An improved regularity result of semi-hyperbolic patch problems for the 2-D isentropic Euler equations

J. Math. Anal. Appl. 467 (2018) 1174–1193

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An improved regularity result of semi-hyperbolic patch problems for the 2-D isentropic Euler equations Yanbo Hu a,∗ , Tong Li b a b

Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, PR China Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

a r t i c l e

i n f o

Article history: Received 21 March 2018 Available online 30 July 2018 Submitted by H. Liu Keywords: Compressible Euler equations Riemann problem Semi-hyperbolic patch Sonic curve Characteristic decomposition Bootstrap

a b s t r a c t This paper investigates the regularity of a semi-hyperbolic patch problem arising from the Riemann problem for the 2-D isentropic Euler equations. We show that 1 1 the solution is uniformly C 1, 6 up to the sonic curve and the sonic curve is C 1, 6 , which improve the C 1 -regularity of Song et al. [20]. We introduce a novel set of change variables which allow us to establish higher regularity results based on the ideas of characteristic decomposition and the bootstrap method. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The two-dimensional isentropic compressible Euler equations reads that ⎧ ⎪ ⎨ ρt + (ρu)x + (ρv)y = 0, (ρu)t + (ρu2 + p)x + (ρuv)y = 0, ⎪ ⎩ (ρv) + (ρuv) + (ρv 2 + p) = 0, t x y

(1.1)

where ρ is the density, (u, v) is the velocity and p is the pressure given by the polytropic gas equation p(ρ) = Aργ , A > 0 is a constant can be scaled to be one, γ > 1 is the adiabatic gas constant. In this paper, we consider the semi-hyperbolic patch problems arising from the two-dimensional Riemann problem for (1.1). The study of two-dimensional Riemann problem to the Euler equations was initiated by Zhang and Zheng [25]. Based on the generalized characteristic analysis method and numerical experiments, they provided a set of conjectures on the configuration of solutions. However, the construction of a global solution in rigorous theory is considerably more complex due to the existence of transonic and small-scale * Corresponding author. E-mail addresses: [email protected] (Y. Hu), [email protected] (T. Li). https://doi.org/10.1016/j.jmaa.2018.07.064 0022-247X/© 2018 Elsevier Inc. All rights reserved.

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structures [12,29]. The semi-hyperbolic patches are one of such kind of structures, which appear frequently in the two-dimensional Riemann problem for the Euler equations and the related models, see [5,12,29] for more details. A semi-hyperbolic patch is a region in which a family of characteristics starts on a sonic curve and ends on either a sonic curve or a transonic shock wave [21]. This type of regions may also occur in the transonic flow over an airfoil [2,3], in rarefaction wave reflection along a compressive corner [17], and in Guderley shock reflection of the von Neumann triple point paradox [22,23]. The study of the internal structure of semi-hyperbolic patches of solutions locally is quite meaningful and is an essential step for constructing global solutions of mixed-type equations in future work. The investigation of the semi-hyperbolic patches of solutions in theoretical was started by Song and Zheng [21] for the pressure-gradient system. The semi-hyperbolic patch problems for the isentropic and isothermal Euler equations were studied, respectively, by Li and Zheng [16] and Hu et al. [7]. In [19], Song provided a different viewpoint to understand this kind of problems for the pressure-gradient system. The regularity of the semi-hyperbolic patch problems for the pressure-gradient and isentropic Euler systems were presented in [20,24]. We also refer the reader to [6,26–28] for the construction of classical sonic–supersonic solutions to the Euler system. The above works [6,7,16,19–21,24,26–28] on building smooth solutions are based on the idea of characteristic decomposition which is a powerful tool revealed in [4], see, e.g., [1,9–11,13–15,18] for more applications. The purpose of this paper is to develop an improved regularity result of semi-hyperbolic patch problems for the two-dimensional isentropic Euler equations (1.1). We show that the global solution is uniformly 1 1 C 1, 6 up to the degenerate sonic boundary and the sonic curve is C 1, 6 -continuous. In [20], Song, Wang and Zheng verified that the semi-hyperbolic patch problem of (1.1) admits a global solution up to the sonic boundary and that the sonic boundary has C 1 -regularity. We point out that the work [20] relies heavily on the known relation from [16] that ∂¯+ c + ∂¯− c = 0 on the sonic curve. In the current paper, we establish our regularity results without using such a relation on the sonic curve by introducing a novel set of change variables. In particular, these change variables allow us to establish higher regularity of the solution and of the sonic boundary. The approach in this paper originates from our recent work [8], where the regularity of the isothermal Euler system was studied. The rest of the paper is organized as follows. Section 2 is devoted to delivering some preliminaries, including the variables of inclination angles and their characteristic decompositions, describing our problem and the main results in this paper. We establish the uniform boundedness of (∂¯+ c, ∂¯− c) in Section 3. In Section 4, we introduce a partial hodograph transformation and show the regularity of solutions in the partial hodograph coordinates. In the final Section 5, we complete the proof of our main theorem. 2. Preliminaries and the main results 2.1. Characteristic decompositions in angle variables Note that system (1.1) admits self-similar solutions. In terms of self-similar variables (ξ, η) = (x/t, y/t), (1.1) changes to ⎧ U ρξ + V ρη + ρ(uξ + vη ) = 0, ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎨ U u + V u + c2 = 0, ξ

η

γ−1

ξ ⎪   ⎪ ⎪ ⎪ 2 ⎪ c ⎪ = 0, ⎩ U vξ + V vη + γ−1

(2.1)

η



where (U, V ) = (u − ξ, v − η) is the pseudo-velocity and c = p (ρ) is the sound speed. Assuming the flow is irrotational, i.e. uη = vξ , we obtain a system in terms of (u, v)

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(c2 − U 2 )uξ − U V (uη + vξ ) + (c2 − V 2 )vη = 0, uη − vξ = 0,

(2.2)

together with the pseudo-Bernoulli law U2 + V 2 c2 + = −φ, γ−1 2

φξ = U,

φη = V,

(2.3)

where φ is the pseudo-velocity potential. The positive/negative characteristics of (2.2) are defined by the integral curves of the following equations √ U V ± c U 2 + V 2 − c2 dη = Γ : =: Λ± , dξ U 2 − c2 ±

(2.4)

where Λ± are the positive/negative eigenvalues of (2.2). It is clear that system (2.2) is of mixed-type: supersonic for U 2 + V 2 > c2 , subsonic for U 2 + V 2 < c2 . We call a point (ξ, η) is the sonic point if it satisfies U 2 (ξ, η) + V 2 (ξ, η) = c2 (ξ, η).

(2.5)

The sonic curve is defined as the set {(ξ, η) : U 2 + V 2 = c2 }. By a standard calculation, system (2.2) can be written to the characteristic form ∂ + u + Λ− ∂ + v = 0, ∂ − u + Λ+ ∂ − v = 0,

∂ ± = ∂ξ + Λ± ∂η .

(2.6)

In [14], Li and Zheng invented the inclination angle variables of characteristics as the dependent variables, which was used successfully in many problems, see, e.g. [1,6,8,10,15,16,18,26,28]. As in [14,16], we introduce the inclination angle variables of characteristics tan α = Λ+ ,

tan β = Λ− ,

(2.7)

and denote σ=

α+β , 2

δ=

α−β , 2

(2.8)

where σ is the pseudo-streamline angle and δ is the pseudo-Mach angle. Then one has u=ξ−c

cos σ , sin δ

v =η−c

sin σ , sin δ

(2.9)

and the sonic curve is now {(ξ, η) : sin δ(ξ, η) = 1}. Moreover, we introduce scaled characteristic fields ∂¯+ = cos α∂ξ + sin α∂η ,

∂¯− = cos β∂ξ + sin β∂η ,

(2.10)

then ∂ξ =

cos σ sin δ(∂¯+ + ∂¯− ) − sin σ cos δ(∂¯+ − ∂¯− ) , sin(2δ)

∂η =

sin σ sin δ(∂¯+ + ∂¯− ) + cos σ cos δ(∂¯+ − ∂¯− ) . sin(2δ)

It follows from (2.6), (2.9) and (2.10) that

(2.11)

Y. Hu, T. Li / J. Math. Anal. Appl. 467 (2018) 1174–1193



∂¯+ σ +

cos2 δ ¯+ κ+1−cos2 δ ∂ δ

=

sin2 δ c

∂¯− σ −

cos2 δ ¯− κ+1−cos2 δ ∂ δ

= − sinc

2

·

κ+1−2 cos2 δ κ+1−cos2 δ ,

δ

·

κ+1−2 cos2 δ κ+1−cos2 δ ,

1177

(2.12)

where κ = (γ − 1)/2. For later calculations, we list the following equations sin2 δ sin δ(κ + sin2 δ) R− , ∂¯+ δ = κ cos δ c

sin2 δ sin δ(κ + sin2 δ) ∂¯− δ = S− κ cos δ c

(2.13)

and the characteristic decomposition for c

 ⎧ sin(2δ) (κ+1)(R+S) κ+2 sin2 δ − ⎪ ¯ ⎪ S , ⎨ ∂ R = R − c + 2κ cos2 δ − κ

 ⎪ (κ+1)(R+S) ⎪ κ+2 sin2 δ ⎩ ∂¯+ S = S − sin(2δ) + 2κ cos2 δ − R , c κ

(2.14)

where R = ∂¯+ c/c and S = ∂¯− c/c. The detailed derivation of equations (2.12)–(2.14) can be found in [16]. 2.2. Description of our problem and the main results Now we give a description of the semi-hyperbolic patch problem of (1.1). Let R14 (η) be a planar rarefaction wave of system (1.1) in the self-similar coordinates (ξ, η), connecting two given constant states T T 1 4 ((ρ 1 , 0, v1 ) ) and ((ρ 4 , 0, 0) ) (ρ1 > ρ4 > 0, v1 > 0), defined by

R14

⎧ ⎪ η = v + c(ρ), (η4 ≤ η ≤ η1 ) ⎪ ⎨ ρ c(s) : v = ρ s ds, (0 < ρ4 ≤ ρ ≤ ρ1 ) 4 ⎪ ⎪ ⎩u = u = u = 0 1

(2.15)

4

in the domain ξ > 0, where ηi = vi + ci , i = 1, 4. From (2.15) and the polytropic gas equation of state p = Aργ , we have the expression of R14 c=

κη + c4 , κ+1

u = 0,

v=

η − c4 , η ∈ [η4 , η1 ]. κ+1

Denote the point (ξ, η) = (0, v1 +c1 ) by A, which is a sonic point by (2.5). We use Γ+ ¯(ξ) to represent A : η =η the positive characteristic curve passing though A in R14 . Thanks to (2.4) and (2.15), the function η = η¯(ξ) is defined by 2(κ + 1)(κη + c4 )ξ d¯ η = , dξ (κη + c4 )2 − (κ + 1)2 ξ 2

η¯|ξ=0 = v1 + c1 .

We set ξB = η¯−1 (η4 ) and denote the point (ξ, η) = (ξB , c4 ) by B which is the intersection point of Γ+ A with

the bottom boundary of R14 , see Fig. 1. Then the boundary data (ρ, u, v) = (¯ ρ, u ¯, v¯)(ξ) on BA are

: ρ¯(ξ) = BA



κ¯ η (ξ) + c4 √ Aγ(κ + 1)

 κ1 ,

u ¯(ξ) = 0,

v¯(ξ) =

η¯(ξ) − c4 , ξ ∈ [0, ξB ]. κ+1

(2.16)

: η = ϕ(ξ) (ξC ≤ ξ ≤ ξB ) be a smooth curve. We assign the Let ξC ∈ (0, ξB ] be a real number and BC

boundary data for (ρ, u, v) on BC, (ρ, u, v)(ξ, ϕ(ξ)) = (ˆ ρ, u ˆ, vˆ)(ξ) such that

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Fig. 1. The semi-hyperbolic patch.

ϕ(ξ) ∈ C 3 , ϕ (ξ) > 0, (ˆ ρ, u ˆ, vˆ)(ξ) ∈ C 2 , (ˆ ρ, u ˆ, vˆ)(ξB ) = (ρ4 , 0, 0),   2 2 2

: ϕ (ξ) = (ˆu−ξ)(ˆv−ϕ)−ˆc (ˆu2−ξ)2 +(ˆv−ϕ) −ˆc (ξ), [(ˆ u − ξ)2 + (ˆ v − ϕ)2 ]ξ = cˆ2 |ξC , BC (ˆ u−ξ) −ˆ c C u ˆ (ξ) +

(2.17)



(ˆ u−ξ)(ˆ v −ϕ)+ˆ c (ˆ u−ξ)2 +(ˆ v −ϕ)2 −ˆ c2 (ξ)ˆ v  (ξ) (ˆ u−ξ)2 −ˆ c2 2

=0

 for any ξ ∈ [ξC , ξB ], where cˆ = p (ˆ ρ). The last condition of (2.17) is the compatibility condition of

boundary data on BC with the equations (2.6). Moreover, the boundary conditions (2.17) imply that the

is a smooth convex negative characteristic curve and the point C is a sonic point. The smooth curve BC semi-hyperbolic patch problem is described as follows: assuming the boundary conditions (2.16) and (2.17), we look for a smooth solution in curvilinear triangle ABC and a simple wave solution in curvilinear triangle

is sonic curve while CD

is an envelope, see Fig. 1. CBD, where AC The existence of smooth solutions to the semi-hyperbolic patch problem in the region ABC was verified by Li and Zheng [16]. In [20], Song, Wang and Zheng shown that there exists a global solution in the region

and the sonic curve AC

is C 1 -continuous. In this paper, we improve ABC up to the sonic boundary AC the regularity results of [20]. The conclusion of the paper can be stated as follows. Theorem 1. Assume that arctan ϕ (ξC ) ∈ (− π2 , 0). The degenerate Goursat boundary value problem (2.1)

(2.16)–(2.17) admits a global smooth solution (ρ, u, v) ∈ C 2 in the region ABC with the sonic boundary AC. 1, 16

Moreover, the solution (ρ, u, v)(ξ, η) is uniformly C up to the sonic boundary AC and the sonic curve

is C 1, 16 -continuous. AC The major difficulty to establish the uniform regularity of solutions arises from the hyperbolic degeneracy of the Euler system at the sonic curves. As pointed out in [20], the key is to prove that R + S approach to zero on the sonic curve at least with a rate of cos δ. The previous work [20] relies on the known relation from [16] that R + S = 0 on the sonic curve. We do not use the results of [20] since we plan to establish the regularity theory without using such a relation. In order to acquire our results, we introduce a partial  H),  which play a crucial role in our analysis, see hodograph transformation and a novel change variables (G, (4.18). Equipped with this change variables and the bootstrap method, we not only show that the function 1 (R + S)/ cos δ is uniformly bounded as obtained in [20], but also show that it is uniformly C 3 -continuous up to the sonic boundary in the partial hodograph plane, see Lemma 4.4. Finally, after returning to the self-similar variables, we establish the uniform regularity of smooth solutions of system (2.1) up to the sonic

boundary AC.

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3. The uniform boundedness of (R, S) We first present the properties of solutions established in [16]. Proposition 3.1 ([16]). Suppose that the boundary conditions (2.16)–(2.17) hold. The boundary data in terms of angle variables (α, β, c) satisfy ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

π 2

⎧ ⎪ ⎪ ⎨

≤ α |BA

≤ π,

β |BA

= 0, c |BA

=

κ¯ η (ξ)+c4 κ+1 ,

⎪ ⎪ ⎩

π 2

≤ α |BC

≤ π + βC ,

βC ≤ β |BC

≤ 0,



∂¯+ α |BA

> 0, − ∂¯ β | > 0, BC

c |BC ˆ(ξ),

= c

∂¯+ c |BA

> 0, − ∂¯ c | < 0,

(3.1)

±∂¯± c > 0.

(3.2)

BC

and the smooth solution (α, β, c)(ξ, η) in the region ABC satisfies α≥

π , 2

β ≤ 0,

π π ≤δ≤ , 4 2

±∂¯± α > 0,

±∂¯± β < 0,

Moreover, in the whole region ABC, the function R and S satisfy 0 < R ≤ M0 ,

0 < −S ≤ M0

(3.3)

for some uniform positive constant M0 .

For any point We next show that R and −S have a positive lower bound up to the sonic boundary AC.

(ξ, η) in the region ABC, we draw the positive and negative characteristic curves up to the boundaries BC

at points B2 and B1 , respectively, see Fig. 2. and BA Lemma 3.1. Suppose that conditions (3.1) hold. For any point (ξ, η) in the region ABC, the functions R and S satisfy 0 < me−ˆκd ≤ R(ξ, η) ≤ M0 ,

0 < me−ˆκd ≤ −S(ξ, η) ≤ M0 ,

(3.4)

where d is the diameter of the domain ABC and √ 1 + 2κ · min{min R, min(−S)} > 0, m= √

1

2 2 1+κ BB BB

κ ˆ=

3κ + 4 . 2κc4

Proof. The uniform upper boundedness of R and −S in (3.4) follows from (3.3). To establish the positive lower bound of R and −S, we introduce new variables R = sin δ

 κ + sin2 δR > 0,

S = − sin δ



κ + sin2 δS > 0.

Then by (3.2), it suffices to show R(ξ, η) ≥ me ˆ −ˆκd ,

S(ξ, η) ≥ me ˆ −ˆκd

for any point (ξ, η) in the region ABC, where m ˆ = min{min R, min S} > 0. 1 BB

2 BB

From (2.14), we derive the equations for (R, S) ∂¯− R = RF,

−∂¯+ S = −SF,

(3.5)

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Fig. 2. Case 2.

where F =

(κ + 1)(R − S) (3κ + 4 sin2 δ)  . − 2c(κ + sin2 δ) 2κ sin δ κ + sin2 δ cos2 δ

Here we use the scaled characteristic directions ∂¯− and −∂¯+ since they are both pointing away from the

The proof is divided into two cases. sonic boundary AC. Case 1. For any point P ∈ ABC, if R ≥ S entirely in the region P B1 BB2 , then −∂¯+ ln S ≤

3κ + 4 3κ + 4 sin2 δ ≤ =κ ˆ 2 2c4 κ 2c(κ + sin δ)

in P B1 BB2 .

(3.6)

Here we have used the fact c ≥ c4 by (3.2). We integrate (3.6) from P to B2 to obtain R|P ≥ S|P ≥ S|B2 e−ˆκ(ξB2 −ξP ) ≥ me ˆ −ˆκd . Case 2. If there exists a point, say P1 , in P B1 BB2 such that R < S at P1 , then from the point P1 , we ∗  draw a negative characteristic curve, called Γ− 1 , up to the boundary BB1 at a point P1 , see Fig. 2. If F ≤ 0 − always holds on Γ1 , then we have by (3.5) ∂¯− R ≤ 0, on Γ− 1 , which implies that S|P1 ≥ R|P1 ≥ R|P1∗ ≥ m ˆ ≥ me ˆ −ˆκd . If F > 0 for some point on Γ− 1 , then there exists a neighborhood N1 of P1 so that F < 0 holds for every − point in N1 ∩ Γ1 and F = 0 at Q1 := ∂N1 ∩ Γ− 1 . We see that ∂¯− R ≤ 0

on N1 ∩ Γ− 1,

from which one has S|P1 ≥ R|P1 ≥ R|Q1 > S|Q1 .

(3.7)

The last inequality holds by F |Q1 = 0. We now draw a positive characteristic curve from the point Q1, ∗  called Γ+ 1 , up to the boundary BB2 at a point Q1 . We further discuss the case in the following two subcases.

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Subcase 2a. If R ≥ S always holds on Γ+ 1 , then from the equation of S in (3.5) we have −∂¯+ ln S ≤

3κ + 4 sin2 δ ≤κ ˆ on Γ+ 1. 2c(κ + sin2 δ)

Integrating the above from Q1 to Q∗1 gives S Q1 ≥ S Q∗1 e−ˆκ(ξQ1 −ξQ1 ) ≥ me ˆ −ˆκd , ∗

which together with (3.7) yields S|P1 ≥ R|P1 ≥ me ˆ −ˆκd . Subcase 2b. If, R < S holds for some point on Γ+ 1 , then there exists a neighborhood N2 of Q1 so that R > S + holds for every point in N2 ∩ Γ1 and R = S at P2 := ∂N2 ∩ Γ+ 1 . Then we integrate the equation for S from Q1 to P2 and use (3.7) to get S|P1 ≥ R|P1 ≥ S Q1 ≥ S P2 e−ˆκ(ξP2 −ξQ1 ) = RP2 e−ˆκ(ξP2 −ξQ1 ) .

(3.8)

∗  Now, from P2 , we draw a negative characteristic curve, called Γ− 2 , up to the boundary BB1 at a point P2 . Note that F < 0 at P2 . If F ≤ 0 holds on the whole segment Γ− 2 , then

RP2 ≥ RP2∗ ≥ m, ˆ which along with (3.8) gets S|P1 ≥ R|P1 ≥ me ˆ −ˆκ(ξP2 −ξQ1 ) ≥ me ˆ −ˆκd Otherwise, we have RP2 ≥ RQ2 ,

(3.9)

−  where Q2 is a point on Γ− 2 such that F < 0 on P2 Q2 ∩ Γ2 and F = 0 at Q2 . Combining (3.8) and (3.9) leads to

S|P1 ≥ R|P1 ≥ RQ2 e−ˆκ(ξP2 −ξQ1 ) . Notice by F = 0 at Q2 that RQ2 > S Q2 . We next draw a positive characteristic curve from Q2 and repeat the above processes to complete the proof. 2 Remark 1. Lemma 3.1 shows that the lower bound of (R, −S)(ξ, η) is independent of the distance from the

This fact is critical to our analysis in the following. point (ξ, η) to the sonic boundary AC. 4. Regularity in the partial hodograph coordinates

Consider the level curves We analyze the need for the regularity of the sonic boundary AC. ε (ξ, η) = 1 − sin δ = ε, where ε is a small positive constant. In view of (2.11) and (2.13), we compute

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εξ = − cos δδξ = =

sin σ cos δ(∂¯+ δ − ∂¯− δ) − cos σ sin δ(∂¯+ δ + ∂¯− δ) 2 sin δ

cos σ sin2 δ cos σ sin δ(κ + sin2 δ) R + S sin σ(κ + sin2 δ) (R − S) + − · , 2κ c 2κ cos δ

(4.1)

and εη = − cos δδη = =−

cos σ cos δ(∂¯− δ − ∂¯+ δ) − sin σ sin δ(∂¯+ δ + ∂¯− δ) 2 sin δ

sin σ sin2 δ sin σ sin δ(κ + sin2 δ) R + S cos σ(κ + sin2 δ) (R − S) + − · , 2κ c 2κ cos δ

(4.2)

from which one has  (εξ )2 + (εη )2 =

(κ + sin2 δ)(R − S) 2κ

2

 +

sin δ(κ + sin2 δ) R + S sin2 δ − · c 2κ cos δ

2 ,

(4.3)

which combined with (3.4) yields

0<

(R − S)2 ≤ (εξ )2 + (εη )2 ≤ 4



(κ + 1)M κ

2

 +

2 κ+1 1 · |W | , + c4 2κ

(4.4)

where W = (R + S)/ cos δ. It is clear from (4.4) that we need to show the uniform boundedness of W to

In order to obtain higher regularity of AC

than that in [20], we further need establish the regularity of AC. to derive the uniform regularity of W , see Lemma 4.4. 4.1. Equations in partial hodograph coordinates We introduce a partial hodograph transformation (ξ, η) → (z, t) by defining t = cos δ(ξ, η),

z = −φ(ξ, η).

(4.5)

The definition of φ is given in (2.3). We note by (3.2) that z ≥ c24 /(γ − 1) > 0. From (2.11) and (2.13), the Jacobian of this transformation is J :=

(κ + sin2 δ)(S − R) ∂(z, t) ∂¯− δ − ∂¯+ δ = sin δ(U δη − V δξ ) = = < 0, ∂(ξ, η) 2c sin δ 2κc cos δ

(4.6)

in the whole region ABC. Under the new coordinates (t, z), we find that    (1 − t2 )3 ct (κ + 1 − t2 )(1 − t2 )R + ¯ − ∂t + √ ∂z , ∂ =− κt c 1 − t2    2 2 (1 − t2 )3 (κ + 1 − t ct )(1 − t )S − − ∂t + √ ∂¯ = − ∂z , κt c 1 − t2 where the sound speed c = c(z, t) =



2κ(1 − t2 )z/(κ + 1 − t2 ). According to (2.14) and (4.7) leads to

(4.7)

Y. Hu, T. Li / J. Math. Anal. Appl. 467 (2018) 1174–1193

⎧ ⎪ ⎪ ⎨ Rt −

cf t2 S−tg Rz

⎪ ⎪ ⎩ St −

cf t2 R−tg Sz

= =

√ −f R 1−t2 S−tg √ −f S 1−t2 R−tg

−

−

2t2

2t2

√ 1−t2 c

√ 1−t2 c

+

κ+1 2κ

+

·

κ+1 2κ

·

R+S t R+S t

1183

 − ,  2 − κ+2−2t Rt , κ κ+2−2t2 St κ

(4.8)

where f = f (t) =



κ  , 2 (κ + 1 − t ) (1 − t2 )3

g = g(z, t) =

κ . 2(κ + 1 − t2 )z

It is easy to see that the functions c(z, t), f (t) and g(z, t) have uniform positive lower and upper bounds in the whole region A B  C  . The region A B  C  is used to denote the region ABC in the (z, t) plane. Set  = 1/R, R

S = −1/S,

which are uniform bounded up to the sonic curve by (3.4). System (4.8) can be rewritten as ⎧ ⎪  ⎪ ⎨ Rt +

 cf t2 S   Rz 1+tg S

⎪ ⎪ ⎩ St −

 cf t2 R   Sz 1−tg R

= =

√ f 1−t2  1+tg S √ f 1−t2  1−tg R





κ+1 2κ κ+1 2κ

· ·

 S  R− t  R  S− t

+ −

2t2

2t2

√ 2 1−t2    RS + κ+2−2t Rt c κ

√ 2 1−t2    RS − κ+2−2t St c κ

 ,



(4.9) .

We denote λ− = −

 cf t2 R ,  1 − tg R

cf t2 S , 1 + tg S

λ+ =

which are the eigenvalues of (4.9), and introduce ∂± = ∂t + λ± ∂z ,

 − ∂− R,  G = ∂+ R

 H = ∂+ S − ∂− S.

Then one has λ+ − λ− =

 + S)  cf (R · t2 ,   (1 − tg R)(1 + tg S)

z = R

G , λ+ − λ−

Sz =

H . λ + − λ−

(4.10)

Furthermore, we use the commutator relation [13] ∂− ∂+ − ∂+ ∂− =

∂− λ+ − ∂+ λ− (∂+ − ∂− ), λ+ − λ−

to arrive at ∂+ G =

∂− λ+ − ∂+ λ−  − ∂− ∂+ R),  G + (∂+ ∂+ R λ + − λ−

(4.11)

∂− H =

∂− λ+ − ∂+ λ−  H + (∂+ ∂− S − ∂− ∂− S). λ+ − λ−

(4.12)

Doing a tedious but straightforward calculation, one finds that 2 ∂− λ+ − ∂+ λ− = + h(z, t), λ + − λ− t

(4.13)

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where √ √

 − S)  S f 1 − t2 (κ + 1)g(R 2t3 1 − t2 g R h(z, t) = +   2κ c (1 − tg R)(1 + tg S)  − S + 2tg R S  2(κ + 1 − t2 )2 + 2(1 − t2 ) t(κ + 2 − 2t2 ) R + + · t  + S κ (1 − t2 )(κ1 − t2 )2 R  − S + 2tg R S S (κ + 1)g R f cgt3 R · − . 2     κ + 1 − t (1 − tg R)(1 + tg S) 2z(1 − tg R)(1 + tg S)

+

Moreover, we directly compute  z = g1 R z + g2 Sz + g3 , (∂¯+ R)

(4.14)

where g1 (z, t) =

 (κ + 1)g S − t(κ + 2 − t2 )(1 + tg S) 1 −  2t 2(κ + 1 − t2 )(1 − t2 )(1 + tg S) √

√  f 1 − t2 2t2 1 − t2  κ + 2 − 2t2 S+ t , + c κ 1 + tg S

  1 (κ + 1)g S − t(κ + 2 − t2 )(1 + tg S) 2t2 f (1 − t2 )R + + 2 2   2t 2(κ + 1 − t )(1 − t )(1 + tg S) c(1 + tg S) √ √

 f g 1 − t2 κ + 1   2t3 1 − t2   κ + 2 − 2t2  2 (R − S) + RS + Rt , − 2 2κ c κ (1 + tg S) √ √

 f g 1 − t2 S κ + 1   2t3 1 − t2   κ + 2 − 2t2  2 g3 (z, t) = (R − S) + RS + Rt 2 2κ c κ 2z(1 + tg S)

g2 (z, t) = −

−

S 2κt2 f (1 − t2 )2 R

 c3 (κ + 1 − t2 )(1 + tg S)

,

and  z = h1 Sz + h2 R z + h3 , (∂¯− S) where h1 (z, t) =

 + t(κ + 2 − t2 )(1 − tg R)  (κ + 1)g R 1 +  2t 2(κ + 1 − t2 )(1 − t2 )(1 − tg R) √

√  f 1 − t2 2t2 1 − t2  κ + 2 − 2t2 − R+ t ,  c κ 1 − tg R

 + t(κ + 2 − t2 )(1 − tg R)  (κ + 1)g R 2t2 f (1 − t2 )S 1 − −   2t 2(κ + 1 − t2 )(1 − t2 )(1 − tg R) c(1 − tg R) √ √

 f g 1 − t2 κ + 1   2t3 1 − t2   κ + 2 − 2t2  2 + (S − R) − RS − St ,  2 2κ c κ (1 − tg R)

h2 (z, t) = −

(4.15)

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h3 (z, t) = −

1185

√ √  3 2 2  κ + 1 f g 1 − t2 R  − 2t 1 − t R S − κ + 2 − 2t St 2 (S − R)  2 2κ c κ 2z(1 − tg R)

+

S 2κt2 f (1 − t2 )2 R

 c3 (κ + 1 − t2 )(1 − tg R)

.

Therefore, it follows by (4.10) and (4.14)–(4.15) that  − ∂− ∂+ R  = (λ+ − λ− )(∂+ R)  z = g1 G + g2 H + (λ+ − λ− )g3 , ∂+ ∂+ R  z = h1 H + h2 G + (λ+ − λ− )h3 . ∂+ ∂− S − ∂− ∂− S = (λ+ − λ− )(∂− S) We put the above into (4.11)–(4.12) and employ (4.13) to get 

 H 5 + g˜1 (z, t) G + g˜2 (z, t) + g˜3 (z, t)t2 , 2t 2t   5 ˜ 1 (z, t) H + h ˜ 3 (z, t)t2 , ˜ 2 (z, t) G + h ∂− H = +h 2t 2t ∂+ G =

(4.16) (4.17)

where   1 , g˜1 = h + g1 − 2t   ˜ 1 = h + h1 − 1 , h 2t

g˜2 = 2tg2 ,

g˜3 =

˜ 2 = 2th2 , h

˜3 = h

 + S)  cf (R g ,   3 (1 − tg R)(1 + tg S)  + S)  cf (R h .   3 (1 − tg R)(1 + tg S)

˜ i (i = 1, 2, 3) are Noting the expressions of h, gi , hi (i = 1, 2, 3), we see by (3.3) that the functions g˜i , h + ˜ 2 → −1 as t → 0 . We comment that here we only use the uniformly bounded near t = 0 and g˜2 → −1, h   uniform boundedness of R and S. To deal with our problem, we further introduce two new variables  = G, G t2

 = H. H t2

(4.18)

Then (4.16)–(4.17) can be transformed to = ∂+ G  = ∂− H



  1  + g˜2 (z, t) H + g˜3 (z, t), + g˜1 (z, t) G 2t 2t



  1 ˜ ˜ 3 (z, t), ˜ 2 (z, t) G + h  +h + h1 (z, t) H 2t 2t

or 

   1  − 32   t˜ g1 G + g˜2 H + t˜ ∂+ t G = t g3 , 2     1 ˜ 1H ˜3 . ˜ 2G  = t− 32 th  + 1h  + th ∂− t − 2 H 2 − 12

(4.19)

(4.20)

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Fig. 3. The region of Ων (¯ z ).

4.2. Regularity in partial hodograph plane We now apply (4.19)–(4.20) to derive the regularity of solutions near the line t = 0, i.e., the sonic

boundary AC.  C  . We choose t small such that the point  Let (¯ z , 0) be any fixed point on the degenerate segment A P P (tP , z¯) stays in the domain A B  C  . From the point P , we draw λ+ and λ− -characteristics, called z+ (P )  C  at P and P , respectively. Thanks to the facts that g ˜ 1, h ˜ 3 are  and z− (P ), up to the segment A ˜1 , g˜3 , h 1 2 + ˜ uniformly bounded near t = 0 and g˜2 → −1, h2 → −1 as t → 0 , then for any constant ν ∈ (0, 1], we can choose tP < 1 small enough such that t|˜ g1 (z, t)| ≤ ν8 , ˜ 1 (z, t)| ≤ ν , t|h 8

t|˜ g3 (z, t)| ≤ ν8 , |˜ g2 (z, t)| ≤ 1 + ν, ˜ 3 (z, t)| ≤ ν , |h ˜ 2 (z, t)| ≤ 1 + ν, t|h 8

(4.21)

hold in the whole domain P P1 P2 . We use Ων (¯ z ) to represent the domain bounded by P P1 , P P2 and the positive and negative characteristics starting from (¯ z , 0). For any (z, t) ∈ Ων (¯ z ), we denote a(za , ta ) and b(zb , tb ) the intersection points of the λ− and λ+ -characteristics through (z, t) with the boundaries P P1 and

P P2 , respectively, see Fig. 3. Let  = max K



   max |G(zb , tb )| + 1, max |H(za , ta )| + 1 ,

Ων (¯ z)

Ων (¯ z)

which is well-defined and uniformly bounded in the domain Ων (¯ z ). Now we use the bootstrap method employed in [20,24] to show the key lemma.  C  and ν ∈ (0, 1] be any constant. Then there  z , 0) be any point on the degenerate line A Lemma 4.1. Let (¯  exists a uniform positive constant K such that there hold

 ≤ K,  |tν G|

 ≤ K,  |tν H|

∀ (z, t) ∈ Ων (¯ z ).

(4.22)

Proof. For a fixed ε ∈ (0, tP ), we denote  ε := {(z, t) : ε ≤ t ≤ tP , z− (P ) ≤ z ≤ z+ (P )} ∩ Ων (¯ Ω z)  ε = max{|tν G|,  |tν H|}.   ε ≤ K,  then (4.22) holds. Otherwise, we and K If for any 0 < ε < tP , one has K ε Ω

 ε > K.  assume there exists ε0 ∈ (0, tP ) such that K 0

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a b For any point (zε0 , ε0 ) ∈ Ων (¯ z ), we use the notations z− (t) and z+ (t) to represent the λ− and

P2 at points a(za , ta ) λ+ -characteristics passing through the point (zε0 , ε0 ) and intersecting P P1 and P and b(zb , tb ), respectively. Integrating (4.19) along the λ+ -characteristic from t(≥ ε0 ) to tb and applying the  yields definition of K

    tb  + 1 g˜2 H  + τ g˜3    G τ g˜1 G b 2  1  =  G(zb1, tb ) + (z+ (τ ), τ ) dτ  3 t2   τ2 t2 b

t

tb    + 1+ν |H|  +  b , tb )   ν |G|  G(z 8 2  ≤ + 1 3  τ2 tb2 t

   b , tb )  2 + 3ν  G(z ε + = K 1 0  4 tb2

1 2

1 +ν



ν 8

  tb  b , tb )  2 + 3ν  G(z 1 b   + K (z+ (τ ), τ ) dτ ≤  dτ ε0 1 3  4 2 2 τ +ν tb t

1 t

1 2 +ν

−

1 1 2 +ν

tb

 <

ε 1 (2 + 3ν)K 0 − −ν  ε t− 12 −ν , t 2
from which one finds that a strict inequality  t=ε < K  ε ε−ν , |G| 0 0 0

(4.23)

 ε . A similar argument for the equation (4.20) leads to holds on the line segment {t = ε0 } ∩ Ω 0  t=ε < K  ε ε−ν . |H| 0 0 0

(4.24)

ε  and |tν H|  in the domain Ω Combining (4.23) and (4.24), we conclude that the maximum values of |tν G| 0    can only happen on ε0 < t ≤ tP . This assertion also holds in a larger domain Ωε , where ε < ε0 . Repeating the above processes and extending the domain larger and larger and until the whole domain Ων (¯ z ) completes the proof of the lemma. 2  C  . For any point (¯  C  , let   The above lemma can be extended a small interval on the segment A z , 0) ∈ A ∗ ∗    μ be a small constant such that (¯ z − μ, z¯ + μ) ⊂ A C . Denote Q1 (Q2 ) the intersection point of the negative P1 (P (positive) characteristic through Q1 := (¯ z − μ, 0) (Q2 := (¯ z + μ, 0)) with the boundary P P2 ). Making use of a similar argument showing Lemma 4.1, we can extend the inequality (4.22) to the region Ων (¯ zμ ) ∗Q , Q ∗ and Q ∗P .      Q∗1 , Q Q bounded by the boundaries P Q , Q 1 2 2 2 1 1 2  C  and μ be a small positive constant such that  Lemma 4.2. Let (¯ z , 0) be any point on the degenerate line A    depending on  (¯ z − μ, z¯ + μ) ⊂ A C . Then, for any constant ν ∈ (0, 1], there exists a positive constant K the interval (¯ z − μ, z¯ + μ) such that there hold

 ≤ K,  |tν G|  := We next show that the function W

 ≤ K,  |tν H|  S  R− t

zμ ). ∀ (z, t) ∈ Ων (¯

(4.25)

is uniformly bounded.

 is uniformly bounded up to Lemma 4.3. Suppose that the assumptions of Theorem 1 hold. The function W    the sonic boundary A C . Proof. It suffices to prove that the lemma holds near t = 0. By (4.9)–(4.10) and (4.18), we directly calculate t = I1 W  + I2 , W

(4.26)

Y. Hu, T. Li / J. Math. Anal. Appl. 467 (2018) 1174–1193

1188

where I1 =

 + 2t(κ + 2 − t2 ) + 2g(κ + 1 − t2 )(1 − t2 )(R  − S + tg R S)  (κ + 1)g(S − R) ,   2(1 − tg R)(1 + tg S)(κ + 1 − t2 )(1 − t2 )

√

     + S)   f 1 − t2 (κ + 2 − 2t2 )(R S(1 − tg R)  ν ) + R(1 + tg S) (Ht  ν ) t1−ν . I2 = − (Gt    + S  + S κ(1 − tg R)(1 + tg S) R R We choose t small enough and ν = 1/2 in Lemma 4.2 such that I1 and I2 are uniformly bounded, which  by (4.26). 2 lead directly to the uniform boundedness of W  and S and use (3.3) to obtain that W = (R + S)/t is uniformly bounded We recall the definitions of R    in the whole domain A B C , that is, there exists a uniform constant K1 such that    R + S    t (z, t) ≤ K1

∀ (z, t) ∈ A B  C  .

(4.27)

The above inequality implies that |∂+ R| and |∂− S| are uniformly bounded in the whole domain A B  C  by (4.8). In addition, applying (4.10) again yields Rz = −

  (1 − tg R)(1 + tg S)  G, 2  (R  + S)  cf R

Sz =

  (1 − tg R)(1 + tg S)  H. 2  + S)  cf S (R

(4.28)

Combining (4.27) and (4.28), it follows by (4.8) that the two derivatives |Rt | and |St | are uniformly bounded in the whole domain A B  C  including the degenerate line t = 0. Therefore, the two functions R and −S

with at least a rate of cos δ. approach a common value on the degenerate curve AC C .  We next show the uniform regularity of R(z, t), S(z, t) and W (z, t) up to A Lemma 4.4. Under the assumptions of Theorem 1, the functions R(z, t), S(z, t) and W (z, t) are uniformly 1 C .  C 3 continuous in the whole domain A B  C  , including the degenerate line A  C  . Suppose that  Proof. Let (z1 , 0) and (z2 , 0) (z1 < z2 ) be any two points on the degenerate curve A (zm , tm ) is the intersection point of the positive and negative characteristics starting from (z1 , 0) and (z2 , 0) respectively. We recall equations (4.8) to get the characteristic curves z ±



d + dt z



cf = − S−tg t2 ,

d − dt z

cf = − R−tg t2 ,

z − |t=0 = z2 ,

z + |t=0 = z1 , which along with (3.4) acquires 1

Ktm ≤ |zm − zi | 3 ≤ Ktm ,

i = 1, 2

(4.29)

for some positive constants K and K. Since |∂+ R| and |∂− S| are uniform bounded, we integrate ∂+ R from (z1 , 0) to (zm , tm ) and ∂− S from (z2 , 0) to (zm , tm ) to arrive |R(z1 , 0) − R(zm , tm )| ≤ K2 tm , for some uniform constant K2 . Thus we obtain

|S(z2 , 0) − S(zm , tm )| ≤ K2 tm

Y. Hu, T. Li / J. Math. Anal. Appl. 467 (2018) 1174–1193

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|R(z1 , 0) − R(z2 , 0)| = |R(z1 , 0) − S(z2 , 0)| ≤ |R(z1 , 0) − R(zm , tm )| + |R(zm , tm ) − S(zm , tm )| + |S(zm , tm ) − S(z2 , 0)| ≤ (2K2 + K1 )tm ≤

1 1 2K2 + K1 2K2 + K1 |z2 − z1 | 3 = |(z2 , 0) − (z1 , 0)| 3 , K K

1 C .  which indicates that R is C 3 continuous on A We next consider any two points (z1 , t1 ) and (z2 , t2 ) (z1 ≤ z2 , 0 ≤ t1 ≤ t2 ) in the region A B  C  . If z1 = z2 , then using the uniform boundedness of Rt arrives at

|R(z1 , t1 ) − R(z2 , t2 )| = |R(z1 , t1 ) − R(z1 , t2 )| ≤ max |Rt | · |t2 − t1 |.    AB C

If z1 < z2 , the analysis is divided into two cases: Case I. t1 ≥ (z2 − z1 ). Choosing ν = 1/2 in (4.25) and using (4.28) yields |R(z2 , t2 ) − R(z1 , t1 )| ≤ |R(z2 , t2 ) − R(z2 , t1 )| + |R(z2 , t1 ) − R(z1 , t1 )| |Rt | · |t2 − t1 | + |Rz | · |z2 − z1 | ≤ max    AB C

− 12

|Rt | · |t2 − t1 | + Kt1 ≤ max    AB C

· |z2 − z1 | 1

1

≤ max |Rt | · |t2 − t1 | + K · |z2 − z1 | 2 ≤ K|(z2 , t2 ) − (z1 , t1 )| 3    AB C

for some uniformly constant K. Case II. t1 < (z2 − z1 ). There exists a uniform constant K > 0 such that |R(z2 , t2 ) − R(z1 , t1 )| ≤|R(z2 , t2 ) − R(z2 , t1 )| + |R(z2 , t1 ) − R(z2 , 0)| + |R(z2 , 0) − R(z1 , 0)| + |R(z1 , 0) − R(z1 , t1 )| ≤ max |Rt | · |t2 − t1 | + max |Rt | · t1 +       AB C

AB C

≤2 max |Rt |(|t2 − t1 | + |z2 − z1 |) +    AB C

1 2K2 + K1 |z2 − z1 | 3 + max |Rt | · t1 A B  C  K

1 1 2K2 + K1 |z2 − z1 | 3 ≤ K|(z2 , t2 ) − (z1 , t1 )| 3 K

Hence the function R(z, t) is uniformly C 3 continuous in the whole domain A B  C  . The uniformly C 3 continuity of S can be obtained analogously. For the function W , we derive the equations ∂− W and ∂+ W from (4.8) as follows: 1

(κ + 1)W (W − g)(R + S − 2tg) (κ + 2 − t2 )tW − (κ + 1 − t2 )(1 − t2 ) 2(κ + 1 − t2 )(1 − t2 )(S − tg)(R − tg)

√   2t 1 − t2 (R2 + S 2 − (R + S)tg) (κ + 2 − 2t2 )RS(R + S − 2tg) 2 + +f 1−t c(S − tg)(R − tg) κ(S − tg)(R − tg)

∂+ W =

+ and

cf (S − R) (tSz ), (R − tg)(S − tg)

1

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(κ + 1)W (W − g)(R + S − 2tg) (κ + 2 − t2 )tW − 2 2 (κ + 1 − t )(1 − t ) 2(κ + 1 − t2 )(1 − t2 )(S − tg)(R − tg) 

√  2t 1 − t2 (R2 + S 2 − (R + S)tg) (κ + 2 − 2t2 )RS(R + S − 2tg) 2 + +f 1−t c(S − tg)(R − tg) κ(S − tg)(R − tg)

∂− W =

−

cf (S − R) (tRz ), (R − tg)(S − tg)

which indicate by (4.27)–(4.28) and Lemma 4.2 that |∂− W | and |∂+ W | are uniform bounded. Therefore, 1 we reproduce the same argument as above for R to get the uniform C 3 -continuity of W . The proof of the lemma is complete. 2 5. Proof of Theorem 1 Based on the results in the previous sections, we now complete the proof of Theorem 1, that is, we show 1

and the that the solution (ρ, u, v)(ξ, η) in the region ABC is uniformly C 1, 6 up to the sonic boundary AC 1 1, 6

sonic curve AC is C -continuous. We divide the proof into four steps. Step I. We first claim that the map (ξ, η) → (z, t) is an one-to-one mapping. If not, assume (ξ1 , η1 ) and (ξ2 , η2 ) are two distinct points in the domain ABC such that cos δ(ξ1 , η1 ) = cos δ(ξ2 , η2 ) and −φ(ξ1 , η1 ) = −φ(ξ2 , η2 ), which means (ξ1 , η1 ) and (ξ2 , η2 ) are two points on a level curve ε (ξ, η) = 1 − sin δ(ξ, η) = ε ≥ 0. Moreover, we see by (4.27) that the equalities (4.1) and (4.2) are well defined in the whole domain ABC, from which we calculate  ∇φ ·

(εη , −εξ )

=

sin σ cos σ , −c −c sin δ sin δ

 · (εη , −εξ )

 2  cos σ(κ + sin2 δ)(R − S) sin δ sin δ(κ + sin2 δ) R + S − + sin σ − · 2κ c 2κ cos δ

 2  sin σ sin σ(κ + sin2 δ)(R − S) sin δ sin δ(κ + sin2 δ) R + S +c + cos σ − · sin δ 2κ c 2κ cos δ

cos σ =−c sin δ

=



c(κ + sin2 δ)(R − S) > 0, 2κ sin δ

which indicates that the function φ is strictly monotonic along each level curve ε (ξ, η) = ε ≥ 0. That contradicts to the assumption φ(ξ1 , η1 ) = φ(ξ2 , η2 ). 1 Step II. We next show that the function δ(ξ, η) is uniformly C 2 -continuous in the whole domain ABC,

In view of (2.11) and (2.13), we obtain including the sonic boundary AC. cos δδξ =

cos σ sin2 δ cos σ sin δ(κ + sin2 δ) sin σ(κ + sin2 δ) (S − R) − + W, 2κ c 2κ

cos δδη =

sin σ sin2 δ sin σ sin δ(κ + sin2 δ) cos σ(κ + sin2 δ) (R − S) − + W, 2κ c 2κ

(5.1)

from which, we have by applying (3.3) and (4.27) | cos δδξ | + | cos δδη | ≤

2 2(κ + 1)M 2(κ + 1)K1 + ≤C + κ c4 2κ

for a uniform positive constant C, from which one has

(5.2)

Y. Hu, T. Li / J. Math. Anal. Appl. 467 (2018) 1174–1193

|(

1191

π π π 2 −δ − δ)δξ | + |( − δ)δη | ≤ C ≤ 2C, 2 2 sin( π2 − δ)

which means that the function (π/2 − δ)2 is uniformly Lipschitz continuous in terms of (ξ, η), that is, for any two points T1 = (ξ1 , η1 ) and T2 = (ξ2 , η2 ) in the whole domain ABC, there holds  2  2   π  π    2 − δ(T2 ) − 2 − δ(T1 )  ≤ 2C|T2 − T1 |.

(5.3)

Noting the facts (π/2 − δ(T2 )) ≥ 0 and (π/2 − δ(T1 )) ≥ 0, we find that  2  2   π  π    2 − δ(T2 ) − 2 − δ(T1 )           π   π  π π = − δ(T2 ) − − δ(T1 )  ·  − δ(T2 ) + − δ(T1 )  2 2 2 2          π   π  π π ≥ − δ(T2 ) − − δ(T1 )  ·  − δ(T2 ) − − δ(T1 )  = |δ(T2 ) − δ(T1 )|2 , 2 2 2 2 which combined with (5.3) yields |δ(T2 ) − δ(T1 )| ≤

√

1

2C |T2 − T1 | 2 .

(5.4)

1

Thus δ(ξ, η) is uniformly C 2 -continuous in the whole domain ABC. 1 Step III. We claim that, for any function ψ(z, t) ∈ C 3 defined in the whole domain A B  C  , the function 1  η) := ψ(z, t) is uniformly C 6 -continuous in the whole domain ABC in terms of (ξ, η). In fact, for any ψ(ξ, two points T1 = (ξ1 , η1 ) and T2 = (ξ2 , η2 ) in ABC, by Step I, there exists two points T1 = (z1 , t1 ) and T2 = (z2 , t2 ) in the domain A B  C  such that ti = cos δ(ξi , ηi ), zi = −φ(ξi , ηi ) (i = 1, 2). It follows by the 1 assumption ψ(z, t) ∈ C 3 that  2 ) − ψ(T  1 )| = |ψ(T  ) − ψ(T  )| ≤ C|T  − T  | 13 |ψ(T 2 1 2 1

 2  2  16 =C cos δ(T2 ) − cos δ(T1 ) + φ(T2 ) − φ(T1 )

(5.5)

for some uniform constant C. On the other hand, by (5.4), one has    1 δ(T2 ) − δ(T1 )   2 | cos δ(T2 ) − cos δ(T1 )| ≤ 2 sin  ≤ |δ(T2 ) − δ(T1 )| ≤ C|T2 − T1 | . 2 Inserting the above into (5.5) and making use of the fact that φ(ξ, η) is uniformly Lipschitz continuous in the whole domain ABC yields  16

1 2   |ψ(T2 ) − ψ(T1 )| ≤ C C|T2 − T1 | + C|T2 − T1 | ≤ C|T2 − T1 | 6 ,  η) is uniformly C 6 -continuous in the whole domain ABC. which means that ψ(ξ, By Lemma 4.4 and the above result, we obtain that the functions R(ξ, η), S(ξ, η) and W (ξ, η) are uni1 formly C 6 -continuous in terms of (ξ, η) in the whole domain ABC. 1

Y. Hu, T. Li / J. Math. Anal. Appl. 467 (2018) 1174–1193

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Step IV. We now establish the uniform regularity of solution (ρ, u, v)(ξ, η) in the whole region ABC and

To obtain it, we first consider the regularity of σ(ξ, η). Due to (2.12) the regularity of sonic boundary AC. and (2.13), we compute sin(2δ)(S − R) , ∂¯+ σ + ∂¯− σ = 2κ

sin(2δ)(R + S) 2 sin2 δ ∂¯+ σ − ∂¯− σ = − . c 2κ

Putting the above into (2.11) gives σξ = − ση =

sin σ sin δ sin σ cos δ(R + S) − cos σ sin δ(R − S) + , c 2κ

cos σ cos δ(R + S) + sin σ sin δ(R − S) cos σ sin δ − , c 2κ

(5.6)

which imply that the function σ(ξ, η) and consequently sin σ(ξ, η) and cos σ(ξ, η) are uniformly Lipschitz 1 1 continuous in the whole domain ABC. Furthermore, we use the facts (R, S) ∈ C 6 , δ ∈ C 2 and (5.6) to 1 see that σ(ξ, η), cos σ(ξ, η) and sin σ(ξ, η) are uniformly C 1, 6 in the whole domain ABC. From (5.1) and 1 Step III, we obtain that the function sin δ(ξ, η) is uniformly C 1, 6 in the whole domain ABC. 1 By using the pseudo-Bernoulli law (2.3), we acquire that the sound speed c is uniformly C 1, 6 in the  whole domain ABC. Recalling the definition c = p (ρ) and the expressions (u, v)(ξ, η) in (2.9), we finally 1 obtain that (ρ, u, v)(ξ, η) are uniformly C 1, 6 in the whole domain ABC.

From the definitions (4.1) and (4.2), the Now we consider the regularity of the sonic boundary AC. 1 ε ε 6 functions ξ (ξ, η) and η (ξ, η) are uniformly C -continuous in the whole domain ABC. In addition, it is  such that obvious by (3.3) and (4.27) that there exist two uniform positive constants c˜ and C  c˜ ≤ [εξ (ξ, η)]2 + [εη (ξ, η)]2 ≤ C.

is C 1, 16 continuous and the proof of Theorem 1 is complete. Thus the sonic boundary AC Acknowledgments Y.B. Hu was supported by the Zhejiang Provincial Natural Science Foundation (LY17A010019), National Science Foundation of China (11301128 and 11571088) and China Scholarship Council (201708330155). Y.B. Hu would also like to thank the hospitality and the support of Department of Mathematics in University of Iowa, during his visit in 2017–2018. References [1] X. Chen, Y.X. Zheng, The direct approach to the interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J. 59 (2010) 231–256. [2] J.D. Cole, L.P. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, 1986. [3] R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948. [4] Z.H. Dai, T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal. 155 (2000) 277–298. [5] G. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang, Y. Zheng, Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations, SIAM J. Appl. Math. 69 (2008) 720–742. [6] Y.B. Hu, J.Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, 2017, submitted for publication. [7] Y.B. Hu, J.Q. Li, W.C. Sheng, Degenerate Goursat-type boundary value problems arising from the study of twodimensional isothermal Euler equations, Z. Angew. Math. Phys. 63 (2012) 1021–1046. [8] Y.B. Hu, T. Li, The regularity of semi-hyperbolic patch problems for the two-dimensional isothermal Euler equations, 2018, submitted for publication. [9] Y.B. Hu, G.D. Wang, Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations 257 (2014) 1579–1590.

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