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Stability of Steady Multi-Wave Configurations for the Full Euler Equations of Compressible Fluid Flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

In this paper, we focus on the stability of steady four-wave configurations in the two space-dimensional case, consisting of two shocks, one vortex sheet, and one entropy wave, which are the solutions of the Riemann problem in the flow direction; see Figure 1. In this configuration, the vortex sheet and the entropy wave coincide in the Euler coordinates. This is one of the fundamental core multi-wave configurations, as a solution of the standard steady Riemann problem for the two-dimensional Euler equations: (i) For supersonic flow, there are at most eight waves (shocks, vortex sheets, entropy waves, rarefaction waves) that emanate from one single point in the Euler coordinates, which consist of one solution (at most four of these waves) of the Riemann problem in the flow direction and the other solution (at most four of these waves) of the other Riemann problem in the opposite direction, while the later Riemann problem can also be reduced into the standard Riemann problem in the flow direction by the coordinate transformation (x, y) → (−x, −y) and the velocity transformation (u, v) → (−u, −v), which are invariant for the Euler equations (1.1).
(ii) Vortex sheets and entropy waves are new key fundamental waves in the multidimensional case, which are normally very sensitive in terms of perturbations as observed in numerical simulations and physical experiments (cf. [1,2,5,7,10]).
The stability problem involving supersonic flows with a single shock past a Lipschitz wedge has been solved in Chen-Zhang-Zhu [11] (also see Chen-Li [9]). The stability problem involving supersonic flows with vortex sheets and entropy waves over a Lipschitz wall has been solved in Chen-Zhang-Zhu [12]. See also Chen-Kuang-Zhang [8] for the stability of two-dimensional steady supersonic exothermically reacting Euler flow past Lipschitz bending walls.
The case of an initial configuration involving two shocks is treated in [21], by using the method of front tracking, for more general equations, under the finiteness and stability conditions. We think that, with the estimates of Riemann solutions involving more than Figure 1. An unperturbed four-wave configuration, consisting of two shocks S 1 and S 4 , one vortex sheet C 2 , and one entropy wave C 3 two strong waves, the estimates on the reflection coefficients of wave interactions should play a similar role so that the method of front tracking may be used.
In this paper, it is proved that steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, then develop a modified Glimm difference scheme similar to those in [11,12] from the original Glimm scheme in [19] for one-dimensional hyperbolic conservation laws, and further identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves carefully. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the vortextsheets/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.
This paper is organized as follows: In §2, we first formulate the stability of multiwave configurations as the Cauchy problem (initial value problem) in the flow direction for the Euler equations (1.1) and then state the main theorem of this paper. In §3, some fundamental properties of system (1.1) and the analysis of the Riemann solutions are presented, which are used in the subsequent sections. In §4, we make estimates on the wave interactions, especially between the strong and weak waves, and identify the key feature of the Euler equations that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the vortex sheet/entropy wave or the shock wave. In §5, we develop a modified Glimm difference scheme, based on the ones in [11,12], to construct a family of approximate solutions, and establish necessary estimates that will be used later to obtain its convergence to an entropy solution of the Cauchy problem (1.1) and (2.1). In §6, we show the convergence of the approximate solutions to an entropy solution, close to the background solution of steady four-wave configuration.

Formulation of the Problem and Main Theorem
In this section, we formulate the stability problem for the steady four-wave configurations as the Cauchy problem (initial value problem) in the flow direction for the Euler equations (1.1) and then state the main theorem of this paper.
2.1. Stability problem. We focus on the stability problem of the four-wave configurations consisting of two strong shocks, one strong vortex sheet, and one entropy wave for the supersonic Euler flows governed by system (1.1) for U = (u, v, p, ρ). More precisely, we consider a background solution U : R 2 + → R that consists of four constant states: where u j > c j for all j ∈ {a, m 1 , m 2 , b} with the sonic speed of state U j : and state U m1 connects to U b by a strong 1-shock of speed σ 10 , U m1 connects to U m2 by a strong 2-vortex sheet and a strong 3-entropy wave of strengths (σ 20 , σ 30 ), and U a connects to U m2 by a strong 4-shock of speed σ 40 ; see Figure 2. We are interested in the stability of the background solution U of steady four-wave configuration, under small BV perturbations of the incoming flow as the initial data, to see whether it leads to entropy solutions containing similar strong four-wave configurations, close to the background solution U . That is, the stability problem can be formulated as the following Cauchy problem (initial value problem) for the Euler equations (1.1) with the Cauchy data: where U 0 ∈ BV(R) is a small perturbation function close to U (·, 0) in BV .
The main theorem of this paper is the following: Theorem 2.1 (Existence and Stability). There exist > 0 and C > 0 such that, if then there are four functions: such that (i) U is a global entropy solution of system (1.1) in R 2 + , satisfying the initial condition (2.1); (ii) Curves {y = χ i (x)}, i = 1, 2, 3, 4, are a strong 1-shock, a combined strong 2-vortex sheet and 3-entropy wave (χ 2,3 := χ 2 = χ 3 ), and a strong 4-shock, respectively, all emanating from the origin, with In §2- §6, we prove this main theorem and related properties of the global solution in BV .

Riemann Problems and Solutions
This section includes some fundamental properties of system (1.1) and some analysis of the Riemann solutions, which will be used in the subsequent sections; see also [11,12].
3.1. Euler equations. With U = (u, v, p, ρ), the Euler system can be written in the following conservation form: so that the eigenvalues of (3.2) are the roots of the fourth order polynomial: (3.4) which are solutions of the equation: where c = γp/ρ is the sonic speed. If the flow is supersonic, i.e. u 2 +v 2 > c 2 , system (1.1) is hyperbolic. In particular, when u > c, system (1.1) has the following four eigenvalues in the x-direction: with four corresponding linearly independent eigenvectors: for j = 1, 4, where κ j are chosen to ensure that r j · ∇λ j = 1 for j = 1, 4, since the first and fourth characteristic fields are always genuinely nonlinear, and the second and third are linearly degenerate.
We focus on the region, {u > c}, in the state space, especially in the neighborhoods of U j in the background solution.
We first consider self-similar solutions of (1.1): which connect to a state U 0 = (u 0 , v 0 , p 0 , v 0 ). We find that which implies First, for the cases i = 2, 3, we obtain dp = 0, vdu − udv = 0. (3.10) This yields the following curves C i (U 0 ) in the phase space through U 0 : which describe compressible vortex sheets (i = 2) and entropy waves (i = 3). More precisely, we have a vortex sheet governed by with strength σ 2 and slope v 0 u 0 , which is determined by and an entropy wave governed by with strength σ 3 and slope v 0 u 0 , which is determined by For j = 1, 4, we obtain the jth rarefaction wave curve R j (U 0 ), j = 1, 4, in the phase space through U 0 : where the jump symbol [ · ] stands for the value of the front state minus that of the back state. We find that where u 0 > c for small shocks. For s i , i = 2, 3, in (3.15)-(3.18), we obtain the same C i (U 0 ), i = 2, 3, defined in (3.12)-(3.13), since the corresponding fields are linearly degenerate.
On the other hand, for s j , j = 1, 4, in (3.15)-(3.18), we obtain the jth shock wave curve S j (U 0 ), j = 1, 4, through U 0 : where ρ 0 < ρ is equivalent to the entropy condition (3.8) on the shock wave. We also know that S j (U 0 ) agrees with R j (U 0 ) up to second order and that The entropy inequality (3.8) is equivalent to the following: 3.3. Riemann problems. We consider the Riemann problem for (1.1): where U a and U b are constant states, regarded as the above and below state with respect to line y = y 0 .
3.3.1. Riemann problem only involving weak waves. Following Lax [20], we can parameterize any physically admissible wave curve in a neighborhood of a constant state U 0 .
, the Riemann problem (3.24) admits a unique admissible solution consisting of four elementary waves. In addition, state U 2 can be represented by From now on, we denote {U 1 , U 2 } = (α 1 , α 2 , α 3 , α 4 ) as a compact way to write the representation of (3.25).
We also note that the renormalization factors κ j in (3.7) have been used to ensure that r j · ∇λ j = 1 in a neighborhood of any unperturbed state U 0 = (u 0 , 0, p 0 , ρ 0 ) with u 0 > 0, such as U m1 or U m2 : which also holds in a neighborhood of U 0 .
Also, since ∇ U Φ(0, 0, 0, 0; U 1 ) is equal to the identity, by the implicit function theorem, we can findΦ (after possibly shrinking O (U 0 )) such that, in the above situation, we can represent U 1 =Φ(α 4 , α 3 , α 2 , α 1 ; U 2 ). Differentiating the relation: This will be used later in §4. We exploit the symmetries between the shock polar and the reverse shock polar, and the symmetry between the 1-shock polar and the reverse 4-shock polar to allow for more concise arguments.

3.3.2.
Riemann problem involving a strong 1-shock. The results here are based on those in §6.1.4 of [11], with small changes for our requirements.
Proof. We omit the full calculations, which can be found at [11, pp. 299-300]. We use Lemma 3.3, along with the first and fourth Rankine-Hugoniot conditions (3.15) and (3.18), to deduce that P > 0. We then use Lemma 3.3 and the entropy condition (3.23) can be parameterized locally for the state which connects to U 0 by a shock of speed σ 10 from above as Proof. It suffices to solve Then the result follows by the implicit function theorem.
Proof. Note that, by (3.23), Since we have the same Rankine-Hugoniot conditions as in Lemma 3.3, with u m2 and u a taking the roles of u m1 and u b , respectively, the proof follows identically.
the reverse shock polarS 2 (U 0 ) -the set of states that connect to U 0 by a strong 4-shock from below -can be parameterized locally for the state which connects to U 0 by a shock of speed σ 40 as

3.3.4.
Riemann problem involving strong vortex sheets and entropy waves. We now look at the interaction between weak waves and the strong vortex sheet/entropy wave, based on those in §2.5 of [12]. For any U 1 ∈ O (U m1 ) and U 2 ∈ O (U m2 ), we use {U 1 , U 2 } = (0, σ 2 , σ 3 , 0) to denote the strong vortex sheet and entropy wave that connect U 1 to U 2 with strength (σ 2 , σ 3 ). That is, In particular, we have By a straightforward calculation, we have The next property allows us to estimate the strength of reflected weak waves in the interactions between the strong vortex sheet/entropy wave and weak waves: Lemma 3.9. The following holds: Proof. By a direct calculation,

Estimates on the Wave Interactions
In this section, we make estimates on the wave interactions, especially between the strong and weak waves. This is based on those in §3 of [11,12], with new estimates for the strong 4-shock.
Below, M > 0 is a universal constant which is understood to be large, and O (U i ) for i ∈ {a, m1, m2, b} is a universal small neighborhood of U i which is understood to be small. Each of them depends only on the system, which may be different at each occurrence.

4.2.
Estimates on weak wave interactions. We have the following standard proposition; see, for example, [23,Chapter 19] for the proof. Note that, in our analysis of the Glimm functional, we only require the estimates for the cases where the waves are approaching.
Proposition 4.1. Suppose that U 1 , U 2 , and U 3 are three states in a small neighborhood of a given state U 0 with Then

4.3.
Estimates on the interaction between the strong vortex sheet/entropy wave and weak waves. We now derive an interaction estimate between the strong vortex sheet/entropy wave and weak waves. The properties that |K vb1 | < 1 in (4.2) and |K va4 | < 1 in (4.5) will be critical in the proof that the Glimm functional is decreasing. and there is a unique (δ 1 , σ 2 , σ 3 , δ 4 ) such that and Proof. This proof is the same as [12, pp. 1673], with additional terms. We need to solve for (δ 1 , σ 2 , σ 3 , δ 4 ) as a function of (α 1 , α 2 , α 3 , α 4 , β 1 , σ 2 , σ 3 , β 4 , U 1 ) in the following equation: Note that, when α 4 = 0, the unique solution is where We see that K vbi and K bvj are bounded from the formulae above. To deduce that |K vb1 | < 1, we differentiate (4.3) with respect to α 4 to obtain By Lemma 3.8 and another similar calculation, we have We can then deduce the following result by symmetry:

Estimates on the interaction between the strong shocks and weak waves.
We now derive an interaction estimate between the strong shocks and weak waves. The properties that |K 1sa4 | < 1 in (4.6) and |K 4sb1 | < 1 in (4.14) will be critical in the proof that the Glimm functional is decreasing.

4.4.1.
Interaction between the strong 1-shock and weak waves.
We now prove a result for the case where weak waves approach the strong 1-shock from below.

Approximate Solutions and BV Estimates
In this section, we develop a modified Glimm difference scheme, based on the ones in [11,12], to construct a family of approximate solutions and establish necessary estimates that will be used later to obtain its convergence to an entropy solution to (1.1) and (2.1).
Then define U 0 (y) to be the state that connects to U 0 (−4∆y) by a strong 1-shock of strength σ 10 on (−4∆y, 0) which lies in O (U m1 ), and the state that connects to U 0 (0) by a combined strong vortex sheet/entropy wave of strength (σ 20 , σ 30 ) on (0, 4∆y) which lies in O (U m2 ). Thus, for small, U 0,∆x,θ (y) ∈ O (U a ) for y > 4∆y, and U 0,∆x,θ (y) ∈ O (U b ) for y < −4∆y. Now, assume that U ∆x,θ on {0 ≤ x ≤ k∆x} has been defined for k ≥ 0. We solve the family of Riemann problems for n ∈ Z with k + n ≡ 0 (mod 2): T k,n = {k∆x ≤ x ≤ (k + 1)∆x, (n − 1)∆y ≤ y ≤ (n + 1)∆y}, then set U ∆x,θ = U n on T k,n , n ∈ N, and define U ∆x,θ | x=(k+1)∆x = U ∆x,θ ((k + 1)∆x−, a k+1,n ), k + n ≡ 0 (mod 2). Now, as long as we can provide a uniform bound on the solutions and show the Riemann problems involved always have solutions, this algorithm defines a family of approximate solutions globally. Glimm-type functional and its bounds. In this section, we prove the approximate solutions are well defined in Ω ∆x by uniformly bounding them. We first introduce the following lemma, which is a combination of Lemma 6.7 in [11, pp. 305] and Lemma 4.1 in [12, pp. 1677], and follows from that Φ, G 1 , G 2,3 , and G 4 are C 1 functions.
Lemma 5.1. The following bounds of the approximate solutions of the Riemann problems hold: • For any σ j ∈ Oˆ (σ j0 ), j = 2, 3, so that then, when U ∈ O (U m1 ) for someˆ =ˆ ( ) withˆ → 0 as → 0, We now show that U ∆x,θ can be defined globally. Assume that U ∆x,θ has been defined in {x < k∆x}, k ≥ 1, by the steps in §5, and assume the following conditions are satisfied: To see that C 2 (k) holds if C 2 (k − 1) holds, by the discussion earlier, for χ (k) 4 to emanate further down than χ (k−1) 4 , we require θ k > b, which implies that χ , we require θ k < a, which implies that χ We will establish a bound on the total variation of U ∆x,θ on the k-mesh curves to establish C 3 (k) and C 1 (k). {k∆x} × ((n − 1)∆y, (n + 1)∆y) to be in O ǫ (U a ), i.e. (n + θ k )∆y > (n + σ (k−1) 4 ∆x ∆y )∆y > (n + a)∆y Figure 5. Separation of the initial data Definition 5.1. A k-mesh curve is an unbounded piecewise linear curve consisting of line segments between the mesh points, lying in the strip: with each line segment of form P k−1,n−1 P k,n or P k,n P k+1,n+1 .
Clearly, for any k > 0, each k-mesh curve I divides plane R 2 into a part I + and a part I − , where I − is the one containing set {x < 0}. As in [19], we partially order these mesh curves by saying J > I if every point of J is either on I or contained in I + , and we call J an immediate successor to I if J > I and every mesh point of J, except one, is on I. We now define a Glimm-type functional on these mesh curves. where |α 4s1 | and |α 1s4 | are the strengths of the weak 4-shock and 1-shock that emanate from the same point as the strong 1-shock and 4-shock, respectively, and these two weak waves are excluded from L 3 4 and L 2 1 , respectively. Remark 5.1. L(J) + C * 4 j=1 |σ J i − σ i0 | measures the total variation of U | J , owing to Lemma 5.1; each term L i (J) measures the total variation in region (i), and each term |σ J i − σ i0 | measures the magnitude of jumps of U between the regions separated by the large shocks. We also see that Q(J) ≤ L(J) for L(J) small enough, so that Q(J) is equivalent to F s (J) + C * 4 j=1 |σ J i − σ i0 |. Proposition 5.1. Suppose that I and J are two k-mesh curves such that J is an immediate successor of I. Suppose that for someˆ ( ) > 0 determined in Lemma 5.1. Then there exists˜ > 0 such that, if F s (I) ≤˜ , and hence Proof. We make M ≥ 2 larger to ensure that it is bigger than all the O(1)-terms in §4, and that M ≥ ∆y ∆x . With the fixed M from here on, we now define our constants in terms of it. We set˜ .
Let Λ be the diamond that is formed by I and J. We can assume that I = I 0 ∪ I and J = J 0 ∪ J such that ∂Λ = I ∪ J . We divide our proof into different cases, based on where diamond Λ is located.
Case 1: Weak-weak interaction. Suppose that Λ lies in the interior of a region (i); see Figure 6. Then, by Proposition 4.1, when L(I 0 ) ≤˜ ≤ 1 8M . Since K * ij < 2 for all i and j, we have Case 3: Weak waves interact with the strong vortex sheet/entropy wave from above. Suppose that the approximate vortex sheet/entropy wave enters Λ from below; see Figure 7. This case follows by symmetry from the above case, owing to the symmetry between Propositions 4.2 and 4.3, and the symmetry between the coefficients.
Case 4: Weak waves interact with the strong 1-shock from above. Suppose that the strong 1-shock enters Λ from below; see Case 5: Weak waves interact with the strong 1-shock from below. Suppose that the strong 1-shock enters Λ from above; see Figure 8. By Proposition 4.5, we have Case 6: Weak waves interact with the strong 4-shock from below. Suppose that the strong 4-shock enters Λ from above; see Figure 9. By symmetry from Case 4, we conclude that F s (J) ≤ F s (I), due to the symmetry between Propositions 4.4 and 4.7. Figure 9. Cases 6 & 7: Weak waves interact with the strong 4-shock from below and above.
Case 7: Weak waves interact with the strong 4-shock from above. Suppose that the strong 4-shock enters Λ from below; see Figure 9. By symmetry from Case 5, we obtain that F s (J) ≤ F s (I), owing to the symmetry between Propositions 4.5 and 4.6.