SEMI-HYPERBOLIC PATCHES OF SOLUTIONS TO THE TWO-DIMENSIONAL COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS

. We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of ﬂow patches appear frequently in transonic ﬂow problems. In order to use the method of characteristic decomposition to construct such a ﬂow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.


Introduction. The compressible magnetohydrodynamic (MHD) equations has the form
where ρ is the fluid density, p is the pressure, − → U is the velocity of the fluid, E = is the specific total energy, e = e(ρ, S) is the specific internal energy, S is the specific entropy, − → H is the magnetic field, and the constant µ is the magnetic permeability; see [1,19].
We assume that the magnetic field − → H = (0, 0, H), the velocity − → U = (u, v, 0), and the variable (u, v, p, ρ, H) is independent of the space variable z. Then the MHD equations (1) can be simplified as x + (ρuv) y = 0, ρv t + ρuv x + ρv 2 + p + µ 2 H 2 y = 0, ρE + µ 2 H 2 t + ρuE + up + µuH 2 x + ρvE + vp + µvH 2 y = 0, H t + Hu x + Hv y = 0. (2) From the first and the fifth equations of (2), we have which implies that H/ρ is a constant along each stream line. The relation (3) is essentially an expression of the frozen-in law. The Rankine-Hugoniot conditions for (2) are where [(·)] denotes the jump of the variable (·) across the discontinuity surface and (n t , n x , n y ) is a normal to the discontinuity surface. From the first and the fifth equations of (4) we have [H/ρ] = 0 (5) across the shock waves. According to (3) and (5), if H/ρ is a constant at the initial time then system (2) can be written as    ρ t + (ρu) x + (ρv) y = 0, (ρu) t + ρu 2 + p + κρ 2 x + (ρuv) y = 0, (ρv) t + (ρuv) x + ρv 2 + p + κρ 2 y = 0, where κ is a constant. We consider a fluid with the equation of state 2D Riemann problems, which refer to Cauchy problems with special initial data that are constant along each ray from the origin, are interesting and important problems. 2D Riemann problems usually allows us to consider the so-called selfsimilar solutions. The self-similar solutions are the solutions which depend only on the self-similar variables (ξ, η) = ( x t , y t ); see [2,32,34]. Then by self-similar transformation system (6) can be changed into the form where (U, V ) = (u − ξ, v − η) is called the pseudo-flow velocity.

SEMI-HYPERBOLIC PATCHES 945
The eigenvalues of system (8) are determined by which yields Here, w = √ c 2 + b 2 is the magneto-acoustic speed, b = √ 2κρ is the Alfven speed, and c = γρ γ−1 is the sound speed. Thus, system (8) is hyperbolic if and only if U 2 +V 2 > w 2 (pseudosupersonic) and elliptic-hyperbolic if and only if U 2 +V 2 < w 2 (pseudosubsonic).
The wave characteristics C ± are defined as the integral curves of C ± : dη dξ = Λ ± . The direction of the wave characteristics is defined as the tangent direction that forms an acute angle δ with the pseudoflow velocity (U, V ). By simple computation, we see that the C + characteristic direction forms with the pseudoflow direction the angle δ from C + to (U, V ) in the clockwise direction, and the C − characteristic direction forms with the pseudoflow direction the angle δ from C − to (U, V ) in the counterclockwise direction. By computation, we have in which q 2 = U 2 + V 2 . The C + (C − ) characteristic angle is defined as the angle between the C + (C − ) characteristic direction and the positive ξ-axis. We denote by α and β the C + and C − characteristic angle, respectively, where 0 < α − β < π. Let σ be the angle between the pseudoflow velocity and the positive ξ-axis. Then we have Therefore, the relations between (U, V, w) and (σ, δ, w) are sonic curve In transonic flow there are two types of sonic curves, one is called the Keldysh type and the other is call the Tricomi type. In pseudosupersonic flow region adjacent to a Keldysh type sonic curve, all the wave characteristics vanish tangentially into the sonic curve; see Figure 1(I). Keldysh type sonic curves can be happened in solutions to some types of 2D Riemann problems; see for example [3,7,21,33]. In pseudosupersonic flow region adjacent to a Tricomi type sonic curve, one family out of two families of wave characteristics start on the sonic curve; see Figure 1(II). Tricomi type sonic curves can be happened in the transonic flow over an airfoil and Guderley-Mach reflection; see [6,8,12,13,22,25,26,27,29]. Zheng et al. [14,23,34] call the pseudosupersonic flow patch adjacent to a Tricomi type sonic curve the semi-hyperbolic patch. Semi-hyperbolic patches also appear frequently in 2D steady transonic flow problems, see Coruant and Friedrichs [6].
In this paper, we are concerned with semi-hyperbolic patches of solutions to the 2D self-similar compressible MHD equations (8). We consider the following problem.
Problem. Let AB be a C + characteristic curve with the direction of A points to B, and let DA be a C − characteristic curve with the direction of D points to A. The data on AB and DA are given such that points B and D are pseudosonic. Construct a pseudo-supersonic solution of system (8) with the data on AB and DA in a region Ω bounded by AB, DA, and a sonic curve connecting D and B; see Figure 2. This problem is a Goursat-type boundary value problem, since AB and DA are characteristics. The main result is stated as Theorem 3.8 where we obtain the existence of classical pseudosupersonic solution up to a sonic boundary BD to Problem 1. This generalizes the result for the 2D compressible Euler equations for polytropic gases of Li and Zheng [14] to the MHD equations. In order to use the method of characteristic decomposition introduced by Li et al. [16,17] in investigating interactions of rarefaction waves of 2D compressible Euler equations, we derive a group of characteristic decompositions for system (8).
For semi-hyperbolic patches of solutions to some other type of hyperbolic conservation laws, the readers can see [9,10,11]. Song et al. [24,28] also studied the regularity of semi-hyperbolic patches near sonic lines. There are also some other ways to construct such a flow patch. Recently, Zhang and Zheng [30,31] construct semi-hyperbolic flow patches of solutions to 2D steady Euler equations and 2D pressure gradient equations by solving Cauchy problems with given data on the sonic curve.
The rest of paper is organized as follows. Section 2 is concerned with characteristic equations for (8). We derive several characteristic decompositions for the magneto-acoustic speed w and wave characteristic angles α and β. Section 3 is devoted to solve Problem 1.
Lemma 2.1. For the variable w, we have the following characteristic decompositions Proof. This lemma was proved in [4] by a method similar to that of [5,15]; we omit the proof here.
Corollary 1. The second order equations of w in homogeneous form: Proof. This corollary can be obtained by performing a direct calculation and simplification for (37) and (38).

Lemma 2.2.
For the characteristic angles α and β, we have the following characteristic decompositions where and Proof. In view of (28), (31) and (32), we get (45) Combining these with (37) we obtain (46) Due tō The right side of the equation (46) is equal to By a direct calculation, we get Moreover, Combining the above relations, we get (41). The decomposition (42) can be proved similarly.
3. Semi-hyperbolic flow patch. In this part we shall construct a semi-hyperbolic flow patch to the two-dimensional self-similar MHD equations (8).
(47) From (a1)-(a5) and (b1)-(b5) we know that problem (8), (47) is a Goursat problem with AB and DA as the characteristic boundaries. Moreover, from (a3) we know that the direction of the characteristic AB is from A to B, from (b3) we know that the direction of the characteristic DA is from D to A, and from (a1) and (b1) we know that B and D are sonic points; see Figure 2.
Proof. From (56) and (57) we havē which implies that the domain Ω ε is uniformly bounded with respect to ε. Therefore, by (13) and Lemmas 3.2 and 3.3 we can get the uniform C 0 norm estimate of the variable (u, v).
Due to the degeneracy caused by sonic curve, the classical approach to extend local solution to global solution does not work here. In what follows, we shall extend the local solution of the Goursat problem to global solution by solving many "small" Goursat problems in each extension step. Assume that the Goursat problem admits a C 1 solution on Ω ε where 0 < ε < π 2 − δ 0 . Let P Q and RP be respectively a C + and a C − characteristic curves in Ω ε . We prescribe (α, β, w) = α P Q , β P Q , w P Q on P Q, where α P Q , β P Q , w P Q and α RP , β RP , w RP are the value of the solution on P Q and RP , respectively. We then have the following theorem.
the Goursat problem (8), (63) admits a global C 1 solution on a curved quadrilateral domain bounded by P Q, RP , RT , and QT , where RT is the C + characteristic passing through R, QT is the C − characteristic passing through Q, and this solution satisfies δ 0 < δ < π 4 + δ0+ε 2 .
Proof. From Lemma 3.5 we have sup RP∂ + w sin 2 δ ≤ ε and sup P Q −∂−w sin 2 δ ≤ ε . Then by using a method similar to that of Lemma 3.5 we have that the solution of Goursat problem (8), (63) satisfies (48) and Thus, by (28) and (32) we have In what follows we shall show that when arc lengthes of P Q and RP are less than ν 0 , this solution satisfies δ < π 4 + δ0+ε 2 . We shall prove this by contradiction. Suppose that there exists a point I such that δ(I) = π 4 + δ0+ε 2 . The C + characteristic curve passing through I intersects with RP at a point I 1 , the C − characteristic curve passing through I intersects with P Q at a point I 2 ; see Figure 3(left). Through the point I 1 draw a straight line with the slope tan α(I), through the point I 2 draw a straight line with the slope tan β(I), the two straight lines intersect at a point H. By estimate (65) we have II 1 and II 2 lie in the triangle HI1I2 and the arc lengths of II 1 and II 2 are less than the sum of the arc lengthes of HI 1 and HI 2 and I 1 I 2 .
Thus, the arc lengthes of II 1 and II 2 are less than 6ν0 sin( π 2 −δ0−ε) . Then, by (48) we have and β(I) > β(I 2 ) − ε tan 2 ( π 4 + δ0+ε 2 ) Thus, we have δ(I) < π 4 + δ0+ε 2 , which leads to a contradiction. By a method similar to that of Lemmas 3.4 and 3.6 we can get the a priori uniform C 1 norm estimate of the solution to the Goursat problem (8), (63). Hence, by the theory of global classical solutions for quasilinear hyperbolic equations (cf. Li [18]) we can get this lemma. Proof. Let Q 0 = B ε , Q 1 , Q 2 , · · ·, Q n = D ε be n + 1 different points on the level curve δ = δ 0 + ε in turn. The C + characteristic curve through Q i intersects with the C − characteristic curve through Q i+1 at a point P i , where i = 0, 1, · · ·, n − 1; see Figure 3(right). Since∂ + δ > 0 and∂ − δ < 0, the level curve δ = δ 0 + ε is a non-characteristic curve. Hence, P i = Q i and P i = Q i+1 for any i = 0, 1, · · ·, n − 1. From (55) we know that when Q i and Q i+1 are sufficiently close the arc lengths of P i Q i and P i Q i+1 are less then ν 0 . Thus, by Lemma 3.7 we know that the Goursat problem for system (8) with P i Q i and P i Q i+1 as characteristic boundaries admits a unique C 1 solution on a curved quadrilateral domain bounded by P i Q i , P i Q i+1 , R i+1 Q i , and R i+1 Q i+1 , where R i+1 Q i is the C − characteristic curve passing through Q i , R i+1 Q i+1 is the C + characteristic curve passing through Q i+1 .