THE REGULARITY OF A DEGENERATE GOURSAT PROBLEM FOR THE 2-D ISOTHERMAL EULER EQUATIONS

. We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly C 1 , 16 up to the degenerate sonic boundary and that the sonic curve is C 1 , 16 .

1. Introduction. We consider the two-dimensional isothermal compressible Euler equations    ρ t + (ρu) x + (ρv) y = 0, (ρu) t + (ρu 2 + p) x + (ρuv) y = 0, (ρv) t + (ρuv) x + (ρv 2 + p) y = 0, where ρ is the density, (u, v) is the velocity and p is given by the isothermal pressure law Comparing with the isentropic gas p(ρ) = ρ γ (γ > 1) studied in [20], the isothermal gas (2) has its own unique features and challenges. For example, the sonic speed is uniformly unit in the isothermal case which prevents us from getting some important relations in the isentropic case. We are interested in the semi-hyperbolic patch problems originated from the two-dimensional Riemann problem for (1) (2). In the previous paper [7], Hu, Li and Sheng established the global existence of smooth solutions in a semi-hyperbolic region to (1) (2) by solving a degenerate Goursat problem which has a sonic curve as a boundary. In the current paper, we address the regularity of solutions near the sonic boundary and regularity of the sonic boundary.
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 appears frequently in the two-dimensional Riemann problem for the compressible Euler equations in gas dynamics, see [12,28,5]. In addition, semi-hyperbolic patches may also occur in the transonic flow over an airfoil [3,2], 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 semihyperbolic patch problems provides us important information about the parabolic degenerate curves. The results on the semi-hyperbolic patches make it possible to construct global solutions for general nonlinear mixed-type equations in future work. The analysis of the semi-hyperbolic patch problems was initiated by Song and Zheng [21] for the pressure-gradient system. In [16], Li and Zheng investigated the semi-hyperbolic patch problems of the two dimensional isentropic Euler equations. A different viewpoint to study this kind of problems for the pressure-gradient system was proposed by Song [19]. In [24], Wang and Zheng studied the regularity of the semi-hyperbolic patch problems for the pressure-gradient system. We also refer the reader to [6,25,26,27] for the construction of classical sonic-supersonic solutions to the Euler system. The above works [7,6,21,16,19,24,25,26,27] on building smooth solutions are based on the idea of characteristic decomposition which is a powerful tool revealed in [4], see, e.g., [1,9,10,11,13,14,15,18] for more applications.
In the present paper, we show that the smooth solution of the semi-hyperbolic patch problem of the isothermal compressible Euler equations (1) (2) is uniformly C 1, 1 6 up to the degenerate sonic boundary and the sonic curve is C 1, 1 6 -continuous. In [20], the authors explored such regularity problems for the isentropic Euler system. They verified that there exists a global solution up to the sonic boundary and that the sonic boundary has C 1 -regularity. The previous work [20] relies on the known result from [16] that∂ + c +∂ − c = 0 on the sonic curve. However, for the current isothermal case (2), such a relation is not available. To overcome the difficulty, we introduce a novel set of change variables to show our regularity results without using any relations on the sonic curve, which is the main innovation of this article. Moreover, these change variables allow us to establish higher regularity of the solution and of the sonic boundary by using the bootstrap method inspired by [20]. The analysis in this paper was also applied to improve the regularity results for the isentropic Euler systems, see [8].
The rest of the paper is organized as follows. Section 2 is devoted to describing the problem and presenting the main results. In Section 3, we provide the characteristic decompositions of the inclination angles and show some property of solutions in the self-similar plane. In Section 4, we discuss the properties of solutions in a partial hodograph coordinates. Finally, we complete the proof of our main theorem in Section 5.
2. Set up of the problem and the main results. In this section, we set up the semi-hyperbolic patch problem of (1) (2) and then present the main results of this paper.
We first provide some preliminaries for (1) with (2). System (1) admits selfsimilar solutions. We introduce the self-similar variables (ξ, η) = (x/t, y/t) and where (U, V ) = (u − ξ, v − η) is the pseudo-velocity. Assume the flow is ir-rotational with u η = v ξ , then we can decouple a subsystem in terms of (u, v) from (3) The density can be restored from the momentum equations of (3) or from the Bernoulli law where φ is the pseudo-velocity potential.
It is easy to calculate that the positive/negative eigenvalues of (4) are from which we see that system (4) is of mixed-type: supersonic for The set of points at which U 2 + V 2 = 1 is called the sonic curve. Now we give a description of the problem that is investigated in this paper. Suppose that 1 ((ρ 1 , 0, v 1 ) T ) and 4 ((ρ 4 , 0, 0) T ) are two constant states satisfying ρ 1 > ρ 4 > 0, v 1 > 0. Let R 14 (η) be a planar rarefaction wave of (1) (2) in the selfsimilar coordinates (ξ, η), connecting these two constant states 1 and 4 , defined by which is a sonic point by (7). According to (6) and (8), the positive characteristic curve passing though A in R 14 is We set ξ B = √ 1 − e −v1 ∈ (0, 1) and denote the point (ξ, η) = (ξ B , 1) by B which is the intersection point of Γ + A with the bottom boundary of R 14 , see Fig. 1. From (8), the boundary data (ρ, u, v) = (ρ,ū,v)(ξ) on BA are Let ξ C ∈ (0, ξ B ] be a real number and BC : η = ϕ(ξ) (ξ C ≤ ξ ≤ ξ B ) be a smooth curve. We assign the boundary data for (ρ, u, v) on BC, (ρ, u, v)(ξ, ϕ(ξ)) = (ρ,û,v)(ξ) such that BC : for any ξ ∈ [ξ C , ξ B ]. The last condition of (10) is the compatibility condition of boundary data on BC with system (4). Moreover, by (6), the boundary conditions (10) imply that the smooth curve BC is a smooth convex negative characteristic curve and the point C is a sonic point. The semi-hyperbolic patch problem is described as follows: assuming the boundary conditions (9) and (10) hold, we look for a smooth solution in curvilinear triangle ABC and a simple wave solution in curvilinear triangle CBD, where AC is sonic curve while CD is an envelope, see Fig. 1. The existence of smooth solutions to the semi-hyperbolic patch problem in the region ABC was verified by Hu, Li and Sheng [7]. In this paper we are mainly concerned with the regularity of solutions near the sonic curve AC. The main conclusion of the paper can be stated as follows.
We need to overcome the major difficulty arising from the hyperbolic degeneracy of the Euler system at the sonic curves. The key is to prove that the term (R − S)/ cos δ is uniformly bounded up to the sonic boundary AC, where the variables R, S and δ are defined in Section 3. For the isentropic case studied in [20], the authors took the relation R − S = 0 on the sonic curve as a known result from [16]. In this paper, such a relation is not available. In order to acquire our regularity theory, we introduce a partial hodograph transformation and a novel change variables ( G, H), which play a crucial role in our analysis, see (50). This change variables allows us to prove the uniform boundedness of (R − S)/ cos δ by employing the bootstrap method. Finally, after returning to the self-similar variables, we establish the uniform regularity of smooth solutions of system (3) up to the sonic boundary AC.
3. Properties of solutions in the self-similar plane. In this section, we deliver the characteristic decompositions for the inclination angles of characteristics and then show an important property of solutions.
By standard manipulation, system (4) can be turned into the characteristic form Following [14] and [7], we introduce and denote Here σ is called the pseudo-streamline angle and δ is called the pseudo-Mach angle. Then (u, v) can be expressed in terms of σ, δ and the sonic curve reads now {(ξ, η) : sin δ(ξ, η) = 1}. Introducing the scaled characteristic fields then Now, we obtain the characteristic form in terms of (σ, δ) from (11) and (14)     ∂ For later calculations, we present the equations for (α, β)

YANBO HU AND TONG LI
and the characteristic decomposition of (α, β) The detailed derivation of equations (17) and (19) can be found in [7].
To deal with our problem, we further introduce two new variables By a direct calculation, one has Making use of (19), we can derive the characteristic decomposition of (R, S) which is considerably more concise than equation (19) and plays an important role in our analysis. Here we use the scaled characteristic directions∂ − and −∂ + since they are both pointing away from the sonic boundary AC.

3.2.
Boundedness of (R, S). In this subsection, we show the boundedness of (R, S) defined in (20). We first present the properties of solutions of (17) established in [7].
Based on Proposition 3.1, we now establish the boundedness of (R, S). For any point (ξ, η) in the region ABC, we draw the positive and negative characteristic curves up to the boundaries BC and BA at points B 2 and B 1 , respectively, see where m = min{min Proof. It is easy to see by the definitions of (R, S) and (24) that R > 0 and S > 0 in the region ABC. Denote F = (R − S)/ sin 2 (2δ) − 2 sin(2δ). We next split the proof into two steps.
Step I. We show that R and S have a uniform upper bound in the whole region ABC in (25). The proof is further divided into two cases. Case 1. For any point P ∈ ABC, if S ≥ R entirely in the region P B 1 BB 2 , then by (24) we have F ≤ 0 and Now from any point P 1 ∈ P B 1 BB 2 , we draw a positive characteristic curve up to the boundary BB 2 at a point P 0 . Then noticing the direction of −∂ + pointing toward to BC and using (26) yields Case 2. If there exists a point, say P 1 , in P B 1 BB 2 such that R > S at P 1 , then from the point P 1 , we draw a negative characteristic curve, called Γ − 1 , up to the boundary BB 1 at a point P * Integrating the above from P 1 to P * 1 and noticing the direction of∂ − gives If R < S for some point on Γ − 1 , then there exists a neighborhood N 1 of P 1 so that R > S holds for every point in It follows by integrating the above from P 1 to Q 1 that From the point Q 1 , we now draw a positive characteristic curve, called Γ + 1 , up to the boundary BB 2 at a point Q * 1 . We recall F | Q1 < 0 and discuss it in the following two subcases. Subcase 2a. If F ≤ 0 always holds on Γ + 1 , then we find by the equation for S that −∂ + S ≥ 0 and S| Q1 ≤ S| Q * 1 ≤ M, which together with (27) finishes the proof. Subcase 2b. If, F > 0 holds for some point on Γ + 1 , then there exists a neighborhood N 2 of Q 1 so that F < 0 holds for every point in N 2 ∩Γ + 1 and F = 0 at P 2 := ∂N 2 ∩Γ + 1 . Moreover, we see that −∂ + S ≥ 0 on N 2 ∩ Γ + 1 , and then S| Q1 ≤ S| P2 , which combined with (27) and the definition of F leads to Next, from P 2 , we draw a negative characteristic curve, called Γ − 2 , up to the boundary BB 1 at a point P * 2 . There must hold where Q 2 is a point on Γ − 2 such that R > S on P 2 Q 2 ∩ Γ − 2 and R = S at Q 2 . If (29) holds, then we combine it with (28) to obtain by using the property of positive characteristics otherwise, we put (30) into (28) to find and repeat the above processes to complete the proof.
Step II. We next show the lower bound of R and S in (25). The proof is similar to Step I. For any point P (ξ, η) ∈ ABC, if S ≥ R entirely on the whole region P B 1 BB 2 , then F ≤ 0 and one has from which we find that If R > S holds on the whole segment P B 2 , it follows by the second equation of (22) that −∂ + ln S ≤ 2 sin(2δ) ≤ 2. Thus Otherwise, we choose a point P on P B 2 such that R = S at the point P and R > S on the segment P P . Then R| P ≥ S| P ≥ S P e −2(ξ P −ξ P ) = R P e −2(ξ P −ξ P ) .
From the point P , we draw the negative characteristic curve up to the boundary BB 1 and repeat the processes as Subcases 2a and 2b in Step I to complete the proof.
Remark 1. Lemma 3.1 shows that the lower bound of (R, S)(ξ, η) in (25) is independent of the distance from the point (ξ, η) to the sonic boundary AC. This fact will be used in the analysis of the paper.

4.
Solutions in the partial hodograph coordinates. Let us first analyze what we need to obtain the regularity of the sonic boundary AC. Since the sonic curve is the set of points at which sin δ = 1, we then consider the level curves Here ε is a small positive constant. According to (16), (18) and (21), it follows that Moreover, we have which along with (25) arrives It is obvious that, in order to obtain the regularity of AC, we need to establish a uniform upper bound for the term W := (R − S)/ cos δ.

The new equations.
We introduce a transformation (ξ, η) → (z, t) by defining where φ is the pseudo-velocity potential defined in (5). Thanks to (21) and (25), the Jacobian of this transformation is in the whole region ABC.
In terms of this new coordinates (t, z), one has Inserting the above into (22) yields a new closed system for (R, S) under the coordinates (t, z) Here we still use the notations (R, S)(z, t) to denote (R, S)(ξ(z, t), η(z, t)). From (25), we set R = 1/R and S = 1/S, which are uniformly bounded up to the sonic boundary. Then system (37) changes to Denote the eigenvalues of (38) as and Making use of the following commutator relation [13] we get By a tedious but straightforward calculation, we deduce where f (z, t) = 3t Furthermore, one arrives at where where Hence we obtain by (40) Putting the above and (43) into (41)-(42) gives that is, wherẽ It is clear by Lemma 3.1 that the functionsg 1 andh 1 are uniformly bounded near t = 0. Moreover, we denotẽ It is easily seen by Lemma 3.1 thatg 2 → −1 andh 2 → −1 as t → 0 + . We comment that here we only need the uniform boundedness of R and S. Thus (46)-(47) can be rewritten as In addition, in order to obtain the explicit limit constructs of coefficients for equations, we recall (39) and (40) to introduce Then we finally have 4.2. Properties of solutions in partial hodograph plane. With the help of (51)-(52), we now analyze the properties of solutions near the line t = 0, i.e., the sonic boundary AC.
Denote the region ABC in the self-similar (ξ, η) plane in Fig. 1 by A B C in the (z, t) plane. Let (z, 0) be any point on the degenerate line A C and let t P be a small positive number such that the point P (t P ,z) stays in the domain A B C . We use the notations z + (P ) and z − (P ) to represent the λ + and λ − -characteristics passing through the point P and intersecting A C at P 1 and P 2 , respectively. Since the functionsg 1 ,h 1 are uniformly bounded near t = 0 andg 2 → −1,h 2 → −1 as t → 0 + , then for any constant ν ∈ (0, 1], we can choose t P < 1 small enough such that hold in the whole domain P P 1 P 2 . Denote Ω ν (z) the domain bounded by P P 1 , P P 2 and the positive and negative characteristics starting from (z, 0). For any (z, t) ∈ Ω ν (z), we denote, respectively, a(z a , t a ) and b(z b , t b ) the intersection points of the λ − and λ + -characteristics through (z, t) with the boundaries P P 1 and P P 2 , see | G(z a , t a )| + 1, max which is well-defined and uniformly bounded in the domain Ω ν (z). Now we are ready to prove the following key lemma.
Proof. We apply the bootstrap method used in [20] to prove the lemma. For a fixed ε ∈ (0, t P ), we denote and K ε = max Ωε {|t ν G|, |t ν H|}. If for any 0 < ε < t P , one has K ε ≤ K, then (54) holds. Otherwise, we assume there exists ε 0 ∈ (0, t P ) such that K ε0 > K. For (z ε0 , ε 0 ) ∈ Ω ν (z), we use the notation z a − (t) (z b + (t)) to represent the negative (positive) characteristic curve passing through the point (z ε0 , ε 0 ) and intersecting P P 1 ( P P 2 ) at point a(z a , t a ) (b(z b , t b )). We integrate (51) along the negative characteristic from t(≥ ε 0 ) to t a and apply the definition of K to obtain Hence, on the line segment {t = ε 0 } ∩ Ω ε0 , we have a strict inequality A similar argument for the equation of H shows By combining (56) and (57), we conclude that the maximum values of |t ν G| and |t ν H| in the domain Ω ε0 can only happen on ε 0 < t ≤ t P . This assertion also holds in a larger domain Ω ε , where ε < ε 0 . We repeat the above process and then can extend the domain larger and larger until the whole domain Ω ν (z). The proof of the lemma is completed. Now, for any point (z, 0) ∈ A C , let µ be a small constant such that (z − µ,z + µ) ⊂ A C . Denote Q * 1 (Q * 2 ) the intersection point of the negative (positive) characteristic through Q 1 := (z − µ, 0) (Q 2 := (z + µ, 0)) with the boundary P P 1 ( P P 2 ). By using a similar argument showing Lemma 4.1, we can extend the result (54) in the region Ω ν (z µ ) bounded by the boundaries P Q * 1 , Q * 1 Q 1 , Q 1 Q 2 , Q 2 Q * 2 and Q * 2 P .
We next prove the uniform boundedness of W := R− S t . Proof. It suffices to prove that the lemma holds near t = 0. For any point (z, 0) ∈ A C , we choose the domain P P 1 P 2 as before. By a direct calculation, one finds by (38) that where It is obvious that I 1 is uniformly bounded near t = 0. Furthermore, we can choose ν = 1/2 in Lemma 4.2 to obtain that I 2 is also uniformly bounded near t = 0. Thus, the uniform boundedness of W follows directly from (59).
Recalling the definitions of R and S and applying Lemma 3.1, we see that W = (R − S)/t is uniformly bounded in the whole domain A B C , that is, there exists a uniform constant K 1 such that Moreover, in view of (40) and (50), one arrives at According to (60) and (61), it is easy to check by the equations for R and S (37) that the two derivatives R t and S t are uniformly bounded in the whole domain A B C including the degenerate line t = 0. Therefore, the two functions R and S approach a common value on the degenerate curve AC with at least a rate of cos δ. Finally, we show the following lemma. Proof. Suppose first that (z 1 , 0) and (z 2 , 0) (z 1 < z 2 ) are any two points on the degenerate curve A C . Let (z m , t m ) be the intersection point of the positive characteristic z + and negative characteristic z − starting from (z 1 , 0) and (z 2 , 0) respectively. Recalling (37), the characteristic curves z ± are defined by

from which and (25) one has
for some positive constants K and K. Furthermore, it follows by (60) and system (37) that |∂ − R| and |∂ + S| are uniform bounded. Thus we integrate ∂ + S from (z 1 , 0) to (z m , t m ) and ∂ − R from (z 2 , 0) to (z m , t m ) to find for some uniform constant K 2 . Hence we have We now consider any two points (z 1 , t 1 ) and (z 2 , t 2 ) (z 1 ≤ z 2 , 0 ≤ t 1 ≤ t 2 ) in the region A B C . If z 1 = z 2 , we recall the uniform boundedness of R t to get Next we assume z 1 < z 2 and divide the problem into two cases: Case I. t 1 ≥ (z 2 − z 1 ). In this case, we choose ν = 1/2 in (58) and use (61) and the fact t 1 ≥ (z 2 − z 1 ) to obtain for some uniformly constant K. Case II. t 1 < (z 2 − z 1 ). In this case, we derive for a uniform constant K. In summary, we obtain that the function R(z, t) is uniformly C 1 3 continuous in the whole domain A B C . A similar argument leads to the uniform continuity of S(z, t).
To establish the uniform C 1 3 -continuity of W , we derive the equations ∂ − W and ∂ + W from (37) as follows: which indicate by Lemma 4.3 and (61) that |∂ − W | and |∂ + W | are uniform bounded. We reproduce the same argument as above for R to end the proof of the lemma.

5.
Proof of the main theorem. We now use the results of the previous section to complete the proof of Theorem 1, that is, we show that the solution (ρ, u, v)(ξ, η) in the region ABC is uniformly C 1, 1 6 up to the sonic boundary AC and the sonic curve AC is C 1, 1 6 -continuous. The proof is divided into four steps.