ (B± ) and V = C± be truncated cones. Now for l > 1, set let U = C± lR/2,2lR lR/4,4lR l l l ¯ U = C± and V = C± . Then for any x ∈ U , we see y = x/ l ∈ U . Let ϕ(y) ¯ = φ(ly). ¯ We then get from φ = 0 the scaled equation

y ϕ(y) ¯ = 0. Applying Proposition 1, we have 1− 2 ¯ ¯ l sup |∇x φ(x)| ¯ ≤ C∇x ϕ(x) ¯ ∇x φ(x) = sup |∇y ϕ(y)| L2 (O ) ≤ Cl L2 (D ) . n

y∈U

x∈U l

¯ ≤ Cl − n2 ∇ φ ¯ L2 (D) . Now, for any r± > R ≥ 4R˘ large, we can find So we get supx∈U l |∇ φ| l > 1 such that r± = lR. Therefore, sup

|x−A± |=r±

¯ ≤ C∇ φ ¯ L2 (D) R 2 |r± |− 2 ≤ CR ∇ φ ¯ L2 (D) |r± |− 2 , |∇ φ| n

n

n

which completes the proof. 3.3. Uniqueness of weak solution In this section, we prove uniqueness of weak solution with finite energy to (27) for a given mass flux. Lemma 14. Problem 1 exists at most one weak solution φ¯ up to a constant such that 

¯ 2 dx ≤ Cm2 . |∇ φ|

(50)

D

Proof. Existence has been proved. So we only need to consider the uniqueness. 1 (D)(j = 1, 2) and Step 1. Assume there are two different solutions of Problem 1 φ¯ j ∈ Hloc ¯ ¯ ∇φj L2 (D) ≤ ∞. Let φ = φ1 − φ2 . Then ∇φL2 (D) < ∞, and there hold

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.19 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••



∀v ∈ C0∞ (Rn ),

∇φ · ∇vdx = 0, D

19

(51)

 ∇φ · lds = 0.

(52)

S0 1 (D) solution must belong to H 2 (D), by (51) we see φ = 0 holds Step 2. Since any Hloc loc + be any finite part a.e. in D, as well as the boundary condition ∇φ · n = 0 a.e. on W. Let D− of the nozzle D bounded by wall W, the left cross-section S− and the right cross-section S+ , + . Then S− ∩ S+ = ∅, with l the corresponding unit normal vector of S± pointing out of D− + integrating the equation in D− , using integration by parts and boundary condition on W, one has





0=

∇φ · lds +

S−

∇φ · lds.

(53)

S+

Particularly, if we choose S− to be S0 , then by (52), we conclude that (52) itself holds for any cross-section S0 of D. 



R,R R \C R and ξ R := C R,2R . Now define an auxiliary Step 3. For R < R  , we set C± := C± ± ± ± piecewise affine function

⎧ (φ)ξ R , ⎪ ⎪ − ⎪   ⎪ 2 R ⎪ ⎪ ⎨ (φ)ξ−R R r− − 2 , ˘ φ(x) = 0,   ⎪ ⎪ ⎪ (φ)ξ R R2 r+ − R2 , ⎪ ⎪ + ⎪ ⎩ (φ) R , ξ +

R, x ∈ C− R/2,R

x ∈ C− , R/2 x∈D , R/2,R x ∈ C+ , R x ∈ C+ .

(54)

 Here R should be very large, and (φ)U = |U1 | U φdx is the average of φ over a set U . We also introduce a piecewise smooth cut-off function η ∈ C0 (D) as follows: ⎧ ⎪ ⎨ 0,   r± , η(x) = ln 2R ⎪ ⎩ 1 ln( 2 ),

2R , x ∈ C±

x ∈ ξ±R , x

(55)

∈ DR .

˘ belongs to H 1 (D2R ) and has compact support. It can be approximated in Then v = η2 (φ − φ) 1 H (D) by functions in C0∞ (Rn ), so we can use it as a test function in (51). Step 4. Substituting v into (51), it comes to  |η∇φ|2 dx D



= D

˘ − η2 ∇φ · ∇ φdx

 D

  2η∇φ · ∇η φ − φ˘ dx

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.20 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

20

  R ∇φ · ∇ r± − dx 2



2(ln 2)2 ± R

≤ (φ)ξ R

R/2,R

C±

          1  ˘ +2  φ − φ η∇φ · ∇r± dx  .  R r±  ξ± 

(56)

For the first term on the right-hand side, using (52) to the cross-sections S = r± ± , it is

(φ)ξ R ±

2(ln 2)2

R

R

⎛ ⎜ ⎝

R/2

⎞



⎟ ∇φ · ∇(r± )ds ⎠ dr± = 0.

(57)

r± ±

For the second term of (56), note |∇r± | ≤ 1 and r± > R in ξ±R , using Cauchy-Schwarz inequality, it is bounded by           1  ˘ η∇φ · ∇r± dx  φ − φ   r  R ±  ξ± ⎛

⎞1 ⎛

1⎜ ≤C ⎜ R⎝

⎟ ⎜ ⎜ |η∇φ|2 dx ⎟ ⎠ ⎝



2

R ξ±

⎛



≤C

1⎝ R



⎞1

2

⎟ ˘ 2 dx ⎟ |φ − φ| ⎠

R ξ±

⎞1

2

⎛

⎜ |η∇φ|2 dx ⎠ ⎜ ⎝

D



⎞1

2

⎟ |φ − (φ)ξ R |2 dx ⎟ ⎠ . ±

R ξ±

Therefore we obtain an energy growing inequality  |∇φ|2 dx ≤ C DR

  1  2 |φ − (φ) | dx ≤ C |∇φ|2 dx. R ξ± R 2 +,− +,− R ξ±

(58)

R ξ±

 Step 5. Since D |∇φ|2 dx < ∞, from the above inequality, letting R → ∞, we infer |∇φ| = 0 in D. So φ¯ 1 differs from φ¯ 2 only by a constant. It completes the proof of Theorem 4.

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.21 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

21

4. Low Mach number limit In this section, we will introduce the approximate problems of Problem 1 and Problem 2 to the incompressible case and the compressible case, and then obtain the existence of the solutions of the approximate problems by the variational approach. 4.1. Compressible potential flow Now let us consider the compressible case. By (13)3 , we can introduce the velocity potential φ (ε) for the compressible case that ∇φ (ε) = uε .

(59)

Therefore, Problem 2 can be reformulated into the following problem. Problem 4. Find function φ (ε) such that    ε ⎧ (ε) 2 (ε) = 0, x ∈ D, ⎪ ⎨ div(ε) ρ |∇φ | , ϕ ∇φ ∂φ x ∈ W, ∂n = 0, ⎪   ⎩ ε |∇φ (ε) |2 , ϕ ∂φ (ε) ds = m. ρ S0 ∂l

(60)

By the straightforward computation, (60)1 can be rewritten as n 

aijε ∂ij φ (ε) +

n 

ij =1

biε ∂i φ (ε) = 0,

(61)

i=1

where  aijε

=ρ

ε

ε 2 ∂i φ (ε) ∂j φ (ε) δij − p˜  (ρ ε )

 ,

and

biε =

ε 2 ρ ε ∂i ϕ . p˜  (ρ ε )

(62)

For 0 < ε < 1 and M ε < θ < 1, we have that 0 < λ1 |ξ |2 ≤

n 

aijε ξi ξj ≤ λ2 |ξ |2 ,

(63)

ij =1

where constants λ1 and λ2 do not depend on ε. Since equation (61) is strictly elliptic if and only if M < 1. We do not know whether equation (61) is elliptic or not before solving it. Therefore we need to introduce the cut-off to truncate the coefficients of equation (61). For 0 < ε0 < 1 and 0 < θ < 1, we introduce q˚θε0 (ϕ) = inf0<ε<ε0 qθε (ϕ), and cut-off function on the phase plane

qˆ ε



⎧ 2 if |q| ≤ q˚θε0 (ϕ), ⎪ ⎨ q − 2ϕ function if q˚θε0 (ϕ) ≤ |q| ≤ q˚θε0 (ϕ), q 2 , ϕ = monotone   smooth  ⎪ 2 ⎩ sup q˚θε0 (ϕ) − 2ϕ (x) if |q| ≥ q˚θε0 (ϕ), x∈D 

(64)

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T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

22

where θ  := 1+θ 2 . (ε) Let ρˆ satisfy qˆ ε (q 2 , ϕ) + h(ε) (ρˆ (ε) ) = h(ε) (1), 2

(65)

  ε 2 qˆ ε (q 2 , ϕ) ˜ ρˆ ε = ρˆ ε (q 2 , ϕ) = h˜ −1 h(1) . − 2

(66)

which is equivalent to that

ε := ∂ ρˆ ε (, ϕ), and ρˆ ε := ∂ ρˆ ε (, ϕ). We denote ρˆ ϕ ∂ ∂ϕ Then, Problem 4 is reformulated into Problem 5 as follows.

Problem 5. Find function φ (ε) which satisfies   ⎧ div ρˆ ε (|∇φ (ε) |2 , ϕ)∇φ (ε) = 0, x ∈ D, ⎪ ⎨ (ε) ∂φ x ∈ W, ∂n = 0, ⎪ ⎩ ∂φ (ε) ε (ε) 2 S0 ρˆ (|∇φ | , ϕ) ∂l ds = m.

(67)

Straightforward calculation yields that (67)1 can be rewritten as n 

aˆ ijε (∇φ (ε) , ϕ)∂ij φ (ε)

i,j =1

+

n 

bˆiε (∇φ (ε) , ϕ)∂i φ (ε) = 0,

i=1

where   aˆ ijε ∇φ (ε) , ϕ

  ε (|∇φ (ε) |2 , ϕ)∂ φ (ε) ∂ φ (ε)   qˆ i j (ε) 2 = ρˆ |∇φ | , ϕ δij − (cε )2   ε (|∇φ (ε) |2 , ϕ)∂ φ (ε) ∂ φ (ε)   ε 2 qˆ i j (ε) (ε) 2 , = ρˆ |∇φ | , ϕ δij − p˜  (ρˆ (ε) ) (ε)

(68)

and bˆiε (∇φ (ε) , ϕ) =

ε 2 ρˆ (ε) qˆϕε (|∇φ (ε) |2 , ϕ)∂i ϕ p˜  (ρˆ (ε) )

.

(69)

Moreover, λˆ 1 |ξ |2 ≤

n  i,j =1

  aˆ ijε ∇φ (ε) , ϕ ξi ξj ≤ λˆ 2 |ξ |2 ,

|bˆiε ∂i φ (ε) | ≤ C|∂i ϕ|,

(70)

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[m1+; v1.304; Prn:4/02/2020; 13:47] P.23 (1-41)

T.-Y. Wang, J. Zhang / J. Differential Equations ••• (••••) •••–•••

23

where constants C, λˆ 1 , and λˆ 2 depend only on the subsonic truncation parameters θ and ε0 , and do not depend on solution φ (ε) . Finally, we remark that a solution of Problem 5, where the density ρˆ (ε) is derived from the new density-speed relation (65), is also a solution of the original potential flow equation in the Problem 4 when |∇φ (ε) | < q˚θε0 . To solve Problem 5, we follow the idea used in [18] to introduce a variational formulation. Denote 1 G (, ϕ) = 2

 ρˆ (ε) (λ, ϕ)dλ.

(ε)

0

Formally, (67) is the Euler-Lagrangian equation of the variational problem with respect to the integral that 

  G(ε) |∇φ (ε) |2 , ϕ dx.

For the low Mach number limits, similar to [31,32] we introduce   I (ε) φ, φ¯ = ε −4

        ¯ 2 , ϕ − ∇ φ¯ · ∇φ − ∇ φ¯ dx. G(ε) |∇φ|2 , ϕ − G(ε) |∇ φ|

(71)

D

Then Problem 5 becomes: Problem 6. Find a minimizer φ˜ (ε) ∈ V such that ¯ = min I (ε) (φ¯ + ε 2 φ, ˜ φ). ¯ I (ε) (φ¯ + ε 2 φ˜ (ε) , φ) ˜ V φ∈

Here, φ˜ =

φ−φ¯ ε2

and φ˜ (ε) =

φ (ε) −φ¯ . ε2

Lemma 15. The minimizer of Problem 6 satisfies (67). Proof. Let ϑ ∈ Cc∞ (Rn ). We can get     I (ε) φ (ε) + τ ϑ, φ¯ − I (ε) φ (ε) , φ¯ lim τ →0 τ   (ε)      −4 |∇φ (ε) + τ ∇ϑ|2 , ϕ − G(ε) |∇φ (ε) |2 , ϕ − τ ∇ φ¯ · ∇ϑ dx ε D G = lim τ →0 τ ⎞ ⎛ 1    ε −4 ⎝ d (ε)  G |∇φ (ε) + sτ ∇ϑ|2 , ϕ ds − τ ∇ φ¯ · ∇ϑ ⎠ dx = lim τ →0 τ ds D

0

(72)

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24

ε −4 = lim τ →0 τ ⎛



= ε −4 ⎝

 D

⎛ ⎝

1

(ε)

2τ G



⎞



|∇φ (ε) + sτ ∇ϑ|2 , ϕ ∇(φ (ε) + sτ ϑ) · ∇ϑds − τ ∇ φ¯ · ∇ϑ ⎠ dx

0

⎞    (ε) ¯ ⎠. 2G |∇φ (ε) |2 , ϕ ∇φ (ε) · ∇ϑdx − ∇ϑ · ∇ φdx

D

D

The first variation of I

 (ε)

0=



φ (ε) , φ¯ with ϑ is

 

(ε)

2G



  |∇φ (ε) |2 , ϕ ∇φ (ε) · ∇ϑ − ∇ φ¯ · ∇ϑ dx.

(73)

D

The second term in the integrand above vanishes due to the definition of the incompressible flow   ¯ Then, the first variation of I (ε) φ (ε) , φ¯ associated with ϑ is φ.  0=

(ε)

2G D



=



 |∇φ (ε) |2 , ϕ ∇φ (ε) · ∇ϑdx

  ρˆ (ε) |∇φ (ε) |2 , ϕ ∇φ (ε) · ∇ϑdx.

D

Hence this remark follows from the identity above. 4.2. Unique existence of the minimizer of Problem 6 First, for the existence of a minimizer of Problem 6, we have the following theorem. Theorem 16. Problem 6 admits a unique minimizer φ˜ (ε) ∈ V, which satisfies that 

|∇ φ˜ (ε) |2 dx ≤ Cm2 ,

(74)

D

where constant C depends only on φ¯ and D and does not depend on ε. Proof. The proof is divided into four steps. ¯ is coercive with respect to φ˜ in V, i.e., we will show ˜ φ) ¯ = I (ε) (φ, φ) Step 1. I (ε) (φ¯ + ε 2 φ, that   C ¯ ≥ 1 |∇ φ| ˜ 2 dx − C3 |∇ φ| ¯ 2 dx. I (ε) (φ, φ) (75) 4 D

Let (ε) ¯ I1 (φ, φ)

D

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=:ε

−4

 

25

        ¯ 2 , ϕ − 2G(ε) |∇ φ| ¯ 2 , ϕ ∇ φ¯ · ∇φ − ∇ φ¯ dx, G(ε) |∇φ|2 , ϕ − G(ε) |∇ φ| 

D

and (ε) ¯ =: ε −4 I2 (φ, φ)

 

(ε)

2G

     ¯ 2 , ϕ − 1 ∇ φ¯ · ∇φ − ∇ φ¯ dx. |∇ φ|

(76)

D

¯ = I (ε) (φ, φ) ¯ + I (ε) (φ, φ). ¯ Obviously, I (ε) (φ, φ) 1 2 (ε) ¯ is coercive with respect to φ˜ in V. First, we will show that I1 (φ, φ)   We denote p = (p1 , · · · , pn ), F (ε) (p) = G(ε) |p|2 , ϕ . Then by direct computation, we can get that         ¯ 2 , ϕ − 2G(ε) |∇ φ| ¯ 2 , ϕ ∇ φ¯ · ∇φ − ∇ φ¯ G(ε) |∇φ|2 , ϕ − G(ε) |∇ φ|        = F (ε) (∇φ) − F (ε) ∇ φ¯ − ∇F (ε) ∇ φ¯ · ∇φ − ∇ φ¯ =

n 1 

  ¯ j (φ − φ). ¯ t∂pi pj F (ε) t∇ φ¯ + (1 − t)∇φ dt∂i (φ − φ)∂

i,j =1 0 2 F (ε) is uniformly positive. In fact, we have It is easy to check ∂pp

  2 F (ε) (p) ∂pp

i,j

= aˆ ijε (∇φ (ε) , ϕ).

(77)

2 F . As a consequence, we have From property (2), we get the uniformly positivity of ∂pp

C1 2



 C˜ 1 (ε) 2 ¯ ˜ ˜ 2 dx. |∇ φ| dx ≤ I1 (φ, φ) ≤ |∇ φ| 2

(ε) ¯ Note that Now, let us consider I2 (φ, φ).

      −2 (ε) ¯ 2 , ϕ − 1  2G |∇ φ| ε       ¯ 2 , ϕ − 1  = ε −2 ρˆ (ε) |∇ φ|      ¯ 2 , ϕ)   −2 −1 ε 2 qˆ ε (|∇ φ|  ˜ ˜ − 1  h h(1) − = ε 2       2 ε ¯ 2 , ϕ)  −2 −1 ε qˆ (|∇ φ| −1 ˜  ˜ ˜ ˜ −h h(1)  h h(1) − = ε 2   1  ε    2 qˆ ε (|∇ φ| ¯ 2 , ϕ)   −1  ¯ 2 , ϕ)   qˆ (|∇ φ| ε ˜ dt  h˜ h(1) −t =  2 2   0

(78)

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26

  1     2 qˆ ε (|∇ φ| ¯ 2 , ϕ)   −1  ¯ 2 , ϕ)  qˆ ε (|∇ φ| ε  ˜ ≤ dt  . h˜ h(1) −t  2 2   0

Then              (ε) (ε) ¯ = ε −4 ¯ 2 , ϕ − 1 ∇ φ¯ · ∇φ − ∇ φ¯ dx  |I2 (φ, φ)| 2G |∇ φ|     D  ¯ 2 , ϕ)|∇ φ||∇(φ ¯ ¯ − φ)|dx ≤ Cε −2 qˆ ε (|∇ φ|  ≤ C3 D

D

¯ 2 dx + C1 |∇ φ| 4



˜ 2 dx. |∇ φ|

(79)

D

¯ By (78), (79), and the definition of φ˜ together, we get (75), which also implies that I (ε) (φ, φ) is bounded from below, i.e., there exists a constant C > 0 depending only on the data such that ¯ ≥ −Cm2 , I (ε) (φ, φ)

(80)

for all φ − φ¯ ∈ V. ¯ Step 2. Similarly, by (78) and (79) together, we can also show the upper bound of I (ε) (φ, φ) as: I

(ε)

¯ ≤ (φ, φ)

  C1 + 2C˜ 1 2 ˜ ¯ 2 dx. |∇ φ| dx + C3 |∇ φ| 4 D

(81)

D

¯ = I (ε) (φ¯ + ε 2 φ, ˜ φ) ¯ is finite for any φ˜ ∈ V. It means that I (ε) (φ, φ) ¯ is uniformly convex in the space V. Step 3. We will prove I (ε) (φ, φ) (ε) ¯ is linear respect to φ˜ for any φ˜ 1 , φ˜ 2 ∈ V, we have that Since I2 (φ, φ)       φ1 + φ2 ¯ I (ε) φ1 , φ¯ + I (ε) φ2 , φ¯ − 2I (ε) ,φ 2     (ε)  (ε)  (ε) φ1 + φ2 ¯ ¯ ¯ = I1 φ1 , φ + I1 φ2 , φ − 2I1 ,φ 2     ∇φ1 + ∇φ2 = ε −4 dx F (ε) (∇φ1 ) + F (ε) (∇φ2 ) − 2F (ε) 2 D

≥

C1 C1 ε −4 φ1 − φ2 2 = φ˜ 1 − φ˜ 2 2 , 2 2

(82)

which proves the uniform convexity of I (ε) . Step 4. We are ready to show the unique existence of minimizer φ˜ (ε) ∈ V of Problem 5, which satisfies (74).

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27

˜ φ) ¯ with respect to φ˜ in V. Taking φ˜ 1 and φ˜ 2 Firstly, we show the continuity of I (ε) (φ¯ + ε 2 φ, in V, corresponding to φ1 and φ2 respectively, we have ¯ − I (ε) (φ2 , φ) ¯ I (ε) (φ1 , φ)    = ε −4 G(ε) (|∇φ1 |2 , ϕ) − G(ε) (|∇ φ¯2 |2 , ϕ) − ∇ φ¯ · ∇ (φ1 − φ2 ) dx D

=ε

−4

 

D

−ε −4

 ¯ 2 , ϕ) − 2G(ε) (|∇ φ| ¯ 2 , ϕ)∇ φ(∇φ ¯ ¯ dx G(ε) (|∇φ1 |2 , ϕ) − G(ε) (|∇ φ| 1 − ∇ φ) 

  D

+ε −2

 

 ¯ 2 , ϕ) − 2G(ε) (|∇ φ| ¯ 2 , ϕ)∇ φ(∇φ ¯ ¯ dx − ∇ φ) G(ε) (|∇φ2 |2 , ϕ) − G(ε) (|∇ φ| 2     (ε) ¯ 2 , ϕ) − 1 ∇ φ∇(φ ¯ ε −2 2G (|∇ φ| 1 − φ2 ) dx

D

=ε

−4



⎛ ⎝

n 1  i,j =1 0

D

n 1 

−

i,j =1 0

+ε

−2

      (1 − t)∂pi pj F (ε) t∇φ1 + (1 − t)∇ φ¯ dt∂i φ1 − φ¯ ∂j φ1 − φ¯

(1 − t)∂pi pj F

 

⎞    t∇φ2 + (1 − t)∇ φ¯ dt∂i φ2 − φ¯ ∂j φ2 − φ¯ ⎠ dx

 (ε)





   (ε) ¯ 2 , ϕ) − 1 ∇ φ∇(φ ¯ ε −2 2G (|∇ φ| 1 − φ2 ) dx

D

≤ ε −4

 D

⎞1 ⎛ 2   C C C˜1 1 1 2 −4 2 2 ⎠ ⎝ ¯ ¯ ˜ ˜ |∇(φ1 ) − φ2 )| dx |∇(φ1 − φ)| dx + ε |∇(φ2 − φ)| dx + 2 2 2 D

D

C1 C1 ≤ φ˜ 1 − φ˜ 2 2 + φ˜ 1 − φ˜ 2 . 2 2 Now we can show the existence of the minimizer φ˜ (ε) by the compactness argument via ˜ φ) ¯ with respect to φ˜ in V. applying the continuity of the functional I (ε) (φ¯ + ε 2 φ, Secondly, for the uniqueness, if φ1 and φ2 are two minimizers such that + φ2 ¯ , φ). 2

(83)

φ1 + φ2 C ¯ ≥ 1 φ˜ 1 − φ˜ 2 2 . , φ) 2 2

(84)

¯ + I (ε) (φ2 , φ) ¯ ≤ 2I (ε) ( φ1 I (ε) (φ1 , φ) Then by (82), we know that ¯ + I (ε) (φ2 , φ) ¯ − 2I (ε) ( 0 ≥ I (ε) (φ1 , φ) Therefore

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28

∇φ1 − ∇φ2 = 0. Note that φ˜ 1 , φ˜ 2 ∈ V, so φ˜ 1 = φ˜ 2 . It means that the minimizer is unique in V. ¯ since we know that I (ε) (φ, φ) ¯ ≤ Finally, (74) easily follows from (75) via replacing φ by φ, (ε) ¯ φ). ¯ I (φ, 5. Proof of Theorem 5 By Lemma 15, we know that the minimizer of Problem 6 is a weak solution of Problem 5. In this section, we will first show the minimizer of Problem 6 is a solution of Problem 4 by remove the elliptic cut-off introduced in Problem 5, and then show the low Mach number limit to conclude the proof of Theorem 5. 5.1. C 1,α -Regularity of the minimizer Before removing the elliptic cut-off, let us consider the regularity of the derivatives of the ˜ solutions. Based on Proposition 1, we can show the C α -regularity of ∇ φ. Lemma 17. The minimizer φ˜ (ε) of Problem 6 satisfies that ∇ φ˜ (ε) C α (D) ≤ C|m|,

(85)

C|m| , (1 + |x|)β

(86)

and that |∇ φ˜ (ε) | ≤ where β  = min{ n2 , β +

n q

− 1} and constant C is independent of ε.

Proof. We divide the proof into three steps. Step 1. Let ϕ˘ = ∂k φ˜ (ε) , φ¯  = ∂k φ¯ for k = 1, . . . , n. Then by the straightforward calculation, ϕ˘ satisfies n 

n n       ∂i aˆ ijε ∂j ϕ˘ + ε −2 ∂i aˆ ijε ∂j φ¯  + ∂i (bˆkε ∂i φ (ε) ) = 0.

i,j =1

i,j =1

i=1

Since φ¯ = 0 and φ¯  = 0, we can change the equation above to: n 

n n       ∂i aˆ ijε ∂j ϕ˘ = −ε −2 ∂i fij ∂j φ¯  − ∂i (bˆkε ∂i φ (ε) ),

i,j =1

i,j =1

i=1

where fij := ε −2 (aˆ ijε − δij ) =

(ε) ε (ε) 2 (ε) (ε) ρˆ (ε) − 1 −2 ρˆ qˆ (|∇φ | , ϕ)∂i φ ∂j φ δ − ε . ij ε2 (cε )2

(87)

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29

Now we are going to show the uniform L∞ estimate of fij . For the first term, ρˆ (ε) − 1 ε2   (ε) ρˆ |∇φ (ε) |2 , ϕ − 1 = ε2   2 ε (ε) |2 ,ϕ) ˜ − ε qˆ (|∇φ h˜ −1 h(1) −1 2 = ε2   2 ε (ε) |2 ,ϕ) ˜ ˜ − ε qˆ (|∇φ h˜ −1 h(1) − h˜ −1 (h(1)) 2 = ε2   1 qˆ ε (|∇φ (ε) |2 , ϕ)  ˜ −1  ˜ ε 2 qˆ ε (|∇φ (ε) |2 , ϕ) =− dt. h h(1) − t 2 2 0

For the second term, ε −2

ε (|∇φ (ε) |2 , ϕ)∂ φ (ε) ∂ φ (ε) qˆ qˆ ε (|∇φ (ε) |2 , ϕ)∂i φ (ε) ∂j φ (ε) i j =  . ε 2 p˜  (ρ ε ) (c )

(88)

Therefore, due to the cut-off, we have the uniform L∞ estimate of fij . Also, for i, k = 1, . . . , n, |bˆkε ∂i φ (ε) | ≤ C|∂k ϕ|.

(89)

By the assumption that ∇ϕ ∈ Lq , we know bˆk ∂i φ (ε) are bounded in Lq for q > n. Step 2. Now we can show the interior estimates based on the equations derived in Step 1. ¯ L2 is uniformly For any bounded interior subregion of D, since φ¯  satisfies φ¯  = 0, and ||∇ φ|| bounded, from the standard elliptic estimate, we have that ∂j φ¯  is locally C ∞ and uniformly bounded, i.e., ¯ L2 , ||∂i φ¯  ||L∞ ≤ C||∇ φ|| where constant C does not depend on ε. Then, by Proposition 1, we have the interior L∞ and local Hölder estimate of ∇ φ˜ (ε) . Next for the boundary estimate near W, one can straighten boundary and apply Theorem 8.29 in [24] to replace Proposition 1 to follow the arguments above to show the boundary estimates near W. Then, by scaling argument, we get (86). Step 3. Finally, let us consider the estimates of ∇ φ˜ when |x| is sufficiently large. For a fixed R/2,2R 2R ∩ (B R/2 )c and V = C R/4,4R be truncated cones. ˘ let U = C± = C± ∩ B± R > 4R, ± ± lR/2,2lR lR/4,4lR Now for l > 1, set U l = C± and V l = C± . Then for any x ∈ U l , we see y = x/ l ∈ U . On V , we define 

w(y) = l β ϕ(ly) ˘

and

 σ (y) = l β φ¯  (ly).

(90)

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30

Then on V , w(y) satisfies   n n n n    ∂ ∂  ∂ ∂ β  +1 ˆ ε (ε) ε ∂ w =− ( fij σ) − (l aˆ ij bk ∂i φ ). ∂yi ∂yj ∂yi ∂yj ∂yi

i,j =1

j =1

i=1

i=1

By (50) and (74), we have that 

w(y)2L2 (V ) ≤ Cm2 l 2β −n ,

and



σ (y)2L2 (V ) ≤ Cm2 l 2β −n ,



with l β +1 bˆkε ∂i φ (ε) being bounded by the observation that 



|l β +1 ∇x ϕ(y)|q dy ≤ C.

V

The inequality above is due to the second condition for ϕ in (24) and β  + 1 ≤ β + qn . Since σ also satisfies y σ = 0, the standard elliptic estimate implies that    ∂     ∂y σ  j



L∞ (U )

≤ C||σ ||L2 (V ) ≤ C|m|l β − 2 . n

Applying Proposition 1, we have |w(y)| ≤ C|m|. So, we get |ϕ(ly)| ˘ =

|w(y)| C|m| ≤ β .  lβ l

(91)

Going back to (90), we have for sufficiently large |x|, and for i = 1, . . . , n, |∇ φ˜ (ε) (x)| ≤

C|m|  . |x|β

(92)

It means (86) holds. Now we can follow the standard argument to lift the regularity to show that ∇ φ˜ (ε) is uniformly Hölder continuous (see [24]), i.e., (85) holds. 5.2. Uniqueness of solutions of Problem 5 In this subsection, we will show the uniqueness of the modified flow such that we can remove the cut-off and apply the Bers skill to show the existence of solutions of Problem 4. Lemma 18. Problem 5 exists at most one weak solution φ up to a constant satisfying ∇φ ∈ L2 (D).

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31

Proof. Existence is proved by Lemma 15, Theorem 16, and Lemma 17. ∇φ ∈ L2 (D) follows from (74). So we only need to consider the uniqueness. 1 (D)(j = 1, 2) and Step 1. Assume there are two different solutions of Problem 5 φj ∈ Hloc ∇φj L2 (D) ≤ ∞. Let φ = φ1 − φ2 . Then ∇φL2 (D) < ∞, and there hold

  n

a˜ ij ∂j φ∂i vdx = 0,

∀v ∈ C0∞ (Rn ),

(93)

D i,j =1

  n

S0

a˜ ij ∂j φli ds = 0,

(94)

i,j =1

where 2 a˜ ij = 1

    2 qˆ ε (|∇φ (ε) |2 , ϕ)∂ φ (ε) ∂ φ (ε) ε i j s s s  ds, ρˆ (ε) |∇φs(ε) |2 , ϕ δij − p˜  (ρˆ (ε) )

with φs(ε) = (2 − s)φ1(ε) + (s − 1)φ2(ε) . Moreover, there exists a positive constant λ, such that for any vector ξ ∈ Rn λ˜ 1 |ξ |2 <

n 

a˜ ij ξi ξj < λ˜ 2 |ξ |2 .

(95)

i,j =1 n 1 (D) solution must belong to H 2 (D), by (93) we see Step 2. Since any Hloc i,j =1 ∂i loc n ij ij (a˜ ∂j φ) = 0 holds a.e. in D, as well as the boundary condition i,j =1 a˜ ∂j φ · ni = 0 a.e. + be any finite part of the nozzle D bounded by wall W, the left cross-section S− on W. Let D− and the right cross-section S+ , S− ∩ S+ = ∅, with l the corresponding unit normal vector of S± + + pointing out of D− . Then integrating the equation in D− , using integration by parts and boundary condition on W, one has

0=

  n

S−

a˜ ∂j φ · li ds +

i,j =1

ij

  n

S+

a˜ ij ∂j φ · li ds

(96)

i,j =1

Particularly, if we choose S− to be S0 , then by (94), we conclude that (94) itself holds for any cross-section S0 of D. ˘ into (93), where η from (55) and φ˘ from Step 3. By Lemma 14, substituting v = η2 (φ − φ) (54), it comes to λ˜ 1

 |η∇φ|2 dx D

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32

≤

=

  n D i,j =1   n

η2 a˜ ij ∂j φ∂i φdx

˘ − η a˜ ∂j φ∂i φdx 2 ij

D i,j =1

˘ 2ηa˜ ij ∂j φ∂i η(φ − φ)dx

D i,j =1



2(ln 2)2 ≤ (φ)ξ R ± R

  n

n 

R/2,R

C±

  R a˜ ∂j φ∂i r± − dx 2 ij

i,j =1

      n     1 ij  a˜ η φ − φ˘ ∂j φ∂i r± dx  . +2    R i,j =1 r±  ξ±

(97)

For the first term on the right-hand side, using (94) to the cross-sections S = r± ± , it is

(φ)ξ R ±

R

2(ln 2)2 R

R/2

⎛ ⎜ ⎝



r± ±

⎞ n 

⎟ a˜ ij ∂j φ∂i (r± )ds ⎠ dr± = 0.

(98)

i,j =1

For the second term of (97), note |∂i r± | ≤ 1 and r± > R in ξ±R , using Cauchy-Schwarz inequality for the positive-definite quadratic form (a˜ ij ), it is bounded by       n     1 ij   a˜ η φ − φ˘ ∂j φ∂i r± dx     R i,j =1 r±  ξ± ⎛

⎞1 ⎛

1⎜ ≤C ⎜ R⎝

⎟ ⎜ ⎜ |η∇φ|2 dx ⎟ ⎠ ⎝



2

R ξ±

⎛



1 ≤C ⎝ R



⎞1

2

⎟ ˘ 2 dx ⎟ |φ − φ| ⎠

R ξ±

⎞1

2

⎛

⎜ |η∇φ|2 dx ⎠ ⎜ ⎝

D



⎞1

2

⎟ |φ − (φ)ξ R |2 dx ⎟ ⎠ . ±

R ξ±

Therefore we obtain an energy growing inequality    1  |∇φ|2 dx ≤ C 2 |φ − (φ)ξ R |2 dx ≤ C |∇φ|2 dx. ± R +,− +,− DR

R ξ±

(99)

R ξ±

 Step 5. Since D |∇φ|2 dx < ∞, from the above inequality, letting R → ∞, we infer |∇φ| = 0 in D. So φ1 differs from φ2 only by a constant.

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33

5.3. Proof of Theorem 5 In this subsection, we will conclude the proof of Theorem 5 by showing the solutions of Problem 5 are solutions of Problem 4, and then consider the convergence rate of the low Mach number limit. Proof. Up to now, we have shown that for any given fixed cut-off parameters θ and ε0 , there exists a unique solution of Problem 5, which is denoted as φ (ε) (x; ε0 , θ ). ε It is noticeable that for a given θ ∈ (0, 1), if |∇φ (ε) (x; ε0 , θ )| < q˚θ 0 , then φ (ε) (x; ε0 , θ ) is the unique solution of Problem 5. Note that ¯ ¯ + C(ε0 , θ)ε 2 . |∇φ (ε) (x; ε0 , θ)| = |ε 2 ∇ φ˜ (ε) (x; ε0 , θ) + ∇ φ(x)| ≤ max |∇ φ|

(100)

Then, there exists ε0,θ ≤ ε0 such that |∇φ (ε) (x; ε0 , θ )| < q˚θε0 , for any 0 < ε < ε0,θ . From the definition and uniqueness of φ (ε) , {ε0,θ } is a non-decreasing sequence respect to θ with upper bound ε0 . Then, we introduce ε0,cr = lim0<θ<1 ε0,θ such that for 0 < ε < ε0,cr , there exists a ˜ unique solution φ(x; ε0 ), such that ε0 ¯ |∇φ (ε) (x; ε0 )| = |ε 2 ∇ φ˜ (ε) (x; ε0 ) + ∇ φ(x)| < q˚cr ,

(101)

which means M ε < 1. In this case, the cut-off can be removed such that the solution is a solution of Problem 4. After removing the subsonic cut-off, we will optimize the critical value εcr . For each 0 < ε0 < 1, there exists an ε0,cr , with 0 < ε0,cr ≤ ε0 < 1. Then, the critical value εc = sup0<ε0 <1 ε0,cr satisfies that for any ε ∈ (0, εc ), 0 < M ε < 1 and |∇ ϕ˜ (ε) | is uniform bounded with respect to ε. Finally, let us consider the convergence rate of the low Mach number limit. Note that φ (ε) = φ¯ + ε 2 φ˜ (ε) , so we have u(ε) = u¯ + ε 2 u˜ (ε) .

(102)

Therefore, for the density, by (23) and the straightforward computation, we have ρ (ε) = 1 + O(ε 2 ).

(103)

Finally, for the pressure, we have that pε = = =

˜ p(ρ ˜ ε ) − p(1) ε2  ˜ p˜ ◦ h˜ −1 h(1) −  ˜ − p˜ ◦ h˜ −1 h(1)

Here we take function

ε2 (ε) 2 2 |∇φ | ε2 ¯2 ε 2 |∇ φ| 2

 + ε 2 ϕ − p(1) ˜

− ε 4 ∇ φ˜ (ε) · ∇ φ¯ − ε2

ε 6 |∇ φ˜ (ε) |2 2

 + ε 2 ϕ − p(1) ˜ .

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      ¯2 2 ε 2 |∇ φ˜ (ε) |2 4 2 |∇ φ| −1 ˜ (ε) ˜ ˜ ¯ f (t) := p˜ ◦ h ε t − ∇φ · ∇φ + ε t , h(1) + ϕ − 2 2 which implies p ε =

f (1)−f (0) . ε2

Noticing p¯ = ϕ −

¯2 |∇ φ| 2

=

f  (0) , ε2

one could get

p ε − p¯ ε2 f (1) − f (0) − f  (0) = ε4     2  1  t h˜ −1 h(1) ˜ ¯ 2 − 2ϕ − (1 − t)2 ε 4 ∇ φ˜ (ε) · ∇ φ¯ + ε6 |∇ φ˜ (ε) |2 |∇ φ| − (1−t)ε 2 2    =  (1−t)ε 2  ε6  −1 2 2 4 (ε) ˜ ˜ ¯ ˜ ¯ ˜ (ε) |2 h(1) − |∇ φ| p ˜ ◦ h − 2ϕ − (1 − t) ∇ φ · ∇ φ + |∇ φ ε 2 2 0 

2  1 ¯ 2 − 2ϕ − 2(1 − t)ε2 ∇ φ˜ (ε) · ∇ φ¯ − (1 − t)ε4 |∇ φ˜ (ε) |2 |∇ φ| 2     2  6 ε ε −1 2 2 4 (ε) (ε) 2 ˜ ¯ − 2ϕ − (1 − t) ε ∇ φ˜ · ∇ φ¯ + |∇ φ˜ | +t h˜ |∇ φ| h(1) − (1 − t) 2 2   −2∇ φ˜ (ε) · ∇ φ¯ − ε 2 |∇ φ˜ (ε) |2 dt, −

which is uniformly bounded respect to ε. Here we employ the classical calculus identity f (1) − 1 f (0) − f  (0) = 0 tf  (1 − t)dt. Then, we can conclude that p (ε) = p¯ + O(ε 2 ). It completes the proof of Theorem 5. 6. Compressible flow In this section, we will study the existence, regularity, and uniqueness of Problem 2 for ε = 1, and the subsonic-sonic limit solution. Here by, p satisfies (2) and the Bernoulli law follows (16) as ε = 1. 6.1. Compressible potential flow Now let us consider the compressible case. By (13)3 , we can introduce the velocity potential φc for the compressible case that ∇φc = u.

(104)

Therefore, Problem 2 can be reformulated into the following problem. Problem 7. Find function φc such that     ⎧ 2 ⎪ ⎨ div ρ |∇φc | , ϕ ∇φc = 0, x ∈ D, ∂φc x ∈ W, ∂n =0, ⎪  ∂φc ⎩ 2 S0 ρ |∇φc | , ϕ ∂l ds = m.

(105)

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By the straightforward computation, (105)1 can be rewritten as n 

aij ∂ij φc +

n 

ij =1

bi ∂i φc = 0,

(106)

i=1

where 

 ∂i φc ∂j φc aij = ρ δij − , p  (ρ)

and

bi =

ρ∂i ϕ . p  (ρ)

(107)

For M < 1, we have that n 

0 < λ1 |ξ |2 ≤

aij ξi ξj ≤ λ2 |ξ |2 .

(108)

ij =1

It is easy to see that |u| < qcr holds if and only if the flow is subsonic, i.e., M < 1. Similarly, for each θ ∈ (0, 1), there exists qθ such that |u| ≤ qθ holds if and only if M ≤ θ . Since equation (105) is strictly elliptic if and only if M < 1. We do not know whether equation (105) is elliptic or not before solving it. Therefore we need to introduce the cut-off to truncate the coefficients of equation (105). For 0 < θ < 1, we introduce qθ (ϕ), and cut-off function on the phase plane ⎧ 2 ⎨ q − 2ϕ qˆ q 2 , ϕ = monotone  smooth function  ⎩ supx∈D (qθ  )2 (ϕ) − 2ϕ (x) 



if |q| ≤ qθ (ϕ), if qθ (ϕ) ≤ |q| ≤ qθ  (ϕ), if |q| ≥ qθ  (ϕ),

(109)

where θ  := 1+θ 2 . Let ρˆ satisfy q(q ˆ 2 , ϕ) + h(ρ) ˆ = h(1), 2

(110)

  q(q ˆ 2 , ϕ) ρˆ = ρ(q ˆ 2 , ϕ) = h−1 h(1) − . 2

(111)

which is equivalent to that

∂ ∂ We denote ρˆ := ∂ ρ(, ˆ ϕ), and ρˆϕ := ∂ϕ ρ(, ˆ ϕ). Then, Problem 7 is reformulated into Problem 8 as follows.

Problem 8. Find function φc which satisfies ⎧   2 ⎪ ˆ c | , ϕ)∇φc = 0, ⎨ div ρ(|∇φ ∂φc ∂n = 0,  ⎪ ∂φc 2 ⎩ ˆ c | , ϕ) ∂l ds = m. S0 ρ(|∇φ

x ∈ D, x ∈ W,

(112)

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Straightforward calculation yields that (112)1 can be rewritten as n 

aˆ ij (∇φc , ϕ)∂ij φc +

i,j =1

n 

bˆi (∇φc , ϕ)∂i φc = 0,

i=1

where 

aˆ ij (∇φc , ϕ) = ρˆ |∇φc | , ϕ 2





qˆ (|∇φc |2 , ϕ)∂i φc ∂j φc δij − p  (ρ) ˆ

 ,

(113)

and ρˆ qˆϕ (|∇φc |2 , ϕ)∂i ϕ bˆi (∇φc , ϕ) = . p  (ρ) ˆ

(114)

Moreover, n 

λˆ 1 |ξ |2 ≤

aˆ ij (∇φc , ϕ) ξi ξj ≤ λˆ 2 |ξ |2 ,

|bˆi ∂i φc | ≤ C|∂i ϕ|,

(115)

i,j =1

where constants C, λˆ 1 , and λˆ 2 depend only on the subsonic truncation parameters θ , and are independent of solution φc . Finally, we remark that a solution of Problem 8, where the density ρˆ is derived from the new density-speed relation (110), is also a solution of the original potential flow equation in the Problem 7 when |∇φc | < qθ . To solve Problem 8, we employ the similar idea from soliving Problem 5 to introduce a variational formulation.  ˜ G(, ϕ) = 1  ρ(λ, Let φ = φ¯ + φ, ϕ)dλ and 2 0 ˆ   I φ, φ¯ =

        ¯ 2 , ϕ − ∇ φ¯ · ∇φ − ∇ φ¯ dx. G |∇φ|2 , ϕ − G |∇ φ|

(116)

D

Then Problem 8 becomes: Problem 9. Find a minimizer φ˜ c ∈ V such that ¯ = min I (φ¯ + φ, ˜ φ). ¯ I (φ¯ + φ˜ c , φ) ˜ V φ∈

Here, φ = φ¯ + φ˜ and φc = φ¯ + φ˜ c . Lemma 19. The minimizer of Problem 9 satisfies (112).

(117)

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6.2. Unique existence of the minimizer of Problem 9 First, for the existence of a minimizer of Problem 9, we have the following theorem. Theorem 20. Problem 9 admits a unique minimizer φ˜ c ∈ V, which satisfies that 

|∇ φ˜ c |2 dx ≤ Cm2 ,

(118)

D

where constant C depends only on φ¯ and D.   2G |∇ φ| ¯ 2, Proof. The proof is similar to Theorem 16, which is different from the estimate of    ϕ −1 . Note that      ¯ 2 , ϕ − 1 2G |∇ φ|      ¯ 2 , ϕ − 1 = ρˆ |∇ φ|     ¯ 2 , ϕ)   q(|∇ ˆ φ| − 1 = h−1 h(1) − 2   1     ¯ 2 , ϕ) ¯ 2 , ϕ)   −1    q(|∇ q(|∇ ˆ φ| ˆ φ| dt  h h(1) − t =  2 2   0

  1      2 , ϕ)   ¯  q(|∇ ˆ φ| −1  dt  . h h(1) − t ≤C 2   0

By Lemma 18, we know that the minimizer of Problem 9 is a weak solution of Problem 8. In this section, we will first show the minimizer of Problem 9 is a solution of Problem 7 by removing the elliptic cut-off introduced in Problem 8, and then show the proof of Theorem 6. 6.3. C 1,α -Regularity of the minimizer Before removing the elliptic cut-off, let us consider the regularity of the derivatives of the solutions. Based on Proposition 1, we can show the C α -regularity of ∇ φ˜ c . Lemma 21. The minimizer φ˜ c of Problem 9 satisfies that ∇ φ˜ c C α (D) ≤ C|m|,

(119)

C|m| , (1 + |x|)β

(120)

and that |∇ φ˜ c | ≤

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where β  = min{ n2 , β +

n q

− 1}.

˜ φ¯  = ∂k φ¯ for k = 1, . . . , n. Then by the straightforward calculation, φˇ satProof. Let φˇ = ∂k φ, isfies n 

n n       ∂i aˆ ij ∂j φˇ + ∂i aˆ ij ∂j φ¯  + ∂i (bˆk ∂i φ) = 0.

i,j =1

i,j =1

i=1

Since φ¯ = 0 and φ¯  = 0, we can change the equation above to: n  i,j =1

n n       ∂i aˆ ij ∂j φˇ = ∂i f¯ij ∂j φ¯  − ∂i (bˆk ∂i φ), i,j =1

i=1

where f¯ij := aˆ ij − δij   ρˆ qˆ (|∇φ|2 , ϕ)∂i φ∂j φ = ρˆ − 1 δij − . p  (ρ) ˆ

(121)

Therefore, due to the cut-off, we have the uniform L∞ estimate of f¯ij . Also, for i, k = 1, . . . , n, |bˆk ∂i φ| ≤ C|∂k ϕ|.

(122)

By the assumption that ∇ϕ ∈ Lq , we know bˆk ∂i φ are bounded in Lq for q > n. Next we can refer to the method of Theorem 17. 6.4. Uniqueness of solutions of Problem 8 In this subsection, we will show the uniqueness of the modified flow such that we can remove the cut-off and apply the Bers skill to show the existence of solutions of Problem 7. Lemma 22. Problem 8 exists at most one weak solution φc up to a constant satisfying ∇φc ∈ L2 (D). 6.5. Proof of Theorem 6 In this subsection, we will conclude the proof of Theorem 6 by showing the solutions of Problem 8 are solutions of Problem 7. Proof. Up to now, we have shown that for any given fixed cut-off parameters θ , there exists a unique solution of Problem 8, which is denoted as φ(x; m, θ ). It is noticeable that for a given θ ∈ (0, 1), if |∇φ(x; m, θ )| < qθ , then φ(x; m, θ ) is the unique solution of Problem 7. Let {θi }∞ i=1 be a strictly increasing sequence of positive numbers, such that θi → 1 as i → ∞. For fixed i, due to (120), there exists a maximum interval [0, mi ) such that, for m ∈ [0, mi ),

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|∇φ(x; m, θi )| ≤ qθi . So ∇φ(x; m, θi ) is a solution of Problem 7 for any m ∈ [0, mi ). From the uniqueness of Problem 8, we can see mi ≤ mj for i < j . So {mi }∞ i=1 is an non-decreasing sequence with the upper bounded mmax := |S0max | max(ρcr (ϕ)qcr (ϕ)(x)), x∈D

which implies the convergence of the sequence. As a consequence, we can have m ¯ := lim mi . i→∞

For any m ∈ [0, m), ¯ there exists an index i such that m < mi . The solution of Problem 7 is ∇φ(x; m, θi ), which could be written as ∇φ(x, m). It also implies the solution of Problem 2. Next, we will prove the subsonic-sonic limit case. The strong solutions (ρ  , u , p  ) satisfy Problem 2 as m = m . Hence, Theorem 2.2 in [7] immediately implies the strong convergence of u in D. As a consequence, the density function ρ  (x), which defined by (16), is convergence to ρ(x), ˇ which also apply to the pressure p  (x). This completes the proof of Theorem 6. Acknowledgments The authors would like to thank the referees for helpful suggestions. References [1] T. Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differ. Equ. 10 (1) (2005) 19–44. [2] L. Bers, An existence theorem in two-dimensional gas dynamics, Proc. Symp. Appl. Math. 1 (1949) 41–46. [3] L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Commun. Pure Appl. Math. 7 (1954) 441–504. [4] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surveys in Applied Mathematics, vol. 3, 1958. [5] G.-Q. Chen, C. Christoforou, Y. Zhang, Continuous dependence of entropy solutions to the Euler equations on the adiabatic exponent and Mach number, Arch. Ration. Mech. Anal. 189 (1) (2008) 97–130. [6] G.-Q. Chen, M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Am. Math. Soc. 16 (3) (2003) 461–494. [7] G.-Q. Chen, F.-M. Huang, T.-Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal. 219 (2) (2016) 719–740. [8] G.-Q. Chen, F.-M. Huang, T.-Y. Wang, W. Xiang, Incompressible limit of solutions of multidimensional steady compressible Euler equations, Z. Angew. Math. Phys. (2016) 67–75. [9] G.-Q. Chen, F.-M. Huang, T.-Y. Wang, W. Xiang, Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles, Adv. Math. 346 (2019) 946–1008. [10] J. Cheng, L. Du, Y. Wang, Two-dimensional impinging jets in hydrodynamic rotational flows, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34 (6) (2016) 1355–1386. [11] J. Cheng, L. Du, Y. Wang, On incompressible oblique impinging jet flows, J. Differ. Equ. 265 (10) (2018) 4687–4748. [12] J. Cheng, L. Du, Y. Wang, The existence of steady compressible subsonic impinging jet flows, Arch. Ration. Mech. Anal. 229 (3) (2018) 953–1014.

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