1 CHAPTER 1 The Compressible Euler System in Two Space Dimensions Yuxi Zheng Department of Mathematics The Pennsylvania State University E-mail: yzheng@math.psu.edu Abstract We analyze the self-similar isentropic irrotational Euler system via the hodograph transformation. We diagonalize the system of equations in the phase space. We use these equations to analyze the binary interaction of arbitrary planar rarefaction waves, which includes the classical problem of a wedge of gas expanding into vacuum. Introduction Multi-dimensional systems of conservation laws have been a major research area for many decades and substantial progress has been made in recent years. We expect that there will be great progress in the near future. This chapter is intended to introduce the topic at the first-year graduate student level and end at the research height. The primary system is the Euler system for ideal compressible gases. The issues are structures of solutions as well as the existence and uniqueness of solutions to various initial and boundary value problems. Our focus will be on special initial data that yield solutions with distinctive physical features. These types of initial data are collectively called Riemann data. Typical types of physical features are continuous expansive and compressive waves as well as regular and Mach reflections. In another word, we shall study Riemann problems in two space dimensions and explain features of shock reflections. The mathematical tools are theory of characteristics, elliptic estimates, boundary and corner regularity, fixed point theorems, numerical simulations, and asymptotic analysis. The problems are mathematically challenging and aerodynamically significant. This chapter is organized as follows: The author is partly supported by NSF DMS-0603859.
Yuxi Zheng 1. The compressible Euler system.. The characteristics decomposition of the pseudo-steady case. 3. The hodograph transformation and the interaction of rarefactions. 4. Local solutions for quasi-linear systems. 5. Invariant regions for systems. 6. The pressure gradient system. 7. Open problems. 1 Physical Phenomena and Mathematical Problems 1.1 Euler system in n dimensions We recall that the full (or adiabatic) Euler system for an ideal fluid takes the form: ρ t + (ρu) = 0, (ρu) t + (ρu u + pi) = 0, (1.1) (ρe) t + (ρeu + pu) = 0, where E := 1 u + e, and e is the internal energy. For a polytropic gas, there holds e = 1 p γ 1 ρ, where γ > 1 is the adiabatic gas constant. See Courant and Friedrichs [7]. The notations are that is divergence in R n (n = 1,, 3), u u = (u i u j ) (an n n matrix), and I is the identity matrix. Subscript t denotes partial derivative in t. 1. Phenomena Shock waves can form in compressible gases. We see shock waves forming ahead of a flying bullet, or multiple shocks around an airplane moving at supersonic speed, see separate graphics from Van Dyke [38]. 1.3 Mathematical treatment There are a couple of problems designed as space probes to peek into the mysteries of the one-dimensional and two-dimensional gas dynamics. Riemann s shock tube problem ( 1860 ): Assume that system (1.1) is only one-dimensional, say, the x direction. Assume that the initial data (ρ,u, p) =: U consist of two constant states separated by the position x = 0; i.e., U(0, x) = { U, x < 0, U +, x > 0, (1.)
Two-D Euler System 3 where U and U + are two constant vectors in R n. Find the solutions to (1.1)(1.) in t > 0. This mathematical problem can be implemented physically as follows, see Figure 1.1. Imagine an infinitely long cylindrical tube filled with a gas in two different states, separated by an extremely thin membrane. The membrane is broken at t = 0, and you are there to watch the subsequent motion of the gas. U_ U + Figure 1.1. The setup of Riemann s shock tube experiment. Mach s oblique-shock-reflection problem ( 1878): A shock wave, called incident shock I, of system (1.1) traveling down a flat ramp hits the ground at time t = 0, see Figure 1., where the shock is drawn at time t = 1. Find the subsequent reflection/diffraction patterns of the shock wave. I U 1 U 0 θ w Figure 1.. Mach s oblique shock reflection problem. Riemann s shock tube problem has explicit solutions, which are illustrated in Fig. 1.3. A typical solution consists of a shock, rarefaction wave, and a contact discontinuity. The shock and rarefaction waves can travel in either the positive or negative x directions. A shock wave is represented by a single bar, while a rarefaction wave is represented by a group of parallel bars in part (a) of Figure 1.3. In the evolutionary coordinates (x, t) of part (b) of Figure 1.3, a (centered) rarefaction wave is represented by a group of rays from the origin. A shock wave is a curve or surface across which the variables U has a discontinuity and the pressure behind it is greater than the pressure ahead of it. A contact discontinuity is a curve or surface across which the pressure is continuous but other variables are discontinuous. A rarefaction wave is a continuously
4 Yuxi Zheng changing wave whose pressure is decreasing in the direction of positive time. U_ U m U + (a) A snapshot in the tube (diagram). t R J S x (b) The solution in the (x, t) plane. Figure 1.3. A solution of Riemann s shock tube experiment. Mach s experiment produced many patterns of solutions, illustrated in Figure 1.4, which shows the solution at t = 1. In Figure 1.4, the incident shock is marked with I, the reflected shock is noted by R. When the wedge angle θ w is large, the experiment produces a reflection pattern as shown in part (a) of Figure 1.4, which is called regular reflection. When the wedge angle is small, the reflection pattern is like in part (b) of Figure 1.4, and it is called simple Mach reflection. In part (b), besides the incident and reflected shocks I and R, there is a new shock wave, called Mach stem, denoted by M, and a contact discontinuity marked by the dashed curve. The intersection point of the three shocks is denoted by T which is called a triple point. If the wedge angle is neither large nor small, then it happens that the reflected shock R may develop a kink, the point K, as shown in part (c) of Figure 1.4, where the segment TK is straight. The pattern in part (c) is called a complex Mach reflection. It is also possible, when the wedge lies in the middle, that a fourth pattern occurs, as shown in part (d) of Figure 1.4, and it is called a Double Mach reflection, in which a new shock wave and contact discontinuity emerge from the point K. The term Mach reflection refers to the three cases parts (b)-(d). These illustrations are based on physical experiments and numerical simulations, see [1]. There is little
Two-D Euler System 5 (d) Double Mach reflection (DMR) θ U U θ U T θ U T K θ T T U U U U R M M R R M R M R I 0 1 1 0 1 0 1 0 w w w w Figure 1.4. Regular and Mach reflections. (a) Regular reflection (RR) (b) Simple Mach reflection (SMR) (c) Complex Mach reflection (CMR)
6 Yuxi Zheng theoretical justification available so far. In addition, there may be other types of reflection, see Chapter 7 for Guderley reflection. 1.4 Paradoxes: There are several paradoxes arising from the work of von Neumann, known as von Neumann paradoxes. One of them is: On the one-hand, there is a triple point structure from physical and numerical simulations in the case of reflection of a weak incident shock on a ramp of small angle; While on the other hand, theoretical arguments indicate that it cannot exist. See [39]. 1.5 The 8th millennium priceless problem There are seven open problems proposed by the Clay Mathematics Institute. Here we propose one more problem and call it the 8th millennium priceless problem: i.e., the two-dimensional Riemann problem (see Figure 1.5): Find an entropy(or physical) solution for the adiabatic Euler system with a gamma greater than one and initial data being constant along any ray (or four constants in the four quadrants). For more details, see [5, 45]. y U U 1 x U 3 U 4 Figure 1.5. A common Riemann datum.
Two-D Euler System 7 Characteristic decomposition of the pseudosteady case We present a brief overview of the method of characteristics and generalize it to handle the self-similar two-dimensional Euler system. We characterize its simple waves..1 Riemann problems Consider the full Euler system for an ideal fluid from Section 1. Let (r, θ) be the polar coordinate in the plane. Let the domain Ω R be either the whole plane or sectorial: Ω = {(r, θ) r > 0, θ (θ 1, θ )} for a pair θ 1, θ (0, π), with θ 1 < θ. Consider the initial data: (p, ρ, u, v) t=0 = (p 0, ρ 0, u 0, v 0 )(θ), all θ or θ 1 < θ < θ. That is, the data are independent of the radius r. Riemann problems are Cauchy problems for the Euler system with these initial data, which is the most general form that we shall consider. When the domain has boundary, we will provide boundary conditions such as the no-flow boundary condition. We seek self-similar solutions to the above problem in the variables (ξ, η) = (x/t, y/t). The system then becomes ξρ ξ ηρ η + (ρu) ξ + (ρv) η = 0, ξ(ρu) ξ η(ρu) η + (ρu + p) ξ + (ρuv) η = 0, ξ(ρv) ξ η(ρv) η + (ρvu) ξ + (ρv (.1) + p) η = 0, ξ(ρe) ξ η(ρe) η + (ρeu + pu) ξ + (ρev + pv) η = 0, while the initial condition becomes boundary condition at infinity lim(p, ρ, u, v)(ξ, η) = (p 0, ρ 0, u 0, v 0 )(θ), all θ or θ 1 < θ < θ as we send (ξ, η) to infinity while holding η = ξ tanθ fixed. (Examples of explicit solutions are given later.) In the self-similar plane and for smooth solutions, the system takes the form: 1 ρ sρ + u ξ + v η = 0, s u + 1 ρ p ξ = 0, s v + 1 ρ p (.) η = 0, 1 γp sp + u ξ + v η = 0
8 Yuxi Zheng where s := (u ξ) ξ + (v η) η. We call (u ξ, v η) pseudo-flow directions, as opposed to the other two characteristic directions, called (pseudo-)wave characteristics. We easily derive s (pρ γ ) = 0. (.3) So the entropy pρ γ = A is constant along pseudo-flow lines. For a region Ω whose pseudo-flow lines come from a constant state, we obtain that the entropy is constant in the region. Consider the self-similar vorticity W = v ξ u η in an isentropic region. From the two transport equations s u + γaρ γ ρ ξ = 0, s v + γaρ γ ρ η = 0, we obtain by differentiation that s W W + W(u ξ + v η ) = 0. Using the equation s ρ + ρ(u ξ + v η ) = 0, we obtain s W ρ = W ρ. (.4) Thus zero W at a boundary will result in zero value for W in the region the pseudo-flow characteristics flow in. Hence, for a region whose pseudo-flow lines come from a constant state, the vorticity must be zero everywhere. So the region is irrotational and isentropic. In the physical space (t, x, y), the physical vorticity ω = v x u y satisfies ω t + (uω) x + (vω) y + ( p y ρ ) x ( p x ρ ) y = 0. (.5) The physical vorticity ω has zero source of production in the isentropic case when ( py ρ ) x ( px ρ ) y = 0. The two vorticity are related by tω = v ξ u η = W.. Isentropic system Consider the two-dimensional, isentropic compressible Euler equations ρ t + (ρu) x + (ρv) y = 0, (ρu) t + (ρu + p) x + (ρuv) y = 0, (.6) (ρv) t + (ρuv) x + (ρv + p) y = 0, where ρ is the density, (u, v) is the velocity and p is the pressure taken as the function of density, p(ρ) = Aρ γ, for some constant A and a gas
Two-D Euler System 9 constant γ > 1. We investigate the pseudo steady case of (.6); i. e., the solution depends on the self similar variables (ξ, η) = (x/t, y/t). Then (.6) becomes, ξρ ξ ηρ η + (ρu) ξ + (ρv) η = 0, ξ(ρu) ξ η(ρu) η + (ρu + p) ξ + (ρuv) η = 0, (.7) ξ(ρv) ξ η(ρu) η + (ρuv) ξ + (ρv + p) η = 0. This flow is often used in physical applications and numerical schemes, and two-dimensional Riemann problems as well. We use, instead of ρ, the enthalpy i = c γ 1 = Aγ γ 1 ργ 1 as one of the dependent variables. Let us rewrite (.7) for smooth solutions as (u ξ)i ξ + (v η)i η + κ i (u ξ + v η ) = 0, (u ξ)u ξ + (v η)u η + i ξ = 0, (.8) (u ξ)v ξ + (v η)v η + i η = 0, where κ = (γ 1)/. One may assume further that the flow is ir-rotational: u η = v ξ. (.9) Then, we insert the second and third equations of (.8) into the first one to deduce the system, { (κ i (u ξ) )u ξ (u ξ)(v η)(u η + v ξ ) + (κ i (v η) )v η = 0, u η v ξ = 0, supplemented by the Bernoulli s law (.10) i + 1 ((u ξ) + (v η) ) = ϕ, ϕ ξ = u ξ, ϕ η = v η. (.11).3 Some explicit solutions We present the explicit solutions of planar waves and the Suchkov interaction. Planar elementary waves We list waves of the 1-D Riemann problem in the -D setting. (i) Constant states: (ρ, u, v) = constant for the isentropic Euler. (ii) Backward and forward rarefaction waves: R (ξ) : ξ = u c, du dρ = c ρ, v = constant. (c = p (ρ))
10 Yuxi Zheng η θ s ξ Figure.1: Suchkov explicit solution Suchkov explicit solution (1958 [34]): For an incline angle θ s satisfying tan θ s = 3 γ γ + 1, the solution inside the interaction zone has the explicit form c = (1 + γ 1 sinθ s ξ)tan θ s ; u = (ξ 1 sin θ s )tan θ s ; v = η. The characteristics are straight lines. The vacuum boundary is straight, located at ξ =. See Figure.1. sin θs γ 1 Axially symmetric solutions: From Zhang-Zheng (1997), the exact solutions of the two dimensional compressible Euler equations (.6) for p = Aρ γ, A > 0, 1 < γ < 3, are valid only for γ = and take the form u = x + y t, v = x + y, ρ = r t 8At
Two-D Euler System 11 in r t p 0, and ( u = tp 0 cosθ + ) p 0 r t p 0 sin θ /r, ( v = tp 0 sin θ ) p 0 r t p 0 cosθ /r, r > t p (.1) 0 ρ = ρ 0 where ρ 0 > 0 is an arbitrary parameter, p 0 p (ρ 0 ), and (r,θ) is the polar coordinates x = r cos θ, y = r sin θ. The solutions have the initial data: u(x,y,0) = p (ρ 0 ) sinθ, v(x,y,0) = p (ρ 0 )cosθ, (.13) ρ(x,y,0) = ρ 0. The particle trajectories in the cone r < t p (ρ 0 ) given by { dx dt = u(x,y,t), dy dt = v(x,y,t), r(t 0 ) = r 0, θ(t 0 ) = θ 0, where r 0 t 0 p (ρ 0 ), take the form r = r 0 e θ0 e θ, θ = θ 0 1 ln t t 0 r 0 4t 0p (ρ 0) t < in the polar coordinates and valid in the time interval indicated. The number of revolutions of these spirals approach infinity as r 0 0+ in 0 < t t 0 for fixed t 0 > 0..4 A characteristic decomposition We present a characteristic decomposition of the potential flow equation in the self-similar plane. The decomposition allows for a proof that any wave adjacent to a constant state is a simple wave for the adiabatic Euler system. This result is a generalization of the well-known result on -d steady potential flow and a recent similar result on the pressure gradient system [10]. It has the potential for construction of more complex waves. According to Courant and Friedrichs, simple waves are most important tools for the solutions of flow problems; simple waves and their generalizations apparently have not been sufficiently emphasized in mathematical studies of hyperbolic differential equations ([7], p.40)..4.1 Introduction to the method of characteristics Consider solving { ut + a(t, x)u x = 0, t > 0, x R u(0, x) = u 0 (x). (.14)
1 Yuxi Zheng Define the solution x = x(t; x 0 ) to dx dt = a(t, x), x(0) = x 0 R to be a characteristic curve. Examine a solution u(t, x) to (.14) along a characteristic u(t, x(t; x 0 )) with differentiation: So we have du(t, x(t; x 0 )) dt = u t + u x dx dt = u t + au x = 0. u(t, x(t; x 0 )) = u(0, x(0; x 0 )) = u(0, x 0 ) = u 0 (x 0 ). Assuming that a(t, x) is a smooth and bounded function, so that the characteristics cover the upper plane t > 0, x R exactly once, we have a unique solution. This is the method of characteristics. Consider the scalar conservation law { ut + f(u) x = 0, (.15) u(0, x) = 0. The equation can be rewritten for smooth solutions to be u t +f (u)u x = 0. So for any given smooth solution u(t, x) we have the characteristics as dx dt = f (u(t, x)), x(0) = x 0, denoted by x = x(t; x 0 ) as before. Along the characteristics we find that du(t, x(t; x 0 )) dt thus u(t, x(t; x 0 )) = u 0 (x 0 ), and so the speed of the characteristics f (u) = f (u 0 (x 0 )) are constant along the characteristics, which then implies that the characteristics are straight lines. Assuming f > 0 and u 0 (x) is bounded, continuous and increasing, we have found a unique solution by tracing the characteristics. The one-dimensional wave equation = 0 u tt c u xx = 0 (.16) with constant speed c has an interesting decomposition ( t + c x )( t c x )u = 0, or (.17) ( t c x )( t + c x )u = 0 (.18)
Two-D Euler System 13 known from elementary text books. One can rewrite them as + u = 0, or + u = 0, (.19) where ± = t ± c x. Sometimes, the same fact is written in Riemann invariants for the Riemann invariants t R + c x R = 0, t S c x S = 0 (.0) R := ( t c x )u, S := ( t + c x )u. (.1) Now consider a linear system of equations with constant coefficients u t + Au x = 0, where A is an n n matrix of real numbers with n distinct eigenvalues λ 1 < λ < < λ n and n linearly independent left eigen vectors l i (i = 1,, n). We can multiply the system of equations with l i and obtain (l i u) t + λ i (l i u) x = 0 so the system is diagonalized by the Riemann invariants {l i u} n i=1. For a pair of system of hyperbolic conservation laws [ ] [ ] [ u f(u, v) 0 + =, (.) v g(u, v) 0] t it is known that a pair of Riemann invariants exist so that the system can be rewritten as { t R + λ 1 (u, v) x R = 0, (.3) t S + λ (u, v) x S = 0, where (R, S) are the Riemann invariants and the λ s are the two eigenvalues of the system. These decompositions and Riemann invariants are useful in the construction of solutions, for example, the construction of D Alembert formula, and proof of development of singularities ([19]). An example of the system is the system of isentropic irrotational steady two-dimensional Euler equations for compressible ideal gases { (c u )u x uv(u y + v x ) + (c v )v y = 0, (.4) u y v x = 0 supplemented by Bernoulli s law x c γ 1 + u + v = k, (.5)
14 Yuxi Zheng where γ > 1 is the gas constant while k > 0 is an integration constant. This system has two unknowns (u, v), and by introducing the polar coordinates u = q cosθ, v = q sinθ, (.6) we can find that a pair of Riemann invariants are R i = Θ + ( 1) i q c /(qc)dq, (.7) where c = c (q) from Bernoulli s law. Following the existence of Riemann invariants, any solution adjacent to a constant state is a simple wave. A simple wave means a solution (u, v) that depends on a single parameter rather than the pair parameters (x, y). To find out why systems of three or more equations do not in general have Riemann invariants, we consider u t + A(u)u x = 0 (.8) where A has n distinct eigenvalues λ i and linearly independent left eigenvectors l i (i = 1,, n) with component l i = {l ij } n j=1. We obtain similarly l i u t + λ i l i u x = 0 (i = 1,, n) (.9) When A is a constant matrix and so l i are constants, we take R i = l i u(= Σ j l ij u j ) for the Riemann invariants. We now wish for some R i so that Σ j l ij (u j ) t = (R i ) t (.30) or u R i = l i. (.31) This may not be always possible. But we can choose the l i with non-zero factors ϕ i (u), so (.31) can become For n =, the solvability condition for (.3) is u R i = ϕ i l i (.3) curl (ϕ i l i ) = 0 (.33) which in general has a nonzero solution for ϕ i (u)(i = 1, ). But for n >, there are more compatibility conditions than the number of variables ϕ i, so in general there are no solutions to (.3).
Two-D Euler System 15 Despite the general difficulty, the success for the pressure gradient system stirs the desire to consider the pseudo-steady isentropic irrotational Euler system which has three equations with source terms, (ρu) ξ + (ρv ) η = ρ, (ρu + p(ρ)) ξ + (ρuv ) η = 3ρU, (.34) (ρuv ) ξ + (ρv + p(ρ)) η = 3ρV, where (ξ, η) = (x/t, y/t), (U, V ) = (u ξ, v η) is the pseudo-velocity, and the pressure p = p(ρ) is the function of the density ρ. No explicit forms of the Riemann invariants are found, but decompositions similar to + λ = m λ hold for some m, where λ is an eigenvalue. We use the characteristic decomposition to establish that a wave adjacent to a constant state must be a simple wave for the pseudo-steady irrotational isentropic Euler system. A simple wave for this case is such that one family of pseudo-wave characteristics are straight lines and the physical quantities velocity, speed of sound, pressure, and density are constant along the wave characteristics. Further, using the fact that entropy and vorticity are constant along the pseudo-flow characteristics (the pseudo-flow lines), our irrotational result extends to the adiabatic full Euler system..4. Decomposition Now let c be the speed of sound and (U, V ) = (u ξ, v η) be the pseudo-velocity. We can rewrite the equations of motion (.10)(.11) in a new form [ u v ] ξ [ UV + c U c V c U 1 0][ u v ] η = 0 (.35) to draw as much parallelism to the steady case as possible. We emphasize the mixed use of the variables (U, V ) and (u, v), i. e., (U, V ) is used in the coefficients while (u, v) is used in differentiation. This way we obtain zero on the right-hand side for the system. The eigenvalues are similar as before: The left eigenvectors are And we have similarly dη dξ = Λ ± = UV ± c (U + V c ) U c. (.36) l ± = [1, Λ ]. (.37) ± u + Λ ± v = 0 (.38)
16 Yuxi Zheng where ± = ξ + Λ ± η. (The notation ± = u + λ ± v are reserved for the hodograph plane, see later.) Our Λ ± now depend on more than (U, V ). But, let us regard Λ ± as a simple function of three variables Λ ± = Λ ± (U, V, c ) as given in (.36). Thus we need to build differentiation laws for c. We can directly obtain from (.11) that ( ) ( ) c c + Uu ξ + V v ξ = 0, + Uu η + V v η = 0. (.39) γ 1 γ 1 We have So we move on to compute ξ ± c = (γ 1)(U ± u + V ± v). (.40) ± Λ ± = U Λ ± ± U + V Λ ± ± V + c Λ ± ± c = U Λ ± ( ± u 1) + V Λ ± ( ± v Λ ± ) + c Λ ± ± c = U Λ ± ± u + V Λ ± ± v + c Λ ± ± c U Λ ± V Λ ± Λ ±. η (.41) We need to handle the term U Λ ± + V Λ ± Λ ±. We show it is zero. Recalling that (c U )Λ + UV Λ + c V = 0, (.4) and regarding that Λ depends on the three quantities (U, V, c ) independently, we can easily find Λ U = Λ(UΛ V ) Λ(c U ) + UV, Λ UΛ V V = Λ(c U ) + UV. (.43) Thus Λ U + ΛΛ V = 0. (.44) Therefore we end up with ± Λ ± = [ U Λ ± Λ 1 V Λ ± (γ 1) c Λ ± (U Λ 1 V ) ] ± u.(.45) So the factor ± u is present in ± Λ ±. Thus, if one of the quantities (u, v, c ) is a constant along Λ, so are all the rest. Proposition.1 (Commutator relation). We have + I + I = Λ + + Λ Λ Λ + ( I + I). (.46)
Two-D Euler System 17 Proof. Compute + I = ( ξ + Λ η )( ξ + Λ + η )I = I ξξ + Λ I ξη + Λ + I ξη + Λ Λ + I ηη + ξ Λ + I η + Λ η Λ + I η. (.47) Similarly we have + I = I ξξ +Λ + I ξη +Λ I ξη +Λ Λ + I ηη + ξ Λ I η +Λ + η Λ I η. (.48) The difference is We can use the representation to complete the proof. + I + I = ( Λ + + Λ )I η. (.49) I η = I + I Λ Λ + (.50) Theorem.1 (Characteristic decomposition). There holds + ( u)+ + Λ Λ + ( u) = Λ [ +Λ ] Λ Λ + Λ Λ + Λ Λ + u + Λ + Λ u. + (.51) ± Λ ± = [ U Λ ± Λ 1 V Λ ± (γ 1) c Λ ± (U Λ 1 V )] ± u.(.5) ± Λ = [ U Λ 1 Λ V Λ (γ 1) c Λ (U 1 Λ V )] ± u (UΛ V ) c U. (.53) (Keep formulas at this level for structure recognition, e.g., we can utilize concavity properties of the characteristics.) Proof. We use the commutator relation on u to find + u = + u + + Λ Λ + Λ + Λ ( + u u). (.54) We use the Euler equations ± u+λ ± v = 0 to bring the term + u back to + u as follows. Use the equation to obtain + u = ( Λ + v) = [ Λ + v + Λ + v] = Λ Λ + u Λ + v. (.55)
18 Yuxi Zheng Use the commutator to obtain + v = + v + Λ + + Λ Λ Λ + ( v + v). (.56) Now use the equation to convert all the v back to u: ( + v = + ) u Λ + + ( Λ 1 u 1 ) + u. Λ + Λ Λ + Λ + Λ (.57) Combining the last few steps we obtain + u = Λ + u + Λ Λ Λ Λ + + Λ Λ Λ + + u Λ Λ + Λ + Λ + u+ ( + 1 u 1 ) + u. Λ + Λ (.58) Place that into the first expression we obtain an equation for + u and so we solve for it to yield the decomposition of the theorem. The second equality has been proved in the preceding paragraphs. The proof of the third equality is like this. Similar to (.41), we have + Λ = U Λ + U + V Λ + V + c Λ + c = [ U Λ 1 Λ V Λ (γ 1) c Λ (U 1 Λ V )] + u (.59) ( U Λ + Λ + V Λ ), and similar to (.43) we obtain U Λ + Λ + V Λ = U Λ + 1 Λ c V c U V Λ = (UΛ V ) c U.(.60) This completes the proof. This way we see potential for a direct approach to the gas expansion into the vacuum (see [4, 34]), and a possible way for other patches. Theorem. (Simple wave). (Li, Zhang, and Zheng [7] 006) The solution adjacent to a constant state to the pseudo-steady Euler system is a simple wave, in which one family of characteristics are straight lines along each of which the variables (u, v, c) are constant.
Two-D Euler System 19 3 The hodograph transformation and the interaction of rarefaction waves We analyze the self-similar isentropic irrotational Euler via the hodograph transformation. We diagonalize the system of equations in the phase space. We use these equations to analyze the binary interaction of arbitrary planar rarefaction waves, which includes the classical problem of a wedge of gas expanding into vacuum. 3.1 Primary system Consider the two-dimensional isentropic compressible Euler system ρ t + (ρu) x + (ρv) y = 0, (ρu) t + (ρu + p) x + (ρuv) y = 0, (3.1) (ρv) t + (ρuv) x + (ρv + p) y = 0, where p(ρ) = Aρ γ where A > 0 will be scaled to be one and γ > 1 is the gas constant. Here is a list of our notations: ρ density, p pressure, (u, v) velocity, c = γp/ρ speed of sound, i = c /(γ 1) enthalpy, γ gas constant, (ξ, η) = (x/t, y/t) the self-similar (or pseudo-steady) variables, ϕ pseudovelocity potential, θ wedge half-angle, and U = u ξ, V = v η, κ = (γ 1)/, m = (3 γ)/(γ+1), θ s = arctan m. Letters C, C 1 and C denote generic constants. Our primary system is system (3.1) in the self similar variables (ξ, η): (u ξ)i ξ + (v η)i η + κ i (u ξ + v η ) = 0, (u ξ)u ξ + (v η)u η + i ξ = 0, (3.) (u ξ)v ξ + (v η)v η + i η = 0. We assume further that the flow is ir-rotational: u η = v ξ. (3.3) Then, we insert the second and third equations of (3.) into the first one to deduce the system, { (κ i U )u ξ UV (u η + v ξ ) + (κ i V )v η = 0, (3.4) u η v ξ = 0,
0 Yuxi Zheng supplemented by pseudo-bernoulli s law i + 1 ((u ξ) + (v η) ) = ϕ, ϕ ξ = u ξ, ϕ η = v η. (3.5) The system can then be written as a single second-order equation for ϕ: (κ i ϕ ξ)ϕ ξξ ϕ ξ ϕ η ϕ ξη + (κ i ϕ η)ϕ ηη = ϕ ξ + ϕ η 4κ i, (3.6) where i + 1 ϕ + ϕ = 0. 3. The concept of hodograph transformation The original form of a hodograph transformation is for a homogeneous quasi-linear system of two first-order equations for two known variables (u, v) in two independent variables (x, y). By regarding (x, y) as functions of (u, v) and assuming that the Jacobian does not vanish nor is infinity, one can re-write the system for the unknowns (x, y) in the variables (u, v), which is a linear system if the coefficients of the original system do not depend on (x, y). See the book of Courant and Friedrichs [7]. Specifically, consider the system of two equations of the form, ( ) ( ) u u + A(u, v; x, y) = 0, (3.7) v v x where the coefficient matrix A(u, v; x, y) is ( ) a11 a A(u, v; x, y) = 1. (3.8) a 1 a We introduce the hodograph transformation, y T : (x, y) (u, v). (3.9) Differentiating the identities u = u(x(u, v), y(u, v)), v = v(x(u, v), y(u, v)) with respect to u, v, we solve for u x, u y, v x, v y to find u x = y v /j, u y = x v /j, v x = y u /j, v y = x u /j, j := x u y v x v y u. Then (3.7) is reduced to the system ( ) ( ) yv xv + A(u, v; x, y) y u x u = 0. (3.10) Obviously, if the coefficient matrix A does not depend on (x, y), (3.10) becomes a linear system for the unknowns (x, y).
Two-D Euler System 1 3..1 The hodograph transformation for the pseudo-steady Euler The idea of hodograph transformation does not obviously generalize to other systems such as system (3.4) of more than two simple equations or for inhomogeneous systems. For (3.4), one realizes that the three variables (i, u, v) are functions of (ξ, η), so one can still try to use (u, v) as the independent variables and regard (ξ, η) as functions of (u, v) and ultimately regard i as a function of (u, v). This was done in 1958 in a paper [34] by Pogodin, Suchkov and Ianenko. The implementation is as follows. Let the hodograph transformation be T : (ξ, η) (u, v) (3.11) for (3.) and reverse the roles of (ξ, η) and (u, v). Differentiating the identities ξ = ξ(u(ξ, η), v(ξ, η)), η = η(u(ξ, η), v(ξ, η)) with respect to ξ, η, we find ξ u = v η /J, ξ v = u η /J, η u = v ξ /J, η v = u ξ /J; J = u ξ v η v ξ u η. Inserting these into system (3.4), we obtain { (κ i U )η v + UV (ξ v + η u ) + (κ i V )ξ u = 0, ξ v η u = 0. (3.1) The difficulty here is that i, as a function of u and v, cannot be determined explicitly and point-wise. But we can obtain something else. Differentiating pseudo-bernoulli s law (3.5) with respect to u, v, we obtain ξ u = i u, (3.13) η v = i v. These interesting identities provide an explicit correspondence between the physical plane and the hodograph plane provided that the transformation T is not degenerate. We continue to differentiate (3.13) with respect to u and v: ξ u = 1 + i uu, ξ v = i uv, η u = i uv, η v = 1 + i vv, (3.14) and inserting these into the first equations of (3.1) to obtain, (κ i i u)i vv + i u i v i uv + (κ i i v)i uu = i u + i v 4κ i. (3.15)
Yuxi Zheng This is a very interesting second order partial differential equation for i alone. So the study of ir-rotational, pseudo-steady and isentropic fluid flow can proceed along (3.15). We point out for the case γ = 1 that the dependent variable i = lnρ, instead of i = c /(γ 1), is used [3]. Then we can obtain a similar equation for i, (1 i u)i vv + i u i v i uv + (1 i v)i uu = i u + i v. (3.16) 3.. Steady Euler The steady isentropic and irrotational Euler system of (3.1) has the form { (κ i u )u x uv(u y + v x ) + (κ i v )v y = 0, u y v x = 0, (3.17) where i is given by Bernoulli s law u + v + i = k 0, (3.18) where k 0 is a constant. See [7]. Using the hodograph transformation from (x, y) to (u, v), we obtain a linear system, { (κ i u )y v + uv(x v + y u ) + (κ i v )x u = 0, x v y u = 0. (3.19) The hodograph transformation is valid in the region of non-simple waves. Differentiating Bernoulli s law (3.18), we obtain u = i u, v = i v. (3.0) This can also be obtained formally from (3.13) by regarding the steady flow as the limit of unsteady flow (3.1) in t. We see that (3.0) is a trivial consequence of (3.18). Comparing (3.0) with (3.13), we see that it is much more difficult to convert the hodograph plane of the steady case back into the physical plane than the pseudo-steady case. However, system (3.19) has more advantage over (3.1) of the pseudo steady case because i is expressed in an explicit form by Bernoulli s law (3.18). In sum, the phase space structure of the steady case is trivial and its difficulty is finding the conversion (x, y) as a function of (u, v); while the conversion formula for the pseudo-steady case is explicit, its work is to find its phase space structure.
Two-D Euler System 3 3..3 Similarity to one-dimensional problems The current approach parallels the procedure that is used to find centered rarefaction waves to genuinely nonlinear strictly hyperbolic systems of conservation laws in one space dimension. Recall for a one-dimensional system u t + f(u) x = 0 of n equations, a centered rarefaction wave takes the form ξ = λ k (u) for a k [1, n] and the state variable u satisfies the system of ordinary differential equations (f (u) λ k (u)i)u ξ = 0, whose solutions are rarefaction wave curves in the phase space. The development (or inversion) of the phase space solutions onto the ξ axis requires the monotonicity of λ k (u) along the vector field of the k th right eigenvector r k ; i.e., the genuine nonlinearity. For the self-similar -D Euler system, we have a pair ξ = u + i u, η = v + i v from (3.13) in place of ξ = λ k (u); and the second-order partial differential equation (3.15) in place of the ordinary differential system. For inversion to the physical space, we show that the Jacobian J 1 T of (3.9) does not vanish. 3.3 Characteristics in both planes We assert that the characteristics of (3.4) are mapped into the characteristics of (3.1) by the hodograph transformation (3.11). And there holds λ ± = 1 Λ, (3.1) where, the eigenvalues of (3.4) are Λ ± = (u ξ)(v η) ± c (u ξ) + (v η) c (u ξ) c, (3.) while the eigenvalues of (3.1) are λ ± = i ui v ± c (i u + i v c ) c i. (3.3) v By using (3.13), it is easy to see (3.1). For the correspondence between Λ ± and λ ±, we let η = η(ξ) be a characteristic curve in the (ξ, η) plane with dη dξ = Λ + and be mapped onto a curve v = v(u). Then, using (3.14) and (3.1), we have i.e., Λ = dη dξ = η dv u + η v du, (3.4) dv ξ u + ξ v dv du = ξ uλ + η u ξ v Λ + η v = (1 + i uu)λ + i uv i uv Λ + (1 + i vv ) = i uu + 1 + λ i uv i uv + λ (i vv + 1). du (3.5)
4 Yuxi Zheng We rewrite (3.15) as i uu + 1 + (λ + λ + )i uv + λ λ + (i vv + 1) = 0. (3.6) Then we conclude dv du = λ +. (3.7) Similarly we obtain the correspondence between Λ and λ. We remark that we will establish in the pseudo-steady case that the transform is not degenerate, i.e., J T (u, v; ξ, η) = (u, v) (ξ, η) = u ξv η u η v ξ 0 (3.8) in regions of non-simple waves, to be detailed later. In the direction from (u, v) plane to the (ξ, η) plane, it is more direct to compute J 1 T (u, v; ξ, η) = ξ uη v ξ v η u 0. (3.9) 3.4 Phase space system of equations In this section we use the inclination angles of characteristics as useful variables to rewrite (3.15) in the hodograph plane. We proceed as follows. We first transform the second order equation (3.15) into a firstorder system of equations. Introduce X = i u, Y = i v. (3.30) Then we deduce a 3 3 system of first order equations, κ i Y XY 0 0 1 0 X Y + XY κ i X 0 1 0 0 X Y 0 0 1 i 0 0 0 i u v = X + Y 4κ i 0. (3.31) X This system is equivalent to (3.15) for C 1 solutions if the given datum for Y is compatible with the datum for i v. The characteristic equation is (κ i Y )λ XY λ + κ i X = 0 (3.3) besides the trivial factor λ. This system has three eigenvalues λ 0 = 0, dv du = λ ± = XY ± κi(x + Y κi) κ i Y κi X = XY κi(x + Y κi), (3.33)
Two-D Euler System 5 from which we deduce that (3.31) is hyperbolic if X + Y κi > 0 provided that i > 0 and κi Y 0 (or κi X 0). If κi Y = 0 or κi X = 0, then the solutions are planar rarefaction waves. The three left eigenvectors associated with (3.33) are l 0 = (0, 0, 1), l = (1, λ ±, 0). (3.34) We multiply (3.31) by the left eigen-matrix M = (l +, l, l 0 ) (hereafter the superscript means transpose) from the left hand side to obtain the characteristic form X u + λ Y u + λ + (X v + λ Y v ) = X + Y 4κi κi Y, X u + λ + Y u + λ (X v + λ + Y v ) = X + Y 4κi (3.35) κi Y, i u = X. Introduce the inclination angles α, β ( π/ < α, β < π/) of Λ + and Λ -characteristics by Note that, see (3.1), tan α = Λ +, tanβ = Λ. (3.36) Λ + = 1 λ, Λ = 1 λ +, (3.37) so that λ + = cotβ and λ = cotα, from which we have Proposition We have α+β cos X = c sin α β α+β sin, Y = c sin α β. (3.38) Proof. Because of homogeneity, we can let c = 1. Thus the equations are XY + X + Y 1 1 Y = cotβ, XY X + Y 1 1 Y = cotα. The difference and the product of the two equations are X + Y 1 1 Y = cotα cotβ, X 1 Y 1 = cotαcotβ. Thus we use X 1 = (Y 1)cotαcotβ to remove X to yield Y + (Y 1)cotαcotβ = (1 Y )(cotα cotβ). (3.39)
6 Yuxi Zheng Squaring the equation, multiplying it with sin α sin β, we obtain Y 4 sin (α β) Y (sin α + sin β) + sin (α + β) = 0. We then solve the quadratic equation to find sinα ± sin β Y = ± sin(α β). We choose the plus signs for our preference. We can then use the halfangle formula to yield Y = c sin( α + β )/ sin( α β ). Then we use the sum of the equations of (3.39) to find XY = (Y 1)(cotα + cotβ) to yield a unique X as seen. This completes the proof. We observe that the variables α, β are Riemann invariants for (3.35). In fact, we can write (3.35) as + α = γ + 1 4c β = γ + 1 4c 0 c = γ 1 sin(α β) sin β sin(α β) sin α α+β cos sin α β, [ m tan α β [ m tan α β where we use the notations of directional derivatives, ], ], (3.40) + = u + λ + v, = u + λ v, 0 = u, (3.41) and keep the letter m for m = 3 γ 1 + γ. (3.4) We further introduce the normalized directional derivatives along characteristics, + = (sin β, cosβ) ( u, v ), = (sin α, cosα) ( u, v ). (3.43) Using them, we write (3.40) as, + α = γ + 1 [ sin(α β) m tan α β ] =: G(α, β, c), 4c β = G(α, β, c), 0 c = γ 1 α+β cos sin α β. (3.44)
Two-D Euler System 7 A few remarks are in order. We note that we have an alternative expression for We note further that G(α, β, c) = 1 c tan α β (cos(α β) κ). + c = κ, c = κ. (3.45) This means that the first two equations of (3.44) are essentially decoupled from the third c-equation. Further, system (3.40) is linearly degenerate in the sense of Lax [18]. For the particular case that tan((α β)/) = m for 1 < γ < 3, and α and β are constants, the first two equations are satisfied. In fact, the explicit solutions of Suchkov [37] in the expansion problem of a wedge of gas into vacuum is such a case, see Remark 3.1 in Section 3.6. The variables (α, β) might be called Riemann variables, with signature that system (3.44) is diagonalized. The mapping (X, Y ) (α, β) is bijective as long as system (3.35) is hyperbolic. We summarize the above as follows. Theorem 3.1. The two dimensional pseudo-steady, irrotational, isentropic flow (3.15) can be transformed into a linearly degenerate system of first order partial differential equations (3.40) or (3.44) provided that the transform (X, Y ) (α, β) is invertible, i.e., system (3.35) is hyperbolic. Proof of (3.40). We find easily that + α = sin α + λ. Holding λ as a function of three independent variables (X, Y, c ) in the characteristic equation (3.3) we find X + λy X λ = λ(κ i Y ) XY, λ + 1 Y λ = λλ X, c λ = λ(κ i Y ) XY. We then use + λ = X λ + X + Y λ + Y + c λ + c = X λ ( + X + λ + Y ) + c λ κ (X + λ + Y ). (3.46) We use (3.35) and (3.46) to simplify it to obtain (3.40). The proof of (3.40) is complete. Regarding α and + β, we are unable to obtain explicit expressions for them like (3.44). But we have second-order equations. By direct computations, we first obtain Lemma 3.1 (Commutator relation of ± ). For any quantity I = I(u, v), we have + I + I = λ + + λ λ λ + ( I + I). (3.47)
8 Yuxi Zheng Lemma 3. (Commutator relation of ± ). For any quantity I = I(u, v), we have, + I + I = 1 cos(α β) ( I + sin(α β) + I) + α, (3.48) where + α is given in (3.44). Noting + α = β in (3.44), we can also use β in (3.48). Using these commutator relations, we easily derive: Theorem 3.. Assume that the solution of (3.44) (α, β) C. Then we have + α + W α = Q(α, β, c), + β + W (3.49) + β = Q(α, β, c), where W(α, β, c) and Q(α, β, c) are W(α, β, c) = γ + 1 [ (m tan ω )( 3 tan ω 1 ) cos ω + tan ω], 4c Q(α, β, c) = (γ + 1) 16c sin(ω) ( m tan ω )( 3 tan ω 1 ), ω = α β. Proof. The proof is simple. Recall from (3.45) that (3.50) + c = κ, c = κ. (3.51) Then we apply the commutator relation to obtain (setting I = α in (3.48)) + α = + α + 1 cos(α β) ( α + sin(β α) + α) β. (3.5) Using the expressions of + α and β in (3.44), we compute directly to yield the result in (3.49) and the proof of Theorem 3. is complete. 3.5 Planar rarefaction waves We present in concise form planar rarefaction waves of the Euler system. Let R + 1 (ξ)(η > v 1) and R 1 (ξ)(η < v 1) connect the two states (p 1, ρ 1, u 1, v 1 ) and (p, ρ, u, v 1 ), denoted by (1) and () respectively in Figure 3.1, via a planar wave in the ξ variable. The solution is isentropic
Two-D Euler System 9 so that the entropy S := pρ γ = p 1 ρ γ 1 = p ρ γ is constant in the solution. The solution from (3.4)(3.5) has the formula ξ = u + p (ρ), (ξ = u + c ξ ξ 1 = u 1 + c 1 ) R 1 ± (ξ) : u = u 1 + ρ ρ 1 ρ 1 p (ρ)dρ, (0 ρ ρ ρ 1 ), v = v = v 1, η > v 1 or η < v 1. The sonic curve of these two waves is a straight segment (3.53) η = v 1, ξ (ξ, ξ 1 ), (3.54) which is often called a sonic stem. The (pseudo-wave) characteristics are given by dη/dξ = Λ ± where Λ + = in R + 1 and Since Λ = (η v 1) p (ρ) p (ρ)(η v 1 ), η > v 1. dξ dρ = p (ρ) + ρp (ρ) ρ, i.e., ξ = ξ + γ + 1 γs(ρ (γ 1)/ ρ (γ 1)/ p ), (ρ) γ 1 we obtain d(η v 1 ) dρ = γ + 1 ρ [(η v 1) γsρ γ 1 ], which yields the negative family of characteristics ( ) C + γ(γ+1) 3 γ η v 1 = Sρ(γ 3)/ ρ (γ+1)/, if γ 3, ρ C 6S lnρ, if γ = 3, (3.55) where C is an arbitrary constant and S is the entropy. Note that there is a straight characteristic curve among them (when C = 0, ρ = 0) for γ (1, 3); that is η v 1 = γ 1 (3 γ)(γ + 1) (ξ u ). (3.56) Above this straight curve, the characteristics are convex; and below it they are concave. Further, if ρ = 0, then all the negative family of characteristics converge to the point (ξ, η) = (u, v ) with the same
30 Yuxi Zheng asymptotic slope as in (3.56) for γ (1, 3). For γ > 3, the constant C is always positive and the asymptotic slope is plus infinity. If ρ > 0, then the sonic curve consists of the sonic circle of the state (p, ρ, u, v ) and the sonic stem. The characteristics and sonic curves are drawn in Figure 3.1. + + (1) _ () (1) _ vacuum sonic (a) Adjacent to vacuum (b) No vacuum Figure 3.1: Characteristics in planar rarefaction waves for 1 < γ < 3. Dashed lines are characteristics of the negative family, solid lines are those of the plus family; while dotted lines are sonic curves located at η = v 1. 3.6 The gas expansion problem We now use the hodograph transformation to study the expansion of a wedge of gas into vacuum. The problem was studied in [37, 3,, 4, 3, 8]. Here are some notations in this section: We use θ s [0, π/), m 0, defined by tan θ s = (3 γ)/(γ + 1), m 0 = 1/ m, (3.57)
for 1 γ 3; and θ s 0 for γ > 3. Two-D Euler System 31 3.6.1 Planar rarefaction waves in a given direction. First we prepare our planar rarefaction waves. Given a direction (n 1, n ) with n 1 + n = 1, and a positive constant ρ 1. Consider the initial data for (3.1) { (ρ1, 0, 0), for n 1 x + n y > 0, (ρ, u, v)(x, y, 0) = vacuum, for n 1 x + n y < 0. The solution of (3.1) and (3.58) takes the form, see [5], (ρ 1, 0, 0), ζ > 1, (ρ, u, v)(x, y, t) = (ρ, u, v)(ζ), 1/κ ζ 1, vacuum, ζ < 1/κ, (3.58) (3.59) where ζ = n 1 ξ + n η, (ξ, η) = (x/t, y/t), and the solution (c, u, v) has been normalized so that c 1 = 1. The rarefaction wave solution (ρ, u, v)(ζ) satisfies n 1 ζ = n 1 u + n v + c, κ c u = n 1 κ, n κ c v = n κ. (3.60) Note that this rarefaction wave corresponds to a segment in the hodograph plane, n u n 1 v = 0, n 1 /κ u 0. In particular, when we consider the rarefaction wave propagates in the x-direction, i.e., (n 1, n ) = (1, 0), this wave can be expressed as x/t = u + c, c = κu + 1, v 0, 1/κ u 0. (3.61) That is, in the hodograph (u, v) plane, this rarefaction wave is mapped onto a segment v 0, 1/κ u 0, on which we have 3.6. A wedge of gas i = 1 κ (κu + 1), i u = κu + 1, i uu = κ. (3.6) We place the wedge symmetrically with respect to the x axis and the sharp corner at the origin, as in Figure 3.(a). This problem is then formulated mathematically as seeking the solution of (3.1) with the initial data, (i, u, v)(t, x, y) t=0 = { (i0, u 0, v 0 ), θ < δ < θ, (0, ū, v), otherwise, (3.63)
3 Yuxi Zheng where i 0 > 0, u 0 and v 0 are constant, (ū, v) is the velocity of the wave front, not being specified in the state of vacuum, δ = arctany/x is the polar angle, and θ is the half-angle of the wedge restricted between 0 and π/. This can be considered as a two dimensional Riemann problem for (3.1) with two pieces of initial data (3.63). As we will see below, this problem is actually the interaction of two whole planar rarefaction waves. See Figure 3.(b). We note that the solution we construct is valid for any portion of (3.63) as the solution is hyperbolic. t = 0 y l 1 Vacuum (ρ 1, 0, 0) θ 0 x l (a) Initial Data η R 1 Vacuum Interaction region D o P θ (ρ 1, 0, 0) ξ R (b) Interaction of rarefaction waves Figure 3.: The expansion of a wedge of gas The gas away from the sharp corner expands into the vacuum as planar rarefaction waves R 1 and R of the form (i, u, v)(t, x, y) = (i, u, v)(ζ) (ζ = (n 1 x+n y)/t) where (n 1, n ) is the propagation direction of waves. We assume that initially the gas is at rest, i.e., (u 0, v 0 ) = (0, 0). Other-
Two-D Euler System 33 wise, we replace (u, v) by (u u 0, v v 0 ) and (ξ, η) by (ξ u 0, η v 0 ) in the following computations (see also (3.)). We further assume that the initial sound speed is unit since the transformation (u, v, c, ξ, η) c 0 (u, v, c, ξ, η) with c 0 > 0 can make all variables dimensionless. Then the rarefaction waves R 1, R emitting from the initial discontinuities l 1, l are expressed in (3.60) with (n 1, n ) = (sinθ, cosθ) and (n 1, n ) = (sin θ, cosθ), respectively. These two waves begin to interact at P = (1/ sinθ, 0) in the (ξ, η) plane due to the presence of the sharp corner and a wave interaction region, called the wave interaction region D, is formed to separate from the planar rarefaction waves by k 1, k, k 1 : (1 κ )ξ 1 (κη 1 + 1) = C γ (κη 1 + 1) (κ+1)/κ, (ξ 1 > 0, 1 η 1 1/κ), k : (1 κ )ξ (κη + 1) = C γ (κη + 1) (κ+1)/κ, (ξ > 0, 1/κ η 1), (3.64) where k 1 and k are two characteristics from P, associated with the nonlinear eigenvalues of system (3.), see [5, 41], and the constant C γ is C γ = (3 γ)/(γ 1) (γ + 1) /(γ 1) (3 + γ )γ (γ+1)/(γ 1), (3.65) and {ξ1 = ξ cosθ + η sin θ, η 1 = ξ sinθ + η cosθ, { ξ = ξ cosθ η sin θ, η = ξ sin θ + η cosθ. (3.66) So, the wave interaction region D is bounded by k 1, k and the interface of gas with vacuum. The solution outside D consists of the constant state (i 0, u 0, v 0 ), the vacuum, and the planar rarefaction waves R 1 and R. Problem A. Find a solution of (3.) inside the wave interaction region D, subject to the boundary values on k 1 and k, which are determined continuously from the rarefaction waves R 1 and R. This problem is a Goursat type problem for (3.) since k 1 and k are characteristics. Note also that initial data (3.63) is ir-rotational, we conclude that the flow is always ir-rotational provided that it is continuous. So the irrotationality condition (3.3) holds 3.6.3 A wedge of gas in the hodograph plane Our strategy to solve this problem is to use the hodograph transformation, solve the associated problem in the hodograph plane, and show that the hodograph transformation is invertible.
34 Yuxi Zheng For this purpose, we need to map the wave interaction region D in the (ξ, η) plane into a region Ω in the (u, v) plane. Notice that the mapping of the planar rarefaction waves R 1 and R into (u, v) plane are exactly two segments H 1 : u cosθ + v sin θ = 0, ( sinθ/κ u 0) and H : u cosθ v sin θ = 0, ( sinθ/κ u 0). The boundary values of c on H 1, H, are c H1 = 1 + κv =: c 1 0, c H = 1 + κv =: c 0, (3.67) where v = u sinθ v cosθ and v = u sinθ + v cosθ. Obviously, 0 c 1 0, c 0 1. (3.68) Thus the wave interaction region Ω is bounded by H 1, H and the interface of vacuum connecting D and E in the hodograph (u, v)-plane, see Figure 3.3. We define Ω more precisely to contain the boundaries H 1 and H, but not the vacuum boundary c = 0. D v H 1 + θ wave interaction region Ω 0 u H E Figure 3.3: Wave interaction region in the hodograph plane Boundary conditions. We need to derive the necessary boundary conditions on H 1 and H, respectively. This can be done simply by using coordinate transformations tan α = Λ +, tan β = Λ, λ ± Λ = 1
Two-D Euler System 35 and the characteristics distribution of Section 3.5. Rotating Figure 3.1 clockwise by π/ θ, we see easily that α H1 = θ, β H = θ. Thus the boundary values for α, β on H 1 and H are α H1 = θ, β H = θ. (3.69) The boundary values of c on H 1 and H are given in (3.67). Now Problem A becomes: Problem B. Find a solution (α, β, c) of (3.40) with boundary values (3.69) and (3.67), in the wave interaction region Ω in the hodograph plane. The values of β on H 1 or α on H can be integrated from the system. We estimate the boundary values (3.69) and (3.67). Lemma 3.3 (Boundary data estimate). For the boundary data (3.69) on the boundaries H i, i = 1,, we have the following estimates: (i) If θ < θ s, we have (ii) If θ > θ s, we have θ (α β) Hi θ s. (3.70) θ s (α β) Hi θ. (3.71) Proof. It follows from the convexity of the characteristics. The extreme values of β are determined at the ends of the characteristics, i.e., either the starting value or the ending (at the vacuum) asymptotic value tan β e := γ 1 (3 γ)(γ + 1). This β e is related to m by Then the proof is complete. 3.6.4 Local existence tan π β e = m. The local existence of solutions at the origin (u, v) = (0, 0) follows routinely from the idea [31, Chapter ] or [40]. We need only to check the compatibility condition to this problem, i.e., 1 λ + [ l 0 + K κ cos α+β sin ω ] = 1 λ [ l 0 K κ cos α+β sin ω ] (3.7)
36 Yuxi Zheng at (u, v) = (0, 0), where K = (α, β, c) and l 0 = (0, 0, 1). That is, we need to check if this is true [ ] [ ] α+β 1 cos + c κ = 1 α+β cos c κ. (3.73) λ + sin ω λ sinω This is obviously true by using (3.45). Hence we have Lemma 3.4 (Local existence). There is a δ > 0 such that the C 1 solution of (3.40), (3.67), and (3.69) exists uniquely in the region Ω = {(u, v) Ω; δ < u < 0}, where δ depends only on the C 0 and C 1 norms of α, β on the boundaries H 1 and H. We do not give the proof. For details, see [31, Chapter ], [40], or Section 4. 3.6.5 Statement of main existence Next we will extend the local solution to the whole region Ω. Therefore some a priori estimates on the C 0 and C 1 norms of α, β and i, are needed. The norm of i comes from the norms of α and β, see the third equation of (3.40). Therefore we need only the estimate on α and β. Recall that the derivation of (3.40) is based on the strict hyperbolicity of the flow, i > 0. These will be achieved when we estimate the C 0 norms of α and β, see Subsection 3.6.6. The main existence theorem is stated as follows. Let l be the interface of the gas with vacuum. Theorem 3.3 (Global existence in the hodograph plane). There exists a solution (α, β, i) C 1 to the boundary value problem (3.40) with boundary values (3.67) and (3.69)(Problem B) in Ω. The vacuum interface l exists and is Lipschitz continuous. We prove this theorem by two steps. We estimate the solution itself in Subsection 3.6.6 and then proceed with estimates on the gradients in Subsection 3.6.7. The proof of Theorem 3.3 is also given in Subsection 3.6.7. After we solve Problem B, we establish the inversion of hodograph transformation in Subsection 3.6.8, which establishes the existence of the gas expansion problem, Problem A. Theorem 3.4 (Global existence in the physical plane). There exists a solution (c, u, v) C 1 of (3.) for the gas expansion problem (Problem A) in the wave interaction region D in the physical plane, the (ξ, η)- plane.