Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

For the transonic problems of (2), iterative methods seem to be the most likely choices for constructing global solutions. As a first step, in the present paper we investigate the structure of solutions near a sonic curve for the pseudo-steady full Euler equations (2). This is an essential step for using the iterative process to establish a global transonic solution. Many efforts have been made to understand the structure of solutions near sonic curves for the Euler equations and its simplified models. For the steady isentropic Euler equations, some explicit examples of transonic solutions were presented in [14,23], the existence conditions of continuous sonic-supersonic flows were given in [1], the existence of weak solutions in the compensated-compactness framework was investigated in [4,6,7,32], the existence of subsonic-sonic solutions was provided in [39,43,44], the existence and uniqueness of smooth transonic flows in Laval nozzles was established in [40,41]. We also refer the reader to [5,8,9,11,16,45] for the study of transonic shocks arising in supersonic flow past a blunt body or a bounded nozzle. There are more related references on classical methods for solutions [17,18], on perturbation arguments and linear theory [10,33,36] and on asymptotic models [2,13], etc. In the past few years, the semi-hyperbolic patch problems, a kind of degenerate hyperbolic problems with sonic curves as boundaries, have been explored by applying the idea of characteristic decomposition, see [20,21,30,37] for the Euler system, [38,42] for the pressure-gradient system and [22] for the nonlinear wave system. The characteristic decomposition is a powerful tool for studying some degenerate hyperbolic problems which was revealed by Dai and Zhang [15], see [12,24,27,28,35] for more applications.
It seems successfully to choose a pair of appropriate coordinates to overcome the difficulty caused by the sonic degeneracy of the Euler system. Many of previous results on the steady isentropic irrotational Euler equations used the hodograph method which allows to linearize the equations, see, e.g., [13,14,17,31]. However, it is well-known that the hodograph method is difficult in treating boundary conditions and in transforming back to the original independent variables. In [23], Kuz'min took the stream function Ψ and the potential function Φ as the coordinate system to investigate a perturbation problem for a given transonic solution. Recently, Zhang and Zheng [47,48] employed the coordinate system ( q 2 − c 2 , Φ) to construct local classical sonic-supersonic solutions for the steady and pseudo-steady isentropic irrotational Euler equations. Hu and Li [19] established the existence of sonic-supersonic solutions for the steady full Euler equations by using the angle functions as the coordinate system. The angle functions were first introduced as the dependent variables by Li and Zheng [28].
The purpose of this paper is to construct a classical sonic-supersonic solution for (2). In order to deal with the pseudo-steady full Euler equations, we introduce a novel set of dependent and independent change variables to transform (2) into a new hyperbolic system with a clear singularity-regularity structure. We describe the problem as follows.
This means that Γ is a sonic curve. We look for a classical solution for (2) in the region q > c near Γ.
The main result in this paper is stated in the following theorem. Theorem 1.1. Letθ be the pseudo-flow angle on Γ defined byθ = arctan((v − ϕ)/(û − ξ)). Assume that the curve Γ and the boundary data (ρ,û,v,p) satisfy Then there exists a classical solution for Problem 1.1 in the region q > c near Γ.
Remark 1. The regularity conditions (A 1 ) can be relaxed somewhat by using the concept of modulus of continuity as was done in [48]. The conditions (A 2 ) mean that the pseudo-flow direction (cos θ, sin θ) is neither tangent nor normal to the curve Γ. The conditions (A 2 ) and (A 3 ) are mainly used to ensure that on the curve Γ the pseudo-Mach number M a := q/c is strictly increasing along the pseudoflow direction, which seems reasonable since the classical solution is expected to be constructed from the curve M a = 1 to the region M a > 1. The condition forp in (A 3 ) can be properly relaxed, see Remark 2 in Section 2.
To prove Theorem 1.1, the key is to characterize the degeneracy of the system near the sonic curve by suitable independent and dependent changing variables. For the pseudo-steady isentropic irrotational Euler equations, Zhang and Zheng [48] chose ( q 2 − c 2 , Φ) as the independent quantities and (∂ + c,∂ − c) as the dependent quantities to obtain a tidy first-order hyperbolic system. However, for the full Euler system, it is impossible to use ( q 2 − c 2 , Φ) as the independent quantities due to the non-existence of potential function Φ. Moreover, the characteristic decomposition of c is considerably formidable for the full Euler equations such that (∂ + c,∂ − c) are not suitable to be the dependent quantities any more. Inspired by Hu and Li [19], we introduce the pseudo-Mach angle ω and the pseudo-flow angle θ and then choose (cos ω, θ) as the independent coordinate system. To get suitable dependent variables, we further introduce a novel variable Ξ, which is a function of pseudo-Mach angle ω, entropy S and pseudo-Bernoulli number B. We derive the characteristic decomposition of Ξ under a pair of new weighted directional derivatives (∂ + ,∂ − ). Taking (∂ + Ξ,∂ − Ξ,∂ + B,∂ + S) as the new dependent quantities, we reduce the pseudo-steady full Euler equations (2) to a new closed system (39) in the partial hodograph (cos ω, θ)-plane. This new system has the desired explicitly singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Thanks to the choice of the independent variables, we can transform back the solution to the original self-similar coordinate system and thus construct a local classical sonic-supersonic solution to (2).
The rest of the paper is organized as follows. In Section 2, we introduce a new set of variables to reformulate the problem by the idea of characteristic decomposition. In Section 3, we transform the problem into a new problem under a partial hodograph plane and then solve the new problem by employing the fixed-point method in a weighted metric space. In Section 4, we complete the proof of the main theorem by converting the classical solution in the partial hodograph plane into that in terms of original self-similar plane. Appendix is provided for the algorithmic arguments of characteristic decompositions.
2. The reformulation of the problem. In order to analyze the nonlinear degenerate problem under consideration in this paper, we adopt the pseudo-Mach angle, the pseudo-flow angle, the entropy and the pseudo-Bernoulli quantity as dependent variables. The angle functions as the dependent variables was invented by Li and Zheng [28] and then was applied in many problems, see, e.g. [12,20,24,29,30,35,37]. In this section, we provide the characteristic decompositions in angle variables and reformulate the problem in this new framework.
2.1. Characteristic decompositions in angle variables. For smooth solutions, system (2) can be rewritten as where the primitive variables and the coefficient matrices are The pseudo-velocity (U, V ) = (u−ξ, v −η) is defined as before. The four eigenvalues of (5) are expressed in (3) and the corresponding left eigenvectors are . By a standard calculation, system (5) changes to where S = pρ −γ is the entropy function and B = q 2 2 + c 2 γ−1 is the pseudo-Bernoulli function.
We introduce the pseudo-flow angle θ and the pseudo-Mach angle ω as follows Denote the angle variables Then we combine (3), (7) and (8) to obtain which mean that α, β, θ are, respectively, the inclination angles of positive, negative and zero characteristics. According to (7) and the expression of B, the functions (c, u, v) can be expressed in terms of B, θ, ω where κ = (γ − 1)/2. Moreover, we introduce the normalized directional derivatives along the characteristics In terms of the variables (S, B, ω, θ), system (6) can be transformed into a new form The detailed derivation of (13) is given in Appendix A.1. In order to get rid of B in the non-homogeneous terms in (13), we further introduce the following weighted directional derivatives∂ whereB = √ 2κB. Then system (13) can be rewritten as To derive the characteristic decompositions about the angle variables, we now introduce a new variable Then the last two equations of (15) change to ∂+ θ + sin(2ω)∂ + Ξ = − sin ω κ + sin 2 ω, For obtaining a closed hyperbolic system, we take the weighted directional derivatives of (Ξ, S, B) as the dependent variables. Denote Applying (15) and (16) yields and then∂ In addition, by direct calculations, we have the following commutator relations Thanks to (15), (17)- (20) and (21), we get a hyperbolic system in terms of the variables (X, Y, G, H) +Y Y + cos(2ω)X + cos ω κ + sin 2 ω − 2κH + G .

SONIC-SUPERSONIC SOLUTIONS FOR EULER EQUATIONS 1203
The detailed derivation of (22) is presented in Appendix A.2.
2.2. The boundary conditions. We now transform the boundary data for system (22) from the boundary condition (4). Denotê for all ξ ∈ [ξ 1 , ξ 2 ]. Noting S(ξ, ϕ(ξ)) =Ŝ(ξ) on Γ and using the equation∂ 0 S = 0 gives from which one has∂ By the definition of G and (24), we see that the boundary data of G on Γ is From the definition of H, it follows that Making use of the second equation of (13) and noting the fact ω = π/2 on Γ arrives at∂ which together with the function B(ξ, ϕ(ξ)) =B(ξ) on Γ, one deduces from which one obtains Combining (24) and (27) and doing a simple rearrangement leads tō Inserting the above into (26) gets Now we consider the boundary data (X, Y ) on Γ. In view of (12) and (14), we obtain X + Y = 2 cos ω∂ 0 Ξ which means that X = −Y on Γ. Adding the two equations in (17) gives On the other hand, we subtract the two equations in (17) to find that Thus We next calculate the value∂ 0 (q/c) on Γ. Adding the last two equations of (13) gets∂ from which and the facts∂ + S +∂ − S = 0,∂ + B +∂ − B = 0 on Γ, we have by (26) and (30) which along with the function sin ω(ξ, ϕ(ξ)) = 1 gives from which one has Hencē which means by the assumptions (A 2 ) in (4) that∂ 0 (q/c)| Γ and [Ĥ(ξ) −â 0 (ξ)] have the same symbol for all ξ ∈ [ξ 1 , ξ 2 ]. We employ (28) and (30) to computê Recalling the assumption ( We combine (25), (28) and (30) to obtain the boundary data (G, H, X, Y ) on Γ with The constraints (4) on the boundary data become Then Theorem 1.1 is restated in the next theorem. Remark 2. Since we only need to use the inequalityĤ(ξ)−â 0 (ξ) > 0, the condition forp in (4) can be relaxed to b(ξ) < 0 for all ξ ∈ [ξ 1 , ξ 2 ].
3. The problem in a partial hodograph plane. Since the degeneracy on the sonic curve may cause singularities, we need to single out the feature of governing equations near the sonic curve. For this purpose, we introduce a partial hodograph transformation to solve the problem in the new coordinate system.
3.1. Reformulated problem in a partial hodograph plane. In this subsection, we reformulate the problem into a new problem in the partial hodograph plane.
3.1.1. A partial hodograph transformation. We introduce a partial hodograph transformation (ξ, η) → (t, r) by defining Making use of (17) and (20), the Jacobian of this transformation is where By the definition of Ξ (16) and the first two equations of (15), we obtaiñ Hence, due to (30) and (31), one has Thus J = 0 and then by (35) J = 0 away from the sonic curve.
In terms of the new coordinates (t, r), we derivẽ where Then system (22) can be rewritten as a new system under the coordinates (t, r) We comment that system (39) has an explicitly singularity-regularity structure.
In addition, we see from system (39) that each classical solution should satisfy the following requirements which is the corresponding value of (∂ 0 Ξ)| Γ in the (t, r) plane. Therefore, we solve system (39) with the following boundary conditions: for r ∈ [r 1 , r 2 ]. It is easily checked by (32) and (33) that for some constant positive ε 0 . Then Problem 1.1 can be reformulated into the following new problem in the hodograph plane.
Then the boundary conditions (40) are corresponding to By performing a direct calculation, system (39) changes to where and the detailed expressions of b i (t, r) (i = 1, 2, 3, 4) are listed below. The functions b 1 (t, r) and b 2 (t, r) are Moreover, the eigenvalues of system (44) are We define a region in the plane (t, r) wherer 1 (t) andr 2 (t) are smooth functions satisfyingr 1 (0) = r 1 ,r 2 (0) = r 2 and r 1 (t) <r 2 (t) for t ∈ [0, δ]. The functionsr 1 (t) andr 2 (t) will be determined in the iteration below. Then Problem 3.1 is equivalent to the following problem.
Problem 3.2. Assume (41) holds. We look for a classical solution to system (44) with homogeneous boundary condition (43) in the region D δ for some constant δ > 0.

3.2.2.
A weighted metric space. We establish the local existence of classical solutions to the homogeneous boundary value problem (44) (43) in a weighted metric space inspired by Zhang and Zheng [47]. First we give the definition of admissible functions and strong determinate domain to system (44).

YANBO HU AND TONG LI
Definition 3.2 (Strong determinate domains). We call D δ defined in (46) is a strong determinate domain for system (44) if for any admissible vector function F = (f 1 , f 2 , f 3 , f 4 ) T and for any point (τ, z) ∈ D δ , the curves r i (t; τ, z)(i = 1, 2, 3, 4) stay inside D δ for all 0 ≤ t ≤ τ until the intersection with the line t = 0.
Moreover, we use the notation S M δ to denote the function class which incorporates all continuously differentiable vector functions F = (f 1 , f 2 , f 3 , f 4 ) T : D δ → R 4 satisfying the following properties: where · ∞ denotes the supremum norm on the domain D δ . It is obvious that S M δ is a subset of W M δ and both S M δ and W M δ are subsets of C 0 (D δ , R 4 ). For any One can check that (W M δ , d) is a complete metric space, while the subset (S M δ , d) is not closed in the space (W M δ , d). The existence theorem for Problem 3.2 can be stated as follows.
Theorem 3.1. Let conditions (41) be fulfilled and D δ0 be a strong determinate domain for system (44). Then there exists positive constants δ ∈ (0, δ 0 ) and M such that the problem (44) (43) admits a classical solution in the function class S M δ . 3.2.3. The proof of Theorem 3.1. We use the fixed-point method to show Theorem 3.1. The proof is divided into four steps. In Step 1, we construct an integration iteration mapping in the function class S M δ by linearizing the differential equations (44). In Step 2, we establish a series of a priori estimates for b i and λ i (i = 1, 2, 3, 4). In Step 3, we demonstrate the mapping is a contraction by employing the above estimates, which implies that the iteration sequence converge to a vector function in the limit. Finally, in Step 4, we show that this limit vector function also belongs to S M δ .
Step 1 (The iteration mapping). Let vector function (g,h,x,ỹ) T (t, r) ∈ S M δ . We consider the linearized system of (44) where λ i (t, r) and b i (t, r) (i = 1, 2, 3, 4) are given in (45) and (44), respectively, but with (g,h,x,ỹ) replacing ( G, H, X, Y ). The integral form of (49) is where (τ, z) is any point in a strong determinate domain D δ for system (44), r i (t; τ, z) (i = 1, 2, 3, 4) are defined as in (47). Thus system (51) determines an iteration mapping T : Obviously, the solving Problem 3.2 is equivalent to establish a fixed point for the mapping T in the function class S M δ .
Step 2 (Estimates of coefficients in S M δ ). For further convenience, we hereinafter derive a series of estimates about b i and λ i (i=1,2,3,4). Throughout the paper, the notation K will denote a constant depending only on ε 0 , κ, the bounds of h, g and the C 3 norms of G 0 , H 0 , a 0 , a 1 , which may change from one expression to another.
Furthermore, it is easily seen by the definitions of h, g, a 1 , ψ, φ and (41) that there exists a small constant δ 0 < 1 such that there hold The region D δ0 is defined as in (46). From above and (53), there exists a constant δ 1 ≤ δ 0 such that for (t, r) ∈ D δ1 which mean that the denominators in system (49) away from zero. We next estimate b 3 , and b 1 , b 2 , b 4 can be discussed analogously. Denote b 3 as where t, 0 + a 1 t), Due to (53), it follows that Inserting (56) into (55) and applying (54) yields Moreover, we differentiate b 3 with respect to r to get subsequently, Making use of (53)-(54) and (56)-(57), we obtain and By direct calculations, one derives Putting (60) into (58) and (59) gives Repetition of the same arguments for b 1 , b 2 and b 4 leads to For further use, we provide here the estimates about the eigenvalues λ i . By (45) and (53)-(54), it suggests that Moreover, we have the estimates for ∂ r λ 1 and ∂ rr λ 1 as follows Similar estimates also hold for the eigenvalues λ 3 and λ 4 . Summing up (57) and (61)-(64), we have the following a priori estimates for i = 1, 2, 3, 4.
Step 4 (Properties of the limit function). Notice that (S M δ , d) is not a closed subset in the complete space (W M δ , d), we need to confirm that the limit vector function of the iteration sequence { F (n) }, defined by F (n) = T ( F (n−1) ), is in S M δ . This follows directly from Arzela-Ascoli Theorem and the following lemma. Proof. Suppose that (g,h,x,ỹ) T ∈ S M δ . It follows by Lemma 3.1 that ( G, H, X, Y ) T = T (u, v, w) T also in S M δ . The proof of this lemma is separated into three steps.