The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas

We study the regularity of sonic curves to a two-dimensional Riemann problem for the nonlinear wave system of Chaplygin gas, which is an essential step for the global existence of solutions to the two-dimensional Riemann problems. As a result, we establish the global existence of uniformly smooth solutions in the semi-hyperbolic patches up to the sonic boundary, where the degeneracy of hyperbolicity occurs. Furthermore, we show the C^1 regularity of sonic curves.


Introduction
The purpose of this paper is to provide the regularity of sonic curves arising from a two-dimensional Riemann problem governed by a self-similar nonlinear wave equations. Let density be ρ, velocity (u, v), the pressurep be given as a function of ρ. From the wellknown 2-D compressible Euler system for isentropic flow, when the flow is irrotational and the nonlinear velocity terms are ignored, we can derive the following nonlinear wave system ρ t + m x + n y = 0, m t +p x = 0, (1.1) n t +p y = 0, where (m, n) = (uρ, vρ) are momenta. We refer the readers to [4,14,15] for more information on this system. In this paper we are interested in the 2-D Riemann problem of system (1.1) with Chaplygin gas equation of statep = −1/ρ. The Chaplygin gas, which can be used to depict some dark-energy models in cosmology, has been widely studied. Recently, some interesting and important results have been obtained, especially for the Riemann problem. We refer to [7,8,18,23,28] and the references cited therein for the related results.
For decades, there have been wide and intensive developments in the Riemann problems of conservation laws, in particular the 2-D compressible Euler system and its simple models [21,22]. Such models are the unsteady transonic small disturbance (UTSD) equations [1][2][3], the pressure gradient system [11,17,19,26,31], the potential flow [5,6,10,16], and the nonlinear wave equations [4,14,15,27] and so on. For Riemann problems in two-dimensional flow, those governing equations become quasilinear and mixed types. The type of the flow in the far-field is hyperbolic, while near the origin the type is mixed. The sonic lines, Four semi-hyperbolic patches, marked by the short and light curves and overlapped by some bold ones, in the interaction of two forward and two backward rarefaction waves for the Euler system with a gas constant γ = 1.4. The five closed curves at the center are contour curves of the pseudo-Mach number which mark the subsonic region; bold curves originating from the boundaries are characteristics; the short and light curves in the semi-hyperbolic regions are also characteristics. The solution has two axes of symmetry (Courtesy of Glimm et. al. [13]). located between the elliptic part and the hyperbolic part, are important to obtain the global existence of solutions to the mixed type problem. In this paper we consider the presence of the semi-hyperbolic patches with a transonic shock and a sonic curve in a simple wave environment. The regions identified by the existence of a family of characteristics that start on sonic curves and end on transonic shock curves are called semi-hyperbolic patches. The study on this region was initiated in [26] via the pressure gradient system and is being continued in [20,29,30]. These regions are different from the typical fully hyperbolic regions. As in the result of [13], in several numerical calculations, we frequently encounter this kind of regions in several configurations in the two-dimensional Riemann problems of the Euler system as well as its simplified models ( Figure 1). One of the examples for semi-hyperbolic patches happens when air is accelerated in a planar tunnel over a small inward bulge on one of the walls [9]. A global solution to the boundary value problem of the Riemann problem of a hyperbolic system can be patched together by pieces along characteristic lines, sonic curves, shock waves or other natural boundaries. We believe that it is helpful and essential to understand these patches of solutions locally before resolving the Riemann problems globally.
In [12], Hu and Wang have studied the semi-hyperbolic patches of solutions to the 2-D nonlinear wave system for Chaplygin gas state. They have constructed the global existence of solutions by solving a Goursat-type boundary value problem which has a sonic curve as a degenerate boundary, and verified that the sonic curve is Lipschitz continuous. However, the smoothness of the sonic curve and the exact behavior of solutions near the sonic curve are still open. The main difficulty is the degeneracy of hyperbolicity near the sonic lines. In several numerical simulations, we can see that the demarcation line between hyperbolic regions and elliptic regions is decomposed of sonic curves and a transonic shock. Also the pressure seems to change smoothly across the sonic curves. Hence, we mainly focus on the study of the solutions' precise behaviors near sonic curves, and expect better regularity of sonic curves. As a result, we prove that the semi-hyperbolic patches are uniformly smooth up to the sonic curves, and the sonic curves are C 1 continuous.
Here (1) and (4) denote the constant states of p = p 1 and p = p 4 , respectively, and R 14 = −η is a planar wave. Arc CD is an envelope and the region of OACD is a subsonic region. We are interested in the solutions in a curvilinear triangle ABC and the regularity of the sonic curve AC for the further research in the whole region of OABD.
Using this decomposition, the existence of solutions to (1.6) can be obtained in the hyperbolic region even if we cannot get the Riemann invariants [12,20,26]. Letting At this point, let us give a short description of the problem and known results. In the self-similar coordinates, two constant states (p 1 , m 1 , n 1 ) and (p 4 , m 4 , n 4 ) are given with p 1 < p 4 < 0, and a planar wave connects these two constant states in the domain {ξ > 0, η > 0}. Let the point (0, −p 1 ) be A, which is a sonic point satisfying ξ 2 + η 2 = p 2 . We draw a positive characteristic curve defined by (1.3) passing A in the planar wave, and intersecting the bottom boundary of the planar wave at B. Next we draw a negative characteristic defined by (1.3) from B toward the subsonic region, and let C be the ending point of the negative characteristic on the sonic curve. Upon this configuration in Figure 2, if we assume that the negative characteristic BC is given, then the region ABC bounded by two characteristics AB and BC forms a semi-hyperbolic region, and AC is a sonic curve. Here, the curve BC serves as a spine to support the whole patch.
Let W := (R, S, p), and we also denote the arcs AB and BC by Γ + and Γ − , respectively. For given boundary conditions on Γ ± , the global existence and a priori C 1 estimate of W in the domain ABC have been obtained in [12] as follows: [12]). Assume that two constant states (p 1 , m 1 , n 1 ) and (p 4 , m 4 , n 4 ) are given with p 1 < p 4 < 0 and a planar wave connects these two constant states in the domain ξ > 0. For a given negative characteristic curve Γ − , there exists a classical solution W to the Goursat boundary value problem of (1.6) in the whole domain ABC with a Lipschitz continuous sonic boundary AC if p 4 is close enough to p 1 . Furthermore, ∂ + p and ∂ − p are strictly negative if the boundary data of ∂ + p, ∂ − p on Γ ± are strictly negative, and ∂ + p = ∂ − p on the sonic curve. And ∂ + p, ∂ − p are uniformly bounded in the whole semi-hyperbolic region ABC.
In fact, we do not know exactly how far we can extend the C 1 solution in Theorem 1.1 towards the sonic boundary. In this paper, we improve the results of [12] in the sense that the solutions in the semi-hyperbolic patches are uniformly smooth up to the sonic curve, where the degeneracy of hyperbolicity occurs, and that the sonic curve is C 1 continuous. In fact, the smoothness of sonic curve is a necessary requirement for the global existence of solutions in the region of OABD in Figure 2 for the future research. The main theorem is as follows. To overcome the difficulty of degeneracy, we will employ a new different coordinate system (r, t) with t = r + p(r, θ) in Section 2. Under the new coordinates, we derive new systems for (R, S). The proof of the main theorem is based heavily on the uniform bounds of t δ R r and t δ S r for any 1 < δ < 2. In the following section, we will prove the above main theorem using the bootstrap method to get the uniform estimates as in [19,29]. Especially upon the method of [29], we show that a global smooth solution in [12] is extended up to the sonic boundary by obtaining the convergence rate of ∂ + p and ∂ − p near the sonic curve. With those estimates and the convergence rate, we are able to prove ∂ + p and ∂ − p are uniformly continuous on the sonic curve, and furthermore, the sonic curve is C 1 continuous.
For recent related progresses on the classical solution near the sonic lines for 2-D the pressure gradient systems and Euler systems, we would like to mention the work of Zhang and Zheng [30] first. In [30], for any given smooth curve as a sonic line with any specified non-tangential derivative of the speed of sound, they have constructed classical solutions from the sonic line towards the supersonic side by establishing a convergent iteration scheme. On the contrary, in [29] Q. Wang and Y. Zheng solved a Goursat problem and considered the regularity of solutions from the supersonic domain towards the sonic curve. They considered the problem from a different aspect but also hoped it can be used to build a transonic solution that bridges between a semi-hyperbolic patch to an elliptic domain. For the two-dimensional Euler system, M. Li and Y. Zheng [20] studied the semi-hyperbolic patches of solutions. In a forthcoming paper [25], we will consider the regularity of sonic curves for the Euler system.

2.1.
Derivation of New Systems. We introduce the new coordinates (r, t), where t = r + p(r, θ) and r is a radius-component in the polar coordinates. Then {t = 0} flattens the sonic boundary, and We note that λ −1 goes to zero as t → 0.
The sonic line is defined by the limit of the following level curves {(r, θ) : r+p(r, θ) = ε}. Let such level curves be θ = θ ε (r). Then Here, p θ is negative and bounded in the whole domain. However, where R − S and λ −1 vanish simultaneously as t → 0. It seems necessary to know the rate of vanishing of these two components, more precisely the vanishing rate of R − S with respect to t, in order to know the regularity of the sonic line.
In the coordinate system (r, t), we have Since λ −1 goes to zero as t → 0, we note that S − λ −1 = 0 and R + λ −1 = 0 in the (r, t)-plane when t is sufficiently small. Letting where V = 0 on the sonic line, we have Furthermore, we define Then Here, and by direct calculations, Note that h → 0 as t → 0. Furthermore, We note that functions E, f j for j = 1, 2, 3 are all bounded in bounded regions. Also we can obtain by direct calculations that ∂ + ∂ + S − ∂ − ∂ + S := tg 1 (r, t)R r + tg 2 (r, t)S r + t 2 g 3 (r, t), We note that functions g j for j = 1, 2, 3 are all bounded in bounded regions, too. Since R, S are uniformly bounded and negative in the domain Ω and R = S on the sonic line, we can obtain h(r, t), f j (r, t), g j (r, t)(j = 1, 2, 3) are also uniformly bounded in the small subdomain D. Let |f j (r, t)|, |g j (r, t)|, j = 1, 2 , (2.6) and K 3 = min (r,t)∈D |E|. We can see that K 1 tends to zero and K 2 , K 3 have positive bounds as t B → 0 + . If t B = 0, the domain D degenerates into a point on the sonic line, and two constants K 2 and K 3 satisfy that 2K 2 = |E(r, 0)| = K 3 with K 3 > 2K 2 δ −1 for any δ ∈ (1, 2). Thus, we can let K 3 > 2δ −1 K 2 e 2K 1 t B and K 1 < K 2 by taking suitable small t B .
Let Ω(r, 0) denote a bounded domain surrounded by a positive characteristic and a negative characteristic starting from (r, 0) and the characteristics B C and B A . For any (r, t) ∈ Ω(r, 0), let a and b be intersections of the negative characteristic and the positive characteristic through (r, t) with the boundaries B C and B A , respectively. Let where t a and t b are t-coordinates of points a and b, respectively. As discussed before, M 0 is well-defined and uniform in the domain Ω(r, 0), but depending on δ. We can prove the following result.

7)
where M > 0 is depending on δ. That is, t δ R r and t δ S r are uniformly bounded in the domain Ω(r, 0). Proof. For a fixed ε ∈ (0, t B ), let in Ω ε with M ε depending on ε. If for any ε ∈ (0, t B ), we have M ε ≤ M 0 , then (2.8) hold in the whole domain Ω(r, 0). If not, then there exists ε 0 ∈ (0, t B ) making M ε 0 > M 0 . So we let For (r ε 0 , ε 0 ) ∈ Ω(r, 0), we denote the two characteristics passing through (r ε 0 , ε 0 ) by r a + and r b − , where a, b are the intersection points with B C and B A respectively. Here a = a ε 0 and b = b ε 0 are dependent on ε 0 and for convenience we omit the subscript ε 0 . From the equation we have Integrating this along the negative characteristic passing through (r ε 0 , ε 0 ), we get Thus, from (2.3) and (2.5) we obtain < 0, which is true by the definition of M 0 and the fact of M ε 0 > M 0 . Thus we obtain On the other hand, Thus we obtain

So we have a strict inequality such that
on the line segment t = ε 0 . Similarly, integration of the equation along the positive characteristic makes Also by the definition of H, we get the following estimate According to (2.9) and (2.10), |t δ R r | and |t δ S r | do not have the maximum values on the line segment t = ε 0 , that is, the maximum happens on ε 0 < t ≤ t B . Hence we conclude that this argument also holds in a larger domain Ω ε with ε < ε 0 . Repeating this process, we can extend the domain larger and larger until it reaches the whole domain Ω(r, 0). Thus, we find a constant M depending on δ only and satisfying (2.7). The proof is completed.
Next, for any (r, 0) ∈ A C , we can take a small open interval (r − r 0 , r + r 0 ) ⊂ A C . Suppose the negative and positive characteristics passing through points P (r − r 0 , 0) and Q(r + r 0 , 0) intersect the boundaries B C and B A at P and Q , respectively. Then we have the same estimates in the region of B P P QQ in the following lemma. The proof of it is very similar to that of the previous lemma, so let us omit the proof.  Proof. Let (r, 0) ∈ AC. From (2.2) we have It is obvious that lim t→0 + l 1 (r, t) = 1, and l 2 (r, t) is bounded in the domain D. Moreover, we note that also tends to zero as t → 0 + since V → 0 as t → 0 + . Let Now we rewrite (2.11) in the form of Integration of the above equation from t to t B yields that Thus that is, Because of the arbitrariness of t and the continuity of V , we obtain that |V (r, t)| ≤ M t holds for any t ∈ [0, t B ].
Thus we see that the two derivatives ∂ + p and ∂ − p approach a common value on the sonic curve with a rate of O( √ r + p). Furthermore, we have the following result.
Lemma 2.4. R, S, and V (r, t)/t is uniformly continuous in the domain ABC including the sonic line.
Proof. Let us choose two sonic points P 1 (r 1 , 0), P 2 (r 2 , 0) ∈ AC. Then we draw a positive characteristic γ + passing through P 2 and a negative characteristic γ − passing through P 1 . Let Q(r b , t b ) ∈ Ω be an intersection point of γ + and γ − . We note that Q approaches a sonic point as |r 2 − r 1 | → 0. From (2.1), we see that (2.14) Since V (r, t)/t is uniformly bounded, we have |∂ − R|, |∂ + S| ≤ L for some constant L. Now integrating the first equation of (2.14) along γ − from P 1 to Q gives Similary the second equation of (2.14) along γ + from P 2 to Q gives uniformly continuous in the whole domain, θ ε (r) is uniformly continuous as well. Hence θ (r) is continuous on the sonic line. That is, the sonic line is C 1 continuous.